UvA-DARE (Digital Academic Repository) Crystal symmetries ... · Contents What is this thesis about9 I Combined Charge and Orbital Order in the Chalcogen Crystals11 1 Introduction

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Crystal symmetries in charge-orbital order and topological band theory

Oliveira Silva, A.C.

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CRYSTAL SYMMETRIES

IN CHARGE-ORBITAL ORDER

AND TOPOLOGICAL BAND THEORY

CRYSTAL SYMMETRIES

IN CHARGE-ORBITAL ORDER

AND

TOPOLOGICAL BAND THEORY

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Ana S

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01

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Ana Silva

Ana Silva

Thursday,

24 January 2019

at 10.00 am

You are cordiallyinvited to attendthe public defenceof my PhD thesis

Agnietenkapel,Oudezijds Voorburgwal 231,Amsterdam, The Netherlands

Crystal Symmetriesin Charge-Orbital Order andTopological Band Theory

c© 2018, Ana Cristina Oliveira Silva

Cover design by Luis Silva

Crystal Symmetries

in Charge-Orbital Order and

Topological Band Theory

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus

Prof. dr. ir. K.I.J. Maex

ten overstaan van een door het College voor Promoties

ingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapel

op donderdag 24 januari 2019, te 10:00 uur

door

Ana Cristina Oliveira Silva

geboren te Loures Lisboa

Promotiecommissie:

Promotor: Prof. dr. C.J.M. Schoutens Universiteit van Amsterdam

Co-promotor: Dr. J.van Wezel Universiteit van Amsterdam

Overige leden: Prof. dr. J. van den Brink IFW DresdenProf. dr. M.S. Golden Universiteit van AmsterdamProf. dr. J.-S. Caux Universiteit van AmsterdamDr. C. Ortix Universiteit UtrechtDr. N.J. Robinson Universiteit van Amsterdam

Faculteit: Faculteit der Natuurwetenschappen, Wiskunde en Informatica

The research described in this thesis is part of a research programme which is fi-nanced by the Netherlands Organisation for Scientific Research (NWO) and wascarried out at the Institute of Physics, University of Amsterdam in The Nether-lands.

BELIEF IN INSPIRATION. Artists have an interest in others’ believing in sud-den ideas, so-called inspirations; as if the idea of a work of art, of poetry, the fun-damental thought of a philosophy shines down like a merciful light from heaven.In truth, the good artist’s or thinker’s imagination is continually producing thingsgood, mediocre, and bad, but his power of judgment, highly sharpened and prac-ticed, rejects, selects, joins together; thus we now see from Beethoven’s notebooksthat he gradually assembled the most glorious melodies and, to a degree, selectedthem out of disparate beginnings. The artist who separates less rigorously, lik-ing to rely on his imitative memory, can in some circumstances become a greatimproviser; but artistic improvisation stands low in relation to artistic thoughtsearnestly and laboriously chosen. All great men were great workers, untiring notonly in invention but also in rejecting, sifting, reforming, arranging.

Friedrich Nietzsche, Human, All Too Human, section 155.

List of publications

This thesis is based on the following publications:

[1] Elemental chalcogens as a minimal model for combined charge and or-bital orderAna Silva, Jans Henke and Jasper van WezelPhys. Rev. B 97 045151 (2018)

Presented in Chapter 2, part 1.

[2] The simple-cubic structure of elemental Polonium and its relation tocombined charge and orbital order in other elemental chalcogensAna Silva and Jasper van WezelSciPost Physics 4 028 (2018)

Presented in Chapter 3, part 1.

[3] Surface states and the topological classification of 1D crystalline insu-latorsAna Silva and Jasper van Wezelin preparation for two separate publications: a letter and a long article

Presented in part 2.

Other publications by the author of this thesis:

[4] Andreev spectroscopy of Majorana states in topological superconductorswith multipocket Fermi surfacesAna Silva, Miguel A.N. Araujo and Pedro D. SacramentoEPL 110 37008 (2015)

Contents

What is this thesis about 9

I Combined Charge and Orbital Order in the ChalcogenCrystals 11

1 Introduction and motivation 131.1 A reader’s guide, part I . . . . . . . . . . . . . . . . . . . . . . . 161.2 Central assumptions of the nearly free electron model . . . . . . 171.3 Charge ordering and lattice displacements . . . . . . . . . . . . . 19

2 Combined charge and orbital order in selenium and tellurium 252.1 Intuitive picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Macroscopic order parameter theory . . . . . . . . . . . . . . . . 282.3 Minimal microscopic model . . . . . . . . . . . . . . . . . . . . . 292.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Combined charge and orbital order in polonium 353.1 Minimal microscopic model . . . . . . . . . . . . . . . . . . . . . 353.2 Turning up the temperature . . . . . . . . . . . . . . . . . . . . . 393.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Outlook 45

Appendix A Part I 47A.1 Diagonalisation of the bosonic mean field Hamiltonian . . . . . . 47A.2 Obtaining the spin-orbit coupling matrix . . . . . . . . . . . . . . 49

A.3 The origin of H(2)e-ph . . . . . . . . . . . . . . . . . . . . . . . . . . 50

II Edge states in 1D crystals 61

1 Introduction and motivation 63

5

CONTENTS

1.1 Why should one care about edge states in 1D crystals? . . . . . . 63

1.2 A very brief overview of edge states throughout time . . . . . . . 64

1.2.1 When edge states went by the name of surface states . . . 64

1.2.2 The topology fever . . . . . . . . . . . . . . . . . . . . . . 67

1.3 A reader’s guide, part II . . . . . . . . . . . . . . . . . . . . . . . 69

2 Edge state solutions in the nearly free electron model 73

2.1 An analogy with the ”finite square well problem” . . . . . . . . . 73

2.2 The finite crystal with symmetric interfaces: a review of Maue’swork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.3 The semi-infinite crystal . . . . . . . . . . . . . . . . . . . . . . . 84

2.3.1 From finite to semi-infinite . . . . . . . . . . . . . . . . . 85

2.3.2 Disappearing Bloch states . . . . . . . . . . . . . . . . . . 85

2.3.3 Edge state solutions . . . . . . . . . . . . . . . . . . . . . 87

3 At the interface between two crystals 91

3.1 Disappearing Bloch states . . . . . . . . . . . . . . . . . . . . . . 91

3.2 Edge state solutions . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.3 Searching for a physical meaning in Vl’s sign . . . . . . . . . . . . 104

4 Symmetry as the predictor of edge state solutions 109

4.1 A review on Zak’s symmetry criterion for edge state solutions . . 110

4.2 At the interface between two semi-infinite crystals . . . . . . . . 115

4.2.1 Band symmetry arguments . . . . . . . . . . . . . . . . . 116

4.2.2 Connection with topological invariants . . . . . . . . . . . 117

5 Full symmetry criterion for 1D crystals 123

5.1 The finite crystal with symmetric interfaces . . . . . . . . . . . . 124

5.2 The more general proof . . . . . . . . . . . . . . . . . . . . . . . 130

5.3 The left semi-infinite crystal . . . . . . . . . . . . . . . . . . . . . 133

6 Conclusions and outlook 137

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Outlook: towards higher dimensions . . . . . . . . . . . . . . . . 138

Appendix B Part II 141

B.1 Dictionary between Maue’s and Zak’s criterion for edge state so-lutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Bibliography 153

Contributions to publications 163

Summary 165

6

CONTENTS

Samenvatting 167

7

What is this thesis about

The common theme in this thesis is symmetry. First, the reader will find thatthe constant protagonists here are crystal structures. Since they are given bythe regular patterns formed by the atomic positions, group theory becomes thenatural language in which to categorise them. This provides one way in whichsymmetry presents itself in the thesis.Another way comes by analysing the correspondence between these geometricpatterns and the underlying microscopic interactions. Particularly, this will bedone by studying the genesis of the lattice structures in selenium, tellurium andpolonium crystals. This provides an example on how interactions themselves canmodify the symmetry of the stable crystal structure.In the second part of this thesis, symmetry comes in the form of an almost re-versed approach: given a fixed crystal structure, and with that a fixed symmetry,how are the microscopic states of the system affected, when the symmetry at theboundary is changed? Here, a link with topology will be established, but to atype of topological order that is deeply rooted in symmetry [5].

9

Part I

Combined Charge andOrbital Order in theChalcogen Crystals

11

CHAPTER 1

Introduction and motivation

A charge density wave phase (CDW) defines an ordered state, where charge dis-tribution acquires a periodic modulation throughout the material. The electrondensity becomes position dependent, which lowers the translational symmetry ofthe system. But the extent of symmetry reduction goes beyond the effect of re-shaping the electron density. As soon as electron-phonon coupling is introduced,atomic displacements will also occur, which might induce further symmetry re-duction in the system. Having lattice distortions in combination with a chargemodulation is a common phenomenon in materials that undergo a CDW phasetransition. What is sometimes less obvious is the role that the combined actionof interactions, such as electron-phonon and electron-electron interactions, havein giving rise to further symmetry lowered phases on top of the CDW formation.Their effect can even give rise to a degree of symmetry lowering that is unex-pected from the standard CDW mechanism.

This will be one of the central themes of the following chapters: how the pres-ence of certain interactions can give rise to a generic new paradigm for low-energysymmetry broken states to occur. These are characterised by an order parame-ter that not only relies on the electronic charge density, but also on the orbitaloccupation. That such an order parameter might play a role in CDW materialsis motivated by experimental observation: the bulk transition metal dichalco-genide 1T -TiSe2 has been proposed to harbour a charge density wave transitionthat breaks inversion symmetry in a spiral way [6, 7, 8, 9, 10]. This is in spiteof its parent lattice structure being inversion-symmetric, and the fact that thestandard ordered parameter for a CDW phase is given by the scalar distribu-tion of charge. Indeed this last observation seems paradoxical with the observedcharge ordered pattern. After all, a spiral naturally arises from a vector thatrotates, while its origin propagates in the direction perpendicular to the plane

13

Chapter 1. Introduction and motivation

of rotation. Without a vector ordered parameter to begin with, it becomes lessobvious how such an ordered phase can be realised.

In contrast to the well-known chirality of spins in helical magnets, the forma-tion of spirals within the scalar electronic density will not occur by itself, and,as we will see, is necessarily accompanied by the onset of simultaneous orbitalorder [8, 11, 12]. A crucial ingredient is the presence of multiple charge densitywaves instabilities that originate in different orbital sectors. As a result, the or-bital and charge degrees of freedom are combined into a single order parameter.Similar chiral order has been theoretically suggested to determine material prop-erties of various transition metal dichalcogenides [13, 14, 15], and even cupratehigh-temperature superconductors [16, 17, 18]. But the cooperation betweencharge and orbital degrees of freedom does not need to be restricted to chiralphases. The two degrees of freedom have, for example, also already been pro-posed to combine into a polar order parameter [15], and there is no reason tobelieve this exhausts the list of possible novel phases.

Focussing on the chiral combination of charge and orbitals, scanning-tunnelingmicroscopy experiments are a source of indirect evidence for the presence of thistype of order. An example is the already mentioned transition metal dichalco-genide 1T -TiSe2 [6, 10]. For this material, further support in favour of a spiralcharge and orbital order phase comes from the experimental confirmation ofseveral predictions arising from a Ginzburg-Landau theory of the chiral phasetransition [8, 9, 10]. Nevertheless, it has proven difficult to obtain a direct ex-perimental confirmation of the broken inversion symmetry. The main reason forthis is believed to be the presence of small, nanometer wide, domains of varyinghandedness [10], averaged over by almost all direct bulk probes. A microscopicunderstanding, going beyond the predictions of the macroscopic order parame-ter theory, is thus essential in the search for further experiments able to directlyprobe this novel type of combined charge and orbital order.

A microscopic theory for the chiral state of 1T -TiSe2 would necessarily involveall 22 orbitals in its unit cell. Even if such a model was constructed, its in-herent complexity would obscure a general understanding of how charge andorbital degrees of freedom cooperate to form a single ordered phase, and wouldlikely not be useful as a guide to identifying other possible materials that canharbour electronic spirals or other types of combined charge and orbital order.We therefore take here an alternative approach, and formulate a minimal micro-scopic model for the appearance of spiral chains in the atomic structure of theelemental chalcogens Se and Te, which we propose to be prototype materials forcombined orbital and charge order in general. These materials are well known tohave a chiral crystal structure at ambient conditions. The handedness of a givensample is evidenced by both its diffraction pattern and optical activity [19]. Thecrystal structure can be viewed as short bonds arranged along helices in a simple

14

Figure 1.1: a) The chiral crystal structure of Se and Te can be understood as aspiral arrangement of short bonds in a simple cubic parent lattice. Atoms withdifferent colours indicate the three possible local configurations of short bonds.The chiral unit cell is indicated in pink. b) The short bonds involve chargetransfer between specific orbitals only, causing the chiral crystal lattice to alsobe orbital ordered. Note that since the electronic system is 2/3 filled, the orbitalorder shown consists of the least occupied orbitals. The shaded planes, includedas a guide to the eye, connect like orbitals and are perpendicular to the spiralaxis.

cubic lattice, as shown schematically in Fig.1.1a [19]. Although Se and Te do notexhibit a charge ordering transition at any temperature, the spiral bond order isunderstood as an instability of a simple cubic parent lattice structure [11, 12, 7].The charge ordering transition from the hypothetic simple cubic to the actualchiral phase is known to be of the same type as the chiral transition in 1T -TiSe2 [8]. Owing to the simple lattice structure however, an explicit and easilyaccessible microscopic model can be formulated for Te and Se, elucidating howdifferent types of electron-phonon coupling and Coulomb interactions conspireto form the spiral structure. This model will be discussed in this chapter as aminimal description for combined charge and orbital order in general.

Given the simplicity and generality of this model, we will also be able to analysethe origin of polonium’s crystal structure. Polonium is unique in the periodictable, since it is the only element to crystallise into a simple cubic lattice struc-ture under ambient conditions. Besides it being remarkable that such a looselypacked configuration is favoured in any material, this is also surprising given thattellurium and selenium, the two isoelectronic elements directly above Po in theperiodic table, adopt a trigonal spiral lattice structure [20] (sulphur and oxygenin the same column form molecules rather than crystals, and will be ignored fromhere on).As already preempted, the spiral structure in Se and Te can be understoodas a combined charge and orbital ordered state, in which a spiral pattern ofpreferential occupation of different p-orbitals necessarily accompanies the charge

15


order [1]. Polonium however, is considerably heavier than Se and Te, and rela-tivistic effects may be expected to play a role in determining its ground state.Heuristically, it is clear that the presence of strong spin-orbit coupling, eliminat-ing orbitals as individual degrees of freedom, is at odds with the formation oforbital order. In fact, ab initio calculations of the phonon dispersion in elementalchalcogens indicate that inclusion of relativistic effects suppresses a softening ofthe phonons, and possibly a related structural instability, which would otherwisebe present [21, 22, 23, 24]. The mechanism by which this is accomplished, aswell as the identification of the dominant relativistic effect, being either a Darwinterm, mass-velocity term, or atomic spin-orbit interaction, is still an unsettledand controversial issue [21, 22, 23, 24, 25, 26].In this first part of the thesis, we will see how our minimal microscopic model forthe elemental chalcogens allows for the prediction of the expected lattice struc-ture of Po, as a function of the strength of spin-orbit coupling. We will see thatat weak coupling, the simple cubic structure is unstable towards the formationof combined charge and orbital order, which results in the spiral trigonal latticestructure observed in Se and Te. Upon raising the strength of the spin-orbitcoupling, the instability is suppressed, and the simple cubic structure observedin Po is realised instead. Moreover, we show that taking into account thermalexpansion of the lattice, the structural instability is suppressed at elevated tem-peratures. That is, using parameter values that are realistic for Po, the phononstructure is softened to such an extent as to effectively weaken the role of spin-orbit coupling at high temperatures. As a result, we find a transition betweenthe two known allotropes of polonium, the simple cubic α−Po and the trigonalβ−Po. We argue that this corresponds to the experimentally observed transitionat approximately 348K [27, 25], and conclude that like Se and Te, β−Po has acombined charge and orbital ordered structure (as indicated in the phase dia-gram of figure 3.2). The unusual lowering of the crystal symmetry upon raisingtemperature, and the peculiar phase diagram connecting the structure of Po tothat of Se and Te, are thus found to be due to the intricate interplay betweenspins, orbitals, charges, and lattice deformations in the elemental chalcogens,where none of these degrees of freedom can be neglected.

1.1 A reader’s guide, part I

The first part of this thesis is devoted to the general understanding of charge andorbital ordered phases, for which the chalcogen crystals serve as model materials.Chapter 1 provides a general introduction to the main mechanisms promotinga Peierls transition to a charge ordered phase. This chapter also contains a verybrief overview on the main assumptions of the nearly free electron model (sec-tion 1.2). This is because the argument motivating the Peierls transition at zerotemperature, for a 1D chain of atoms, is presented in this formalism (section1.3). The overview is also justified in anticipation for the second part of thisthesis, where the model is used extensively to obtain part of the results.

16

1.2. Central assumptions of the nearly free electron model

Chapter 2 focusses on the emergence of charge and orbital order in seleniumand tellurium. The trigonal lattice, with spiral chains, is argued to be the conse-quence of the lattice distortions, induced by a Peierls type transition. In section2.2, the argument is made in terms of the minimisation of the Landau free en-ergy. The role of microscopic interactions is studied next, in section 2.3. Weshow that it is possible to devise a minimal Hamiltonian model, exhibiting spiralcharge and orbital order.

The Hamiltonian model presented in chapter 2 is extended to include spin-orbitcoupling in chapter 3. This will allow us to describe not only selenium andtellurium within the same Hamiltonian, but also polonium. The reason whypolonium adopts a simple cubic structure at low temperatures is addressed inthis chapter. We first investigate the origin of this crystal structure at zero tem-perature, which will highlight the spin-orbit coupling as the crucial ingredient(section 3.1), and later consider the evermore crucial role of the phonons, astemperature increases. These observations will allow us to understand the un-usual behaviour of polonium’s phase diagram, both as a function of temperatureand of the strength of the spin-orbit coupling (section 3.2).In section 3.3, a brief discussion on the complex interplay between spin, charge,orbital, and lattice degrees of freedom, in the elemental chalcogen crystals, isprovided. The first part of the thesis ends with the outlook chapter (Chapter4), where we highlight other possible directions that may bring a better under-standing of spiral and orbital ordered states in general.

1.2 Central assumptions of the nearly free elec-tron model

Throughout this thesis, we will often make use of the nearly free electron modelto either explain concepts or obtain particular results. The first of these goalsapplies in this chapter, where we will refer to the nearly free electron when mo-tivating the general idea behind the Peierls transition. In the second part of thethesis, however, the nearly free electron will play a more active role. There, wewill use it explicitly as the first approach to analyse the origin of edge statesand bulk disappearing states at special crystal terminations. For this reason,we briefly review the key assumptions responsible for the particular form of thewave functions and eigenvalues in this model.

Let us say that we are interested in determining the energy bands and the cor-responding eigenstates of a Hamiltonian of the form H = T + V , where T isthe kinetic energy operator and V is the periodic lattice field. In the nearly freeelectron model the strength of the atomic lattice potential is assumed to be weakcompared to the average kinetic energy of the electrons, and thus the problemmay be solved using perturbation theory.

17


In the absence of the periodic field, the system is well described by the free elec-tron model, where the eigenstates are given by plane waves. However, when thelattice is present, solutions need to obey Bloch’s theorem, which requires wavefunctions of the form ψ(x) = eikxu(x), where u(x) is a periodic function with thesame periodicity as the lattice. Following the usual framework of perturbationtheory, the eigenstates in the full system (unperturbed system + perturbationfield) can now be constructed using the basis of the unperturbed states: theplane waves. To assure the same periodicity as the lattice, the periodic part ofthe Bloch wave function may be written as:

u(x) =

+∞∑l=−∞

ul ei2πlx/a , (1.1)

where a is the length of the unit cell. By inserting this series expansion into theSchrodinger equation, one can find the lth coefficient, in (1.1), in terms of ul=0

[28]:

ulu0

=2mVl~2

1

k2 − (k − 2πla )2

. (1.2)

This relation depends on Vl, the Fourier transform of the lattice potential Vc(x)over a unit cell,

Vl =1

a

∫ a

0

Vc(x)e2πla xdx . (1.3)

Given that the potential Vc(x) is a small perturbation to the free electron system,Vl should also be regarded as a small parameter compared to the kinetic energyof the electrons, i.e. in the nearly free electron model it is assumed that

~2

2m

(πa

)2

>> |Vl| . (1.4)

This assumption allows us to interpret equation (1.2) as stating that Bloch so-lutions can be well approximated by single plane waves (since ul becomes muchsmaller in comparison to u0), except when the denominator vanishes, which hap-pens whenever k is close to πl/a.Perhaps the simplest way to motivate what distinguishes states at k = πl

a fromthe remaining states (and consequently why non-degenerate perturbation theoryfails there) is to refer to (1.3). The matrix element involving the lattice potentialand two states with momenta k and k′ is given by [29]:

〈k′|Vc(x)|k〉 ∝∫V (x)ei(k−k

′)xdx (1.5)

18

1.3. Charge ordering and lattice displacements

Since Vc(x) has the same periodicity as the lattice, one can check that the matrixelement in (1.5) can only be non-zero if k and k′ differ by a reciprocal latticevector. Thus, the presence of these matrix elements mixes the states and leadsto eigenstates that are given by linear combination of plane waves instead. Fur-thermore, in the reduced Brillouin zone, the presence of these matrix elementsleads to energy gaps right at k = 0 and k = ±π/a. As the strength of the latticefield goes to zero, the gaps are suppressed, and states at these points, whichin the presence of the lattice had energies belonging to different bands, becomedegenerate.

In the nearly free electron model, the eigenstates are then given by single planewaves, except close to the edges of the Brillouin zones, where linear combinationsof plane wave states have to be taken. Moreover, the energy spectrum is formedby continuous curves as a function of k, that are separated by energy gaps atk = ±π/a and k = 0, in the reduced Brillouin zone scheme.

1.3 Charge ordering and lattice displacements

In this section the intent is to provide motivation as to why a periodic chargemodulation may become the stable ground state of a crystal. To this aim, westart by briefly reviewing the Peierls mechanism [30]. The idea rests in ad-dressing the following problem: given a metal at zero temperature, formed by a1D chain of atoms, will the ground state be one for which all atoms are equallyspaced from each other (see fig.(1.2))?

Figure 1.2: Linear chain of atoms, with a uniform charge density distribution.

Given that the 1D crystal here is meant to represent a metal, the nearly freeelectron model becomes a good approach to describe the electrons in the system.In this framework, the average kinetic energy of the electrons is assumed to bemuch larger than the periodic field set by the lattice. For that reason, the latteris treated as a perturbation parameter, causing the electronic band structure tobe modified only slightly by the appearance of energy gaps at the edges of theBrillouin zone, i.e. at k = πl

a . Let us further assume that the electronic statesare only half filled, fixing the Fermi momentum to be kF = π

2a . Then at zerotemperature, this situation is as given in fig(1.3), where only the band hostingthe energies of the occupied states is shown for simplicity.Recall that in the nearly free electron model, the reason as to why gaps opened

19


Figure 1.3: If the electronic system is only half-filled, then, at zero temperature,the ground state is given by populating electron states with energies given bythe yellow curve.

up precisely at k = πla was because the lattice field coupled only the electronic

states whose momenta differed by a reciprocal lattice vector. This originatedfrom the lattice periodicity. Hence, if the lattice constant is made bigger byletting a → 2a, this would determine the energy gaps to open up at |k| = πl

2a .At the same time, it would imply the existence of a periodic field, such thatVc(x) = Vc(x + 2a). Starting from the original chain of atoms in fig.(1.2), thenew periodicity can be induced by imagining introducing a periodic field thatdisplaces, for example, every second atom in the chain. This would then leadto a new configuration corresponding to a dimerised chain, with periodicity of2a, as see from fig.(1.4). But why would the system be prone to form this newlattice?

Figure 1.4: Dimerised chain of atoms. We can imagine obtaining this configu-ration from fig.(1.2) by slightly moving every second atom to the left.

Having an energy gap at k = πl2a means that now all the energies of the occupied

electron states are separated from the unoccupied ones (see fig.(1.5)). Since thisintroduces a lowering in energy for the occupied states near k = πl

2a , the result is again in energy for the electronic system. For small lattice displacements, it turnsout that this gain in electronic energy is always bigger than the cost in (elastic)

20


energy arising from displacing the ions [30] [29]. Thus, from the perspective ofan energetic gain, the 1D chain given in fig.(1.2) is unstable towards dimerisa-tion, and, as a consequence, the metal is unstable towards becoming an insulator.

Figure 1.5: Gap opening up at |k| = π2a , leading to the lowering of the electronic

energy for the occupied states.

The simple energetic argument above motivates why, at zero temperature, latticedistortions in a 1D metal become a likely scenario. But the onset of charge orderand lattice distortions can be equivalently motivated by looking at the behaviourof response functions [31].To sketch the main points of this argument, let us recall that a 1D free electronsystem, at zero temperature, is unstable towards developing a charge modulation,with period Q = 2kF , even in the presence of a very weak periodic perturbation.This result, which can be obtained in the context of linear response theory, is dueto the divergent response function χ(Q = 2kF ) (the Lindhard response function)(see fig.(1.6)), inducing the redistribution of charge through the relation [32]:

ρind(Q) = −δV (Q)χ(Q) , (1.6)

where δV (Q) is the periodic perturbation field and ρ(Q) is the induced chargedensity. Thus, even if δV (Q) is a weak potential, the divergent χ(Q) determinesa strong charge redistribution.The divergence of the response function can be interpreted as a consequence ofthe particular Fermi surface shape, which allows for perfect nesting, i.e. thereexists a single Q vector that connects all points in one sheet of the Fermi surfaceto another Fermi surface sheet [33].In 1D, the situation is rather special since the Fermi surface consists of only twopoints (kF and −kF ), and thus it is possible to superimpose the two ”sheets”by a translation through a unique Q-vector, equal to 2kF . If the dimensional-ity increases, the regions in each Fermi surface sheet that can be connected to

21


Figure 1.6: Behaviour of the electronic response function χ(Q)/χ(0), as a func-tion of the wavenumber Q, at zero temperature, for each spatial dimension.

one another with the same Q-vector are reduced. Consequently, also the stronginstability at Q = 2kF disappears (see fig. (1.6)). However, if the electronicwave functions allow for anisitropic overlaps, i.e. stronger along particular crys-tallographic directions and weaker in others, then the resulting crystal structurecan allow for linear chains or layered structures to be present. This gives riseto a quasi-one-dimensional motion for the electrons, bringing back the instabil-ity towards the charge redistribution [33, 34, 32]. An example of a material,where the aforementioned behaviour is present, is the Krogmann’s salt (KCP)[34]. There, linear chains of platinum atoms arise because of the strong overlapbetween dz2 orbitals, along the chain’s direction. We note that the presence ofanistropic overlaps, leading to the formation of chains, will also be crucial in thechalcogen materials, studied in the following chapters.Even though the previous systems allow for a quasi-one-dimensional electronmotion, the structure itself is still 3D. The three-dimensional character comesabout because Coulomb interactions between neighbouring chains, or interchaintunnelling, for example, are processes that are present, even if they are not thedominant interactions. It is also due to these same interactions that these ma-terials can sustain a long-range ordered phase (the charge ordering) up to finitetemperatures, since in pure one-dimensional systems, where only short-range in-teractions are present, the fluctuations destroy the long-range order [35]. Thus,for the Frohlich Hamiltonian, consisting of free electrons and phonons, interactingvia the electron-phonon coupling, the picture of charge ordering and correspond-ing lattice distortion is not present at any finite temperature in 1D. Nevertheless,even in 1D the model still provides worthwhile information concerning the role

22


of electron-phonon coupling for the combined onset of a new charge modulationand a displacement of the lattice positions, which is why it is briefly describednext [32].The periodic perturbation causing the induced charge modulation in (1.6) canbe obtained, in a more microscopic description, through the electron-phononcoupling interaction. This can be briefly motivated as follows: the lattice vibra-tions, induced by the phonons, create a polarisation, which as a result gives riseto an external potential for the electronic system; in turn, the electronic systemdevelops an induced charge re-ordering, and this has further consequences forthe lattice vibrations. When the lattice vibrates, the frequency w of these oscil-lations depends on the magnitude of the restoring force. In particular, when theions are displaced about their equilibrium position, the origin of the restoringforce is due to the Coulomb repulsion between the positively charged ions. Theinduced ionic potential field gives rise to the new charge density in (1.6), for theelectronic system. But if this new re-ordering of charge minimises the overallenergy of the system, then it will screen the electronic repulsion between theions. As a result, the frequency w should decrease.

Figure 1.7: Renormalised phonon frequency, as a function of the phonon wavenumber Q, for each spatial dimension.

The explicit calculation from a Hamiltonian that includes the free electron andphonon systems plus the electron-phonon coupling (the Frohlich Hamiltonian)can be found in the literature, and provides a renormalised phonon frequencythat depends not only on the strength of the electron-phonon coupling, but alsoin the electronic response function χ(Q) [32]. This renormalisation of the phononfrequency is what is called the Kohn anomaly [31]. The dependency of w on theelectronic response function is what causes its massive reduction when the di-

23


vergence in the electronic system is present: the renormalised phonon frequencygoes to zero, causing a maximum occupation of the phonon mode at Q = 2kF(see fig.(1.7)) (see fig.(1.7)). The result is a static lattice distortion, with thesame momenta Q, occurring alongside a modulation of charge. In this way,the electron-phonon coupling becomes the main promoter for a combined latticedisplacement and charge modulation, in the zero temperature ground state ofone-dimensional crystals. Furthermore, it also allows us to think of the chargedensity wave instability in terms of a Kohn anomaly.

24

CHAPTER 2

Combined charge and orbital order in seleniumand tellurium

Helices of increased electron density emerging spontaneously in materials con-taining multiple, interacting density waves, are an example of how orbital andcharge degrees of freedom may combine to form a single ordered phase.Here, we present the elemental chalcogens selenium and tellurium as model ma-terials for the development of combined charge and orbital order. We formulateminimal models capturing the formation of spiral structures consisting of orderedoccupied orbitals and increased charge density, both in terms of a macroscopicLandau theory and a microscopic Hamiltonian. Both reproduce the known chi-ral crystal structure and are consistent with its observed thermal evolution andbehaviour under pressure. The combination of microscopic and macroscopicframeworks allows us to distil the essential ingredients in the emergence of com-bined orbital and charge order, and may serve as a general guide to predictingand understanding spontaneous chirality as well as other, more general, types ofcombined charge and orbital order in other materials.

2.1 Intuitive pictureBefore presenting both macroscopic and microscopic models of the chiral chargeorder, we first give an intuitive picture showcasing their basic ingredients. Thestarting point is the simple cubic lattice structure. Both Se and Te crystalspossess the chiral structure shown in Fig.1.1a for any temperature at ambientpressure. Upon melting Te however, the short-ranged chiral order in the fluidturns into a more cubic, metallic phase at a crossover temperature not muchabove the melting point [36, 37, 38]. This observation can be understood as alatent structural phase transition, which is preempted by the material meltingbefore the transition temperature can be reached. In fact, in the element Po,which is iso-electronic to Se and Te and sits just below them in the periodic

25

Chapter 2. Combined charge and orbital order in selenium and tellurium

Figure 2.1: Depiction of the allowed and neglected orbital overlaps in our mini-mal model. The allowed ones (featuring orbitals aligned in a head-to-toe manner)are indicated by the black arrows, while the neglected ones are indicated by theaddition of red crosses to the black arrows. Each atomic site has valence elec-trons occupying the three p orbitals, but to avoid overloading the figure, theyare not all represented simultaneously for all atomic sites of the cubic lattice.

table, strong spin-orbit coupling prevents the formation of chiral order and al-lows the underlying simple cubic lattice structure to remain visible even at lowtemperatures [39, 24].

Within a simple cubic lattice, the four valence electrons (2/3 filling) of elementalchalcogens are distributed among p orbitals that may be chosen to point along thecrystallographic x, y, and z axes. The overlap between neighbouring px orbitalson the x axis is significantly larger than that between neighbouring px orbitals onthe y or z axes, or between p orbitals of different type. Taking this difference tothe extreme limit, we consider a minimal model in which only overlaps betweenorbitals aligned in a head-to-toe manner are non-zero (see fig.(2.1)). Althoughquantitatively unrealistic, it captures all qualitative aspects of the chiral phasetransition.

In that limit, an electron in for example a px orbital can only hop in the x direc-tion, onto a neighbouring px orbital. The simple cubic lattice is thus filled withinterwoven but independent one-dimensional chains running in all three latticedirections. The electronic structure consists of three one-dimensional bands, eachproducing a pair of parallel planar Fermi surfaces in the first Brillouin zone. TheFermi surface is extremely well-nested, and a Peierls-type charge density waveis expected to emerge[30]. In fact, a single nesting vector Q, corresponding to abody diagonal of the cube of intersecting Fermi surfaces, connects each point onany of the Fermi surface sheets to a point within a parallel sheet (see fig.(2.2)).The dominant instability will therefore be towards the formation of charge den-

26

2.1. Intuitive picture

Figure 2.2: Fermi surface within our minimal model. Each pair of parallel planarFermi surfaces, coming from each 1D band, is represented by the differentlycoloured planar sheets. The intersections of the planar surfaces themselves forma cube. As a result, the Fermi surface of the 3D lattice is well nested, withnesting vector Q (identified in the figure by the blue arrow).

sity waves ρj(x) = ρ0+A cos(Q·x) in each of the three orbital sectors (labeled byj), who all share the same propagation direction Q. Here ρ0 is the average chargedensity in the normal state, and A is the amplitude of the charge modulation.

The atomic displacement waves uj(x) = uej sin(Q · x), forming in response tothe charge modulations, have polarisations ej whose direction is determined bythe anisotropy of the local electron-phonon coupling matrix elements[8]. In achain of px orbitals with overlaps only along x, the electron-phonon couplingis maximally anisotropic, and the displacement direction e will be purely alongx. The three orthogonal chains running through each atom act independently,and the actual atomic displacement is the sum of the three contributions uj (seefig.(2.3)).

The charge density wave in each orbital chain can be shifted along its propagationdirection by the addition of a phase: ρj(x) ∝ cos(Q · x + ϕj). A Coulombinteraction between charges in orthogonal orbitals on the same site, will cause thecharge maxima along one chain to prefer to avoid those of other chains, effectivelycoupling the phases in different orbital sectors. The lowest energy configurationthen produces precisely the charge redistribution and lattice deformations shownin Fig.1.1a, which agree with the experimentally observed crystal structure of Seand Te[11, 7]. Each atom in this final structure has a single least occupied porbital. The chiral charge ordered structure is therefore also automatically anorbital ordered phase, as shown in Fig.1.1b.

27


Figure 2.3: The effect of only considering orbital overlaps among nearest neigh-bouring p orbitals aligned in a head-to-toe manner gives rise to an effectiveelectronic system, which prior to interactions, is identical to three independent1D systems, where electrons occupy a particular p orbital. For this reason, wecan refer to the resulting 1D systems as orbital sectors. Given that they areindependent, the displacements of the full system become the sum of all threedisplacements in each orbital sector. On-site Coulomb interactions provide aweak coupling between the charge modulations in different 1D chains and fixestheir relative phase difference. The result is shown here in the rightmost figure.

2.2 Macroscopic order parameter theory

A Landau free energy may be written in terms of the dimensionless order param-eters αj(x) representing the periodic modulation of the charge density within agiven chain of head-to-toe orbitals: ρj(x) = ρ0(1 + αj(x)). If the orbital sectorsare non-interacting, their free energies are independent:

Fj =

∫d3x a(x)α2

j + b(x)α3j + c(x)α4

j . (2.1)

Notice that the presence of the lattice is taken into account by expanding thecoefficients in terms of reciprocal lattice vectors, so that for example a(x) =a0 + a1

∑n e

iGn·x + . . . [40]. Here, Gn denote the shortest reciprocal latticevectors. Terms aj with j > 0 in this sum arise from the electron-phonon couplingin a more microscopic model. The on-site Coulomb interaction provides theinteraction terms

FCoul =∑j

∫d3x A0αjαj+1 , (2.2)

and adds an energy cost to having all the charge density waves with chargemaxima at the same lattice site. The periodic charge distributions can be writtenas αj(x) = ψ0 cos(Q · x + ϕj), with the amplitude ψ0 equal for all three orderparameters, and ϕj a spatial shift of the charge density wave along j. Performing

28

2.3. Minimal microscopic model

the integration over x, the full free energy, per volume, becomes:

F =3

2a0ψ

20 +

9

8c0ψ

40 +

1

4b1ψ

30

∑j

cos(3ϕj)

+1

2A0ψ

20

(cos(ϕ1 − ϕ2) + cos(ϕ2 − ϕ3) + cos(ϕ3 − ϕ1)

). (2.3)

As usual, the temperature dependence is considered by expanding a0 as a func-tion of T − Tc, near the critical temperature. In this way, a0 determines whenthe free energy has a minimum at non-zero values of ψ0, and charge order setsin. This also fixes the value of the first two terms in F , in the charge orderedphase. The final two terms, arising from the electron-phonon coupling and theCoulomb interaction respectively, determine the values of the phases ϕj .This last statement can be understood as follows: in the charge order phase,we can make F lower in energy by having the phases ϕj such that the lasttwo terms in (2.3) are negative; if b1 is positive, this is then accomplished byhaving ϕ1 = nπ/3, where n is an odd integer, with relative phases chosen asϕj − ϕj+1 = ±2π/3, while if b1 is negative the only difference is that n becomesan even integer instead. Note that the amplitude ψ0 is defined to be positive.Physically, the two possible signs of b1 correspond to the charge order being eithersite or bond centred. The two choices in the sign of the relative phase difference,i.e. ϕj − ϕj+1, correspond to left and right handed chiral configurations, one ofwhich is shown in Fig.1.1a. One can see this by drawing the three displacementsthat are induced by the resulting three periodic charge modulations (αj(x)’s) inthe parent simple cubic lattice, just as illustrated in fig.(2.3).Comparing the free energy of (2.3) to the one given for 1T -TiSe2 in Ref.[8], itappears that in spite of the different underlying atomic and electronic configu-rations, the routes to chiral charge and orbital order are largely the same. Theonset of charge order from a disordered state is determined by amplitude termsonly. Electron-phonon coupling then favours locking of the different charge den-sity wave contributions to the lattice. The on-site Coulomb interaction finally,couples density waves in distinct orbital sectors, leading to relative phase shiftsand the emergence of chiral charge and orbital order.

2.3 Minimal microscopic modelTo see how the terms in the Landau free energy emerge from the interplay ofmicroscopic degrees of freedom, we start again from a two-third filled p-shellwithin the simple cubic lattice. Including hopping only between head-to-toeorbitals, the tight-binding Hamiltonian can be written as

HTB = t∑x,j

c†j(x)cj(x + aj) + H.c. . (2.4)

Here c†j(x) creates an electron in orbital j on position x, and aj is the simplecubic lattice vector in direction j. The hopping amplitude t is positive, since

29


overlapping orbital lobes on neighbouring sites have opposite signs. We nextconsider the Coulomb interaction acting between electrons in different orbitalson the same site,

HCoul = V∑x,j

c†j(x)cj(x)c†j+1(x)cj+1(x) . (2.5)

The displacement uj(x) of the atom on position x in the direction of j is written

in terms of the phonon operator b†j(x), taken to be a dispersionless Einstein mode.This is motivated by the value of the nesting vector Q, which sits sufficientlyaway from zero and allows the phonon dispersion relation to be considered as aslowly varying function of the momentum q. We further simplify it by taking itto be approximately constant. The Hamiltonian for the phononic system is thengiven by:

Hboson = ~ω∑q,j

b†j(q)bj(q). (2.6)

To introduce an interdependency between the electronic system and the underly-ing lattice, the electron-phonon coupling interaction He-ph is added to the model.

This is done through two contributions (He-ph = H(1)e-ph + H

(2)e-ph):

H(1)e-ph = α(1)

∑x,j

(uj(x)− uj(x + aj)

)·(

c†j(x)cj(x + aj) + c†j(x + aj)cj(x))

H(2)e-ph = α(2)

∑x,j

(uj(x + aj)− uj(x− aj)

)c†j(x)cj(x). (2.7)

The first type of electron-phonon coupling affects the kinetic energy of electrons,by increasing the hopping amplitude if the interatomic distance decreases, andis similar to the electron-phonon coupling considered in the Frohlich Hamilto-nian [32]. The second process lowers the electronic potential energy in regionsof increased ionic density (this trait is motivated in appendix A.3).The full Hamiltonian is then given by combining all the previous effects,

H = HTB + HCoul + He-ph +Hboson (2.8)

and can be diagonalised in the mean field approximation, using Ansatz averagesthat reflect the ordered states found in the Landau theory analysis:

〈c†j(x)cj(x)〉 = ρ0 +A cos (Q · x + ϕj)

〈c†j(x)cj(x + aj)〉 = σ0 +B cos (Q · (x + aj)/2 + χj)

〈uj(x)〉 = u sin (Q · x + φj) .

(2.9)

30


Here A is the mean field for modulations of the on-site charge density, B forthe bond-density, and u for the atomic positions. Coarse graining of the meanfields A and B would yield the Landau order parameter α, while the values ofcoefficients appearing in the Landau theory can in principle be obtained fromthe microscopic model by calculating loop diagrams [41].

The bosonic part of the mean field Hamiltonian can be diagonalised by introduc-ing a new set of operators defined as γj(q) = bj(q) + vq,jδ(q−Q) + wq,jδ(q +Q)[42]. The explicit computation can be found in the appendix A1. Requir-ing this transformation to bring the bosonic Hamiltonian into diagonal formdetermines the values of vq,j and wq,j . The expectation values of the displace-

ment operators uj can then be computed in the diagonal basis, which yieldsthe relation between atomic displacements and the electronic order parameters:u = 2

√3/~ω(2Bα(1)e(χj−φj) − Aα(2)ei(ϕj−φj)). Demanding the displacement

to be real restricts the phase differences to be integer multiples of π. Further-more, since the lattice displacements need to be consistent with the charge re-ordering (so that the overall energy of the system is at its minimum), this fixesϕj = φj = χj . This ensures the ions move towards the regions of higher electrondensity.By Fourier transforming the fermionic part of the mean field Hamiltonian, weobtain (in the reduced Brillouin zone scheme):

HMFF =

∑k,j

(c†j(k) c†j(k + Q) c†j(k−Q)) HMF

F

cj(k)cj(k + Q)cj(k−Q)

, (2.10)

where HMFF is given by

HMFF =

ε(k) g∗j (k) + F ∗j + J∗j gj(k −Q) + Fj + Jjgj(k) + Fj + Jj ε(k +Q) g∗j (k +Q) + F ∗j + J∗j

g∗j (k −Q) + F ∗j + J∗j gj(k +Q) + Fj + Jj ε(k −Q)

(2.11)

The functions Fj , gj , Jj and εk are defined as follows:

Fj = α(2)u sin(Q · aj)eiφj

gj(k) = α(1)ueiφj (sin(k · aj)− sin((k + Q) · aj))

Jj = AV∑l eiϕl with l 6= j

ε(k) = 2t(cos(k · aj)− 1/2)

. (2.12)

31


Figure 2.4: The ground state phase diagram as a function of the two contributionsto the electron-phonon coupling. The vertical axis shows the order parameterB, while the colours indicate the normalised atomic displacement. Two possibletypes of chiral charge and orbital ordered state are shown schematically in theinsets. Increased bond density is indicated by double lines, while curved arrowsindicate charge transfer from or onto atomic sites. The grey line indicates theregion where the displacement is zero.

The matrix HMFF can then be diagonalised numerically. The phases and ampli-

tudes of the order parameters are determined self-consistently, by requiring thatthe expectation values computed using the particular Ansatz averages (given in(2.9)) match the Ansatz values, numerically determined from the eigenvectors ofthe diagonalised HMF

F .With zero on-site Coulomb interaction, the orbital sectors independently developcharge density waves. The phases are ϕj = njπ/3, with nj integer, which includesboth non-chiral solutions in which the nj are all equal, and chiral ones. For anynon-zero value of the Coulomb interaction, this degeneracy is lifted, and the left-and right-handed chiral charge ordered configurations with nj − nj+1 = ±2π/3become the lowest energy states.For each handedness, the nj may be odd or even multiples of π/3. These solu-tions correspond to either bond-centered or site-centered charge density waves,as indicated in the insets of Fig.2.4. The bond-centered solution dominates forlarge α(1)/α(2), while the site-centered one is consistent with the opposite regime.The atomic structure observed in elemental Se and Te corresponds to the bond-centered charge order [19], with α(1) prevailing.Within the chiral phase, the short bonds in the three orbital chain directionsconnect to form a spiral, in agreement with the experimentally observed structureshown schematically in Fig.2.5, on the left hand side. The displacements in the

32

2.4. Conclusions

Figure 2.5: Charge and orbital order in Se and Te. On the left side is theschematic depiction of the chiral trigonal lattice in Se and Te (in green), emerg-ing out of the parent cubic lattice when charge and orbital order set in. Thespiral chain is highlighted in yellow in the figure. Following the direction of theshort bonds (figure on the right hand side), the least occupied orbital alternatesperiodically between the three p orbitals. Thus, the spiral modulation of chargeoccurs in combination with an orbital ordered phase.

x, y and z directions arise from charge order in chains of px, py, and pz orbitalsrespectively. The modulation of charge density can thus also be seen as a spatialmodulation of orbital occupation, as shown explicitly in Fig.2.5 (on the righthand side). The emergence of orbital order in conjunction with chiral chargeorder is inevitable, since both arise from the same relative phase shifts betweencharge density waves in distinct orbital sectors. The presence of an interactionbetween charge density wave components leading to relative phase shifts, canthus be interpreted as a generic route to combined charge and orbital order,which should be applicable to a wide range of materials hosting multi-componentcharge order.

2.4 Conclusions

Although a phase of combined charge and orbital order has been proposed toexist in the low-temperature phase of 1T -TiSe2[6, 8, 9], the broken inversionsymmetry is yet to be observed directly. In addition, the interplay between thegreat number of atoms within the unit cell of TiSe2 complicate the extraction ofphysical insight from microscopic models[43]. Having a model material, whichharbours a similar charge and orbital ordered state but is structurally simple andwell-understood, is therefore crucial to aid in building a general understanding ofthis novel state of matter, and in allowing the identification of related novel typesof order in other materials. We argue that the elemental chalcogens telluriumand selenium constitute precisely such model materials.

In both the elemental chalcogens and 1T -TiSe2 multiple density wave instabil-

33


ities occur in distinct orbital sectors. These give rise to multiple, differentlypolarised, displacement waves in both materials. The on-site Coulomb repulsionthen causes maxima of different density waves to repel each other. This resultsin shifting of the density waves, thus breaking inversion symmetry and yieldinga chiral crystal structure (as long as no mirror symmetries in the parent latticereduce the chiral order to polar[13, 15, 14]). The density waves originating indistinct orbital sectors, necessarily implies that orbital order accompanies thecharge modulations, creating a combined charge and orbital ordered phase.In contrast to the chalcogens, the propagation vectors for different density wavesin TiSe2 are all distinct. Also, the on-site Coulomb repulsion coupling differentdensity waves, yields only an indirect interaction between bond-centered chargesin the case of TiSe2. Despite these and other differences, including for examplethe different driving mechanisms underlying the density wave formation[42, 44], acommon general mechanism for combining charge and orbital order is identified:as long as multiple density wave instabilities occur in distinct sections of theFermi surface, which correspond to distinct orbital textures, any local interactionbetween orbital sections will cause the combination of charge and orbital orderinto a single ordered phase. We thus predict that a type of combined ordercan be found in any charge ordered material involving multiple orbital sectors,including (families of) materials with much more complicated structures thanthe elemental chalcogens. The theoretical understanding developed here for Seand Te can be used as a guiding principle in looking for such novel states ofmatter.

34

CHAPTER 3

Combined charge and orbital order in polonium

Polonium is the only element to crystallise into a simple cubic structure underambiant conditions. Moreover, at high temperatures it undergoes a structuralphase transition into a less symmetric trigonal configuration. It has long beensuspected that the strong spin-orbit coupling in polonium is involved in bothpeculiarities [21, 22, 23, 24], but the precise mechanism by which it operatesremains controversial [21, 22, 23, 24, 25, 26]. Here, we introduce a single micro-scopic model capable of capturing the atomic structure of all chalcogen crystals:selenium, tellurium, and polonium. We show that the strong spin-orbit couplingin polonium suppresses the trigonal charge and orbital ordered state known to bethe ground state configuration of selenium and tellurium, and allows the simplecubic state to prevail instead. We also confirm a recent suggestion based on abinitio calculations that a small increase in the lattice constant may effectivelydecrease the role of spin-orbit coupling, leading to a re-emergence of the trigo-nal orbital ordered state at high temperatures. We conclude that polonium isa unique element, in which spins, orbitals, electronic charges, and lattice defor-mations all cooperate and collectively cause the emergence of the only elementalcrystal structure with the simplest possible, cubic, lattice.

3.1 Minimal microscopic model

The starting point for constructing a minimal microscopic model capable of de-scribing the lattice instabilities in the entire family of elemental chalcogens, isa simple cubic arrangement of atoms. All chalcogens have four electrons in theouter shell of p-orbitals, so we will consider a tight-binding model taking intoaccount only px, py, and pz-orbitals on each site. Recall that, for convenience,we choose the quantisation axes for the orbitals to coincide with the lattice di-rections. The strongest orbital overlaps then occur in one-dimensional chains ofp-orbitals aligned in a head-to-toe fashion along their long axis. In other words,

35

Chapter 3. Combined charge and orbital order in polonium

the overlaps of for example neighbouring px orbitals on the x-axis are much largerthan those between neighbouring px orbitals on the y or z axes, or between anytwo p orbitals of different type.

A minimal model for the bare electronic structure may thus be constructed bytaking into account a hopping integral t along chains in all three directions, butneglecting all other orbital overlaps, and in particular any inter-chain hopping.Interactions between one-dimensional chains in different direction can then betaken into account by including the Coulomb interaction V between electronsin different p-orbitals on the same site. The resulting model is known to qual-itatively capture the instability in the electronic structure which underlies theformation of combined charge and orbital order in Se and Te[1]. The electronicHamiltonian for this minimal model can be written as the sum of tight bindingand Coulomb terms. These have essentially the same form as presented in theprevious chapter (equations (2.4) and (2.5)), except that now the spin index isincluded. For convenience we rewrite them here again:

HTB = t∑r,j,σ

c†r,j,σ cr+j,j,σ + H.c.

HCoul = V∑

r,j,σ,σ′

c†r,j,σ cr,j,σ c†r,j+1,σ′ cr,j+1,σ′ (3.1)

where c†r,j,σ creates an electron on position r, with spin σ, in a pj-orbital, withj ∈ {x, y, z}. The lattice vectors aj are written using the shorthand notation j.In our simulations, we use the parameter values t = 2.0 eV and V = 39 meV. Wenote that the numerical values of V , α(1) and α(2) were chosen to ensure fasternumerical convergence, in particular because it gives a bigger energy differencebetween the chiral and non-chiral CDW phases. It was numerically checked thatincreasing the value of these parameters does not lead to any additional phasetransition. The values are then justifiable since we do not aim to make anyquantitative predictions, but only intend to describe qualitatively the underly-ing physics of the system.We additionally allow atoms to be displaced by introducing phonons. Since thephonon dispersion is approximately flat in the momentum-space region of inter-est, we employ an Einstein mode of constant energy ~ω = 3.5 meV. This value isconsistent with the values obtained in the ab initio calculations in reference [24].Furthermore, we account for the two different ways in which electrons coupleto the phonons, introduced in the previous chapter: on the one hand, atomicdisplacements alter the interatomic distances, which affects the hopping of elec-

trons between them (H(1)e-ph in the previous chapter); on the other hand, atomic

displacements also alter the local density of ions surrounding a particular site,

which influences the on-site potential energy of electrons (H(2)e-ph in the previous

36


chapter). Hence:

Hkinel-ph = α(1)

∑r,j,σ

(ur,j − ur+j,j

)c†r,j,σ cr+j,j,σ + H.c.

Hpotel-ph = α(2)

∑r,j

(ur+j,j − ur−j,j

)c†r,j,σ cr,j,σ. (3.2)

Here ur,j is the operator corresponding to the j-component of displacement forthe atom on position r. The relative strength of the two types of electron-phonon coupling α(1) and α(2) determines whether a spiral trigonal structureconsisting of site-centered or bond-centered charge density waves is formed in Seand Te. For simplicity, we assume equal values α(1) = α(2) = 0.04 eV for thesecouplings, resulting in a bond-centered spiral state consistent with experimentalobservations.

As we learned from Se and Te, the minimal model formed by the terms con-sidered so far gives rise to three sets of mutually parallel Fermi surface sheets.This situation is extremely well-nested, and, together with the electron-phononcoupling, renders the simple cubic phase unstable towards the formation of threesimultaneous charge density waves, connected to the three sets of planes. Infact, a single, common nesting vector Q = 2π/3a(1, 1, 1) can be chosen such thatevery point on a Fermi surface sheet is connected to a corresponding point ona parallel sheet. The on-site Coulomb interaction, acting between electrons indifferent orbitals, provides a coupling between the density waves, resulting inan overall spiral trigonal structure. Because each charge density wave residesin chains of a particular type of orbital, the trigonal structure is automaticallyorbital ordered as well as charge ordered[1].

In Polonium, we expect relativistic effects to suppress the trigonal β-Po phaseat low temperatures, and instead stabilise the simple cubic α-Po allotrope. Thisis made possible in the minimal model by including spin-orbit coupling:

HSOC = λSOC

∑r,j,j′,σ,σ′

Mjj′σσ′ c†r,j,σ cr,j′,σ′ , (3.3)

where λSOC is the overall strength of the spin-orbit coupling, while M containsthe matrix elements of the operator L · S in the basis of states labelled by orbitalindex j and spin S (this matrix and its construction are provided in appendixA2).

The full Hamiltonian, combining all terms from equations (3.1), (3.2), and (3.3),can be solved numerically within the mean field approximation. This is done byintroducing mean field averages corresponding to charge density, bond density,

37


and displacement waves in each of the three lattice directions:∑σ

〈c†r,j,σ cr,j,σ〉 = ρ0 +A cos (Q · r + ϕj)∑σ

〈c†r,j,σ cr+j,n,σ〉 = σ0 +B cos (Q · (r + n)/2 + ϕj)

〈ur,j〉 = u sin (Q · r + ϕj) . (3.4)

Recall that A is the mean-field amplitude for the on-site charge density varia-tions, while B corresponds to modulations of the bond densities. The atomicdisplacement field is given by u. The wave vector Q is equal for all instabilitiesand is determined by the strongly nested Fermi surface, but the phases ϕj differbetween density waves in different lattice directions j. Taking ϕj = n · 2π/3,the known spiral trigonal lattice structure of Te and Se is recovered for vanish-ing spin-orbit coupling. This relation can be understood as an optimisation ofthe competition between Coulomb and electron-phonon interactions[1], as dis-cussed in the previous chapter, and is assumed to hold also for finite values ofthe spin-orbit coupling.As previously, the phonon part of the mean field Hamiltonian is solved analyt-ically using a Bogoliubov transformation[42], which shows the atomic displace-ments in the presence of given electronic order parameters A and B to be, asbefore, u = 2

√3/~ω(2Bα(1) − Aα(2)). This expression relates the displacement

field u to the amplitudes of site-centered and bond-centered charge modulations.Notice that the size of displacements is inversely proportional to the bare phononfrequency. The fermionic part of the mean field Hamiltonian can, just as in theprevious chapter, be written in matrix form and diagonalised numerically forany given value of u. Iterating this procedure eventually yields self-consistentsolutions for the displacement u and the density modulations A and B.

Without spin-orbit coupling and at zero temperature, the mean field groundstate has a non-zero expectation value for the displacements, and is hence inthe spiral trigonal lattice configuration. As the strength of spin-orbit coupling,λSOC, is increased, a critical point is encountered, beyond which no non-trivialself-consistent solutions exist, as shown in figure 3.1. Intuitively, the disappear-ance of the trigonal state at large λSOC can be understood by realising that itcorresponds to a state of combined charge and orbital order. The strong couplingbetween spin and orbitals destroys the independent orbital degree of freedom,and hence prevents the onset of orbital order. As a result, the simple cubic latticeremains the ground state configuration. Alternatively, the competition betweenspin orbit coupling and density wave order may be phrased in terms of energetics.Large spin orbit coupling causes the Fermi surface to deform and gaps to openup. This obstructs the formation of charge and orbital order, which depends onhaving sufficiently nested Fermi surface available for a charge ordering gap tolower the overall electronic energy.

38

3.2. Turning up the temperature

Figure 3.1: The value of the bond density order parameter B, as a functionof the strength of spin orbit coupling λSOC, which is measured in units of thebandwidth t. The points are self-consistent numerical solutions of the set ofequations in eq. (3.4), while the solid line connecting them is a guide to the eyeonly. The unordered state at high λSOC/t corresponds to the simple cubic latticestructure of α−Po, as shown in the right inset. In the ordered state, the planesperpendicular to the cube’s body diagonal move closer together, by means of acontraction of the thick bonds shown in blue in the left inset. The result is thespiral trigonal lattice structure known to be realised in Se and Te. Because thestructural transition is the result of three simultaneous density wave instabilities,each occurring in chains of distinct orbitals, the trigonal state necessarily is alsoan orbital ordered state. The least occupied orbitals in each trigonal plane arehighlighted in the left inset.

3.2 Turning up the temperature

It is known experimentally that polonium undergoes an unusual structural phasetransition at about 348 K, where the low temperature simple cubic α−Po latticestructure is reduced in symmetry and becomes the high temperature trigonalβ−Po phase[45, 27, 46]. In order to describe this effect in our minimal model,we include the effect of temperature in two places. First, the mean field expecta-tion values all become thermal expectation values, written for the electronic partof the Hamiltonian in terms of Fermi-Dirac distributions. Secondly, and moreimportantly, we take into account the fact that thermal expansion of the latticewill cause a lowering of the bare phonon energy. Owing to the relative softnessof the material, the change in phonon energy in Po is significant, and cannot beneglected[24].

To describe the dependence of phonon energy on temperature, we first ap-

39


proximate the thermal expansion to be linear, so that the lattice constant attemperature T can be written as a(T ) = a0(1 + α∆T ). Here a0 is the latticeconstant at some reference temperature (∆T = 0), and α is the linear thermalexpansion coefficient, which we take to be the experimentally determined valueα = 23.5 × 10−6K−1, obtained at 298K[47]. Taking α to be fixed while vary-ing the temperature is seen to be a reasonable approximation in the region ofinterest by comparing it to the volumetric thermal expansion in Po as obtainedby first principle calculations[23]. Assuming the phonon energy to depend onthe interatomic distance, the expansion of the lattice will cause the bare phononenergies to soften, which we describe by the linear dependence:

~ω = ~ω0 + γ(a(T )− a0)/a0, (3.5)

where ~ω0 is the energy of the bare phonon at the reference temperature where∆T = 0 and a(T ) = a0. Fitting equation (3.5) to experimental data in orderto establish the value of γ is prevented by the fact that polonium’s strong ra-dioactivity leads to a scarcity in relevant experimental data. A rough estimateof γ ≈ −172 meV can nonetheless be obtained by fitting equation (3.5) to abinitio studies of phonon energy versus lattice constant, reported in reference[24].

The lattice expansion affects the fermionic part of the mean field calculationsthrough the inverse proportionality of the displacement u on the bare phononenergy. Looking for self consistent solutions as a function of both temperatureand spin-orbit coupling then leads to the phase diagram shown in figure 3.2. Atzero temperature, sufficiently large values of spin-orbit coupling are seen to ef-fectively prevent the simple cubic structure from distorting into a trigonal phase.Raising the temperature lowers the bare phonon energy however, which makesthe simple cubic structure more unstable, and hence requires ever larger spin-orbit coupling to prevent it from breaking down. As a result, for any fixed valueof the spin-orbit coupling, the lattice may undergo a charge ordering transitioninto the trigonal charge and orbital ordered state, even if the low temperaturephase was simple cubic. This effect is shown once more in figure 3.3 in terms ofthe thermal evolution of the order parameter for fixed values of the spin-orbitcoupling. Notice that the predicted chirality of the combined charge and orbitalordered phase can in principle be observed in x-ray diffraction of optical activityexperiments, while the orbital order itself should yield observable signatures indedicated STM experiments. The phase diagram of figure 3.2 agrees qualitativelywith the evolution of lattice structures throughout the family of elemental chalco-gens. The spin-orbit coupling in elemental Se and Te is weak enough to placethem to the left of the zero-temperature transition point, as indicated schemat-ically by the dashed lines in figure 3.2. Notice that at extremely high tempera-tures, the combined charge and orbital order in these crystals may be expectedto be destroyed by thermal fluctuations. There is no guarantee however that thiswill happen below the melting temperature of the material. In fact, there areexperimental indications that the short-range coordination in molten elemental

40

3.3. Conclusions

Figure 3.2: Phase diagram for elemental chalcogens as a function of temperatureand strength of spin-orbit coupling. A. Schematic phase diagram. At low tem-peratures, increasing the strength of spin orbit coupling leads to a suppressionof the trigonal instability, and hence a stabilisation of the simple cubic lattice.Polonium is expected to fall just to the right of the transition point, and thus tohave a simple cubic ground state, while selenium and tellurium have low spin-orbit interaction, and thus a trigonal ground state structure. At fixed spin-orbitcoupling, starting from the trigonal phase, the melting point (schematically in-dicated by the red dashed line) is encountered before charge and orbital order isdestroyed and the local structure becomes cubic. Starting instead from the sim-ple cubic phase, thermal expansion of the lattice lowers the bare phonon energyand thus shifts the balance of competing interactions in favour of the trigonalphase. B. The transition temperatures between cubic (lower right) and trigonal(upper left) phases, found by self-consistently solving the mean field equations.The error bars indicate the uncertainty in assigning the transition point withinthe precision of our numerical routine, and the solid line is a guide to the eye.

Te changes from trigonal to cubic just above its melting temperature[36, 37, 38].In contrast, polonium has strong spin-orbit coupling, placing it to the right ofthe zero-temperature transition, where the thermal evolution going from zero tohigh temperatures includes a transition from simple cubic to the less symmetrictrigonal phase before the melting point is reached. Although probably impracti-cal, further experimental exploration of the phase diagram of figure 3.2 could inprinciple be achieved by considering different isotopes of Po, in which the changein atomic mass affects the strength of the spin-orbit coupling.

3.3 Conclusions

The unique simple cubic lattice structure of elemental α−Po at ambient con-ditions, as well as its unusual symmetry-lowering structural transition towardsβ−Po at elevated temperatures, can be qualitatively understood in terms of

41


Figure 3.3: The value of the order parameter B as a function of temperature, atvarious fixed values of the spin orbit coupling strength. At low temperature thelattice is simple cubic and order is exponentially suppressed (as indicated by theexponential fits to the data), while high temperatures favour the formation ofcombined charge and orbital order within a trigonal lattice structure (as shownby the linear fits). The transition into the ordered state, qualitatively indicatedby the dotted lines, shifts to progressively higher temperatures for increasingstrength of the spin-orbit coupling.

the minimal microscopic model presented here. That the lattice structures andphase diagrams of the isoelectronic elements Se and Te can be understood withinthe same model without any additional assumptions, firmly establishes the factthat it captures the essential physics in the description of crystalline elementalchalcogens.

The simple cubic ground state of polonium is found in this model to be of adeceptive simplicity. The electronic structure consists of well-nested pieces ofFermi surface, which in the presence of electron-phonon coupling inevitably leadto large peaks in the electronic susceptibility and hence an incipient structuralinstability. The fact that three separate instabilities loom in three distinct orbitalsectors, coupled together by Coulomb interactions, yields a preferred trigonalconfiguration of the lattice, corresponding to a combined charge and orbitalordered state. This novel type of order is in fact realised in Se and Te, whichhave spiral trigonal lattice structures at all temperatures. In polonium however,the additional presence of strong spin-orbit coupling competes with the onset ofcharge and orbital order, which can be understood either in terms of the orbitaldegree of freedom becoming obsolete, or in terms of decreased nesting due togaps opening up at the Fermi energy. The spin-orbit coupling thus preventsthe simple cubic lattice from becoming unstable. At elevated temperatures,the balance is once again shifted in favour of the structural instability, by thesoftening of phonon energies as the lattice expands. The result is a re-emergence

42

3.3. Conclusions

of the spiral trigonal state, but now at high temperatures, sitting above a moresymmetric low-temperature simple cubic phase.The elements selenium, tellurium, and polonium, thus emerge as crystals in whichan intricate balance between all possible degrees of freedom, orbitals, charge,spin, and atomic displacements, determines the structure of the atomic lattice.The fact that multiple degrees of freedom cooperate and compete with each otherprofoundly affects the physics of these deceptively simple materials, as can beclearly seen from the phase diagram across the family of chalcogens. Spin-orbitcoupling competes with the onset of a cooperative charge and orbital orderedphase. This can be undone at high temperatures, but rather than thermal fluc-tuations determining the evolution of the phase diagram along the temperatureaxis, it is the indirect effect coming from the softening of phonons upon thermalexpansion that shifts the balance of power between competing ingredients. Thatsuch a complex interplay can nonetheless be understood in terms of a simpleminimal model, puts forward the family of chalcogens as a textbook case forunderstanding the possible effects of competition, co-existence, and cooperationamong spin, charge, orbital, and lattice degrees of freedom.

43

CHAPTER 4

Outlook

The model devised to understand the crystal structure of the elemental chalcogencrystals does not have to be viewed as constrained to these materials. Systemswhere the Peierls instability has been suggested to play a crucial role shouldshare common features. This was highlighted here by the comparison betweenthe macroscopic model for Se and Te and the one devised for 1T-TiSe2 [8]: bothpresent a charge and orbital ordered phase that relies on the presence of multiplecharge density waves in different orbital sectors; in both it is the electron-phononcoupling that sets the phases to be commensurate with the lattice, and finallyit is the Coulomb repulsion, among the multiple charge density waves, that de-termines the phase shifts. These are ultimately what give rise to the combinedspiral charge and orbital ordered phase, and consequently to the observed crystalstructure.For example, one elemental material whose crystal structure has been suggestedto undergo a Peierls distortion is bismuth (Bi). It has a distorted crystal struc-ture, with only small deviations from a parent simple cubic lattice [30, 48]. Fur-thermore, the outermost valence electrons occupy p−orbitals, which are onlyhalf-filled. Like in Se and Te, the s-valence electrons are well separated in en-ergy from the p-valence electrons [48]. To these similarities adds the fact that,like Po, Bi is a heavy element for which spin-orbit coupling cannot be ignored[48, 49]. And yet, at ambient temperature and normal pressure, Po is simple cu-bic and Bi has a distorted lattice. The question then arises as to why spin-orbitcoupling seems to act as a promoter for different crystal structures in Po andBi. In particularly, it would be interesting to understand how does the interplaybetween spin, orbital and charge degrees of freedom change between the two, andif it would be possible to comprehend the crystal structure of Bi as a product ofa spiral and orbital ordered phase.

45

Chapter 4. Outlook

To motivate looking for chiral and orbital ordered states more generally, it isworthwhile making a comparison with cholesteric liquid crystals. These are sys-tems where the chirality of the order parameter has important consequences intheir interaction with light. For instance, if circularly polarised light is madeto go through a cholesteric liquid crystal, and the incidence is normal to thesample, then only light with opposite handedness to that of the order parameteris transmitted without significant reflection [50]. These are also optically activesystems. This motivates the expectation that also a combined spiral charge andorbital ordered phase should exhibit a similar behaviour, given that the orbitalorder provides an effective director order parameter that is chiral. That a chiraltrigonal crystal structure displays optical activity is not unexpected [19], but tobe able to theoretically relate the chirality of the order parameter to the afore-mentioned behaviour remains to be done. By having a better understanding ofhow the order parameter relates to the optical activity in these systems, onecould perhaps identify more specific signals that could be used to identify chargeand orbital ordered states more generally.

Perhaps other peculiar responses may also be supported by this type of orderedphase. Thus, it is likewise interesting to analyse the response of the system to,for example, an external potential field or an applied current. For instance, ifwe inject current in the material, electron motion should more easily occur alongthe spiral chain. Consequently, a magnetic moment should be generated, andwith that a magnetisation along the spiral axis. If this picture is accurate, thenthe change in magnetisation would lead to a change in angular momentum forthe electronic system, but since there is no applied torque along the direction ofthe spiral axis, to conserve angular momentum, the material should rotate. Itis then an interesting question to be able to theoretically analyse if a combinedspiral and charge ordered state is prone to exhibit the described Einstein-DeHaas effect.

The study of orbital order and its consequences to the properties of materials hasbeen an interesting research field in condensed matter physics [51, 52, 53, 54, 55].In this sense, the comparison with liquid crystals, which have well-known tech-nological applications, serves to stress the potential of orbital order as a validavenue for engineering materials with novel properties.

46

APPENDIX A

Part I

A.1 Diagonalisation of the bosonic mean fieldHamiltonian

In section 2.3, the full Hamiltonian H is treated in the mean-filed approximation.Consequently, all operators are expressed in terms of their mean values and theirfluctuations around it:

O = 〈O〉+ δO . (A.1)

O is intended as a generic operator, and δO = O−〈O〉 identifies the fluctuationsaround the mean value. The approximation is introduced by neglecting all sec-ond order terms (and higher) of the fluctuation operator δO. The mean valuesare set by the Ansatz averages provided in (2.9), particularly by their Fouriertransformed expressions (where N is the size of the system):

〈c†kck+Q〉 = ρ0δ(q) +A

2

(eiϕjδ(q−Q) + e−iϕjδ(q + Q)

)+ 2σ0 cos(k · aj)δ(q)

+B cos((

k +q

2

)· aj)(eiχjδ(q−Q) + e−iχjδ(q + Q)

),

(A.2)

〈uj(q)〉 =u

2i

√N(eiφjδ(q−Q)− e−iφjδ(q−Q)

). (A.3)

As a result of the mean-field approximation, we obtain a Hamiltonian that is

47

Appendix A. Part I

at most quadratic in all operators. We can collect all bosonic operators into asingle Hamiltonian, which we denote by HMF

B :

HMFB = ~w

∑q,j

b†j(q)bj(q)+

+ iN√N

∑j

{uj(Q)

(α2A sin(Q · aj)e−iϕj − 2Bα1 sin

(Q · aj2

)e−iχj

)− uj(−Q)

(α2A sin(Q · aj)eiϕj − 2Bα1 sin

(Q · aj2

)eiχj

)}.

(A.4)

Recall that Q = 2π3a (ex + ey + ez) and aj = aej, with ej one of the three unit

vectors, along the x, y or z direction. HMFB can then be diagonalised by defining

the new operators γj(q):

γj(q) = bj(q) + vq,jδ(q−Q) + wq,jδ(q + Q) ,with (A.5)

vQ,j = w∗−Q,j =iN

~w√N

(α2A sin(Q · aj)e−iϕj−2Bα1 sin

(Q · aj2

)e−iχj

). (A.6)

This allows us to rewrite (A.4) as:

HMFB =

∑q,j

~wγ†j (q)γj(q)− 2~w∑j

|vq,j |2 (A.7)

From (A.7) we can now directly compute the mean value 〈uj(q)〉, by expressinguj(q) in terms of the new operators γj(q). We then obtain:

〈uj(q)〉 = −2(w−q,jδ(q−Q) + vq,jδ(q + Q)) . (A.8)

Comparing the previous equation with (A.3) determines the self-consistent so-lution for the displacement amplitude u. This amplitude is now dependent onthe charge density wave amplitudes A and B, the phase shifts, and the electron-phonon coupling constants α1 and α2:

u =2√

3

~ω(2Bα(1)e(χj−φj) −Aα(2)ei(ϕj−φj)) . (A.9)

48

A.2. Obtaining the spin-orbit coupling matrix

A.2 Obtaining the spin-orbit coupling matrix

The matrix M in the definition of the spin-orbit coupling Hamiltonian HSOC

(section 3.1) is formed by the matrix elements of the operator L · S, expressedin the basis of states labelled by the angular momentum quantum number l andthe spin s, i.e. |l,ml〉 ⊗ |s = 1/2,ms〉. The p orbital states (equation (A.11))can then be written as the real linear combinations of the spherical harmonicsY mll (θ, φ) ≡ 〈r|l,ml〉 (given in the same notation as in [56]), for which |r〉 denotesthe projection onto the position representation:

Y −11 =

√3

8π sin(θ)e−iφ

Y 01 =

√3

4π cos(θ)

Y 11 = −

√3

8π sin(θ)eiφ

(A.10)

|px,σ〉 = 1√2

(|l = 1,ml = −1〉 − |l = 1,ml = 1〉

)⊗ |s = 1/2,ms = σ〉

|py,σ〉 = i√2

(|l = 1,ml = −1〉+ |l = 1,ml = 1〉

)⊗ |s = 1/2,ms = σ〉

|pz,σ〉 = |l = 1,ml = 0〉 ⊗ |s = 1/2,ms = σ〉 .(A.11)

σ represents the eigenvalue of the Sz spin operator, and thus takes only one ofthe two possible values: ms = 1/2 or ms = −1/2. In this basis, the convenient

way to express the L · S operator is in terms of the raising and lowering angularand spin momentum operators. If we then set ~=1,

L · S =1

2

(L+S− + L−S+ + 2LzSz

). (A.12)

We note that L± and S± are, respectively, the raising and lowering angular mo-mentum and spin operators. It is now straightforward to obtain the matrix ele-ments in M , since we know how each operator acts on the |l,ml〉⊗ |s = 1/2,ms〉

49

Appendix A. Part I

states,

Sz(|l,ml〉 ⊗ |s,ms〉) = ms(|l,ml〉 ⊗ |s,ms〉)

S±(|l,ml〉 ⊗ |s,ms〉) =√s(s+ 1)−ms(ms ± 1)(|l,ml〉 ⊗ |s,ms ± 1〉)

Lz(|l,ml〉 ⊗ |s,ms〉) = ml(|l,ml〉 ⊗ |s,ms〉)

L±(|l,ml〉 ⊗ |s,ms〉) =√l(l + 1)−ml(ml ± 1)(|l,ml ± 1〉 ⊗ |s,ms〉) .

(A.13)

The end result is the spin-orbit Hamiltonian matrix written as follows:

M = λSOC

0 0 −i 0 0 10 0 0 i −1 0i 0 0 0 0 −i0 −i 0 0 −i 00 −1 0 i 0 01 0 i 0 0 0

.

A.3 The origin of H(2)e-ph

The second term contributing to the electron-phonon coupling in section 2.3

(equation (2.7)) is given by H(2)e-ph, and affects the electron potential energy.

Namely this term ensures that, for three atoms in a row, having the two outer-most ions moving closer to the central one comes with an increase in the distri-bution of electron charge around the central ion.

Our aim is now to give a concrete example of how such a term may arise. To doso, we will consider a simplified problem, where we only take into account threepositive and three negative charges. The question is how the outermost positiveions change in distance with respect to the central one, given a change in thedistribution of negative charges. This will then explain the form of the effective

interaction H(2)e-ph included in the model for Te and Se. In more precise terms,

we start by computing the optimal distance between the positive ions that aresurrounded by three equally negative charges. Thus, if qi labels the strengthof the negative point charges, we start by considering that q1 = q2 = q3. Wecan then think of our system as three hydrogen atoms that are brought togethersufficiently close, such that interactions among their constituents can no longerbe ignored. For this reason, the problem shares close similarities to the quantummechanical approach to the formation of molecules, and the route taken here tosolve the problem will also bear close resemblance to the perturbative methods

50

A.3. The origin of H(2)e-ph

found in molecular physics [57].To simulate an asymmetric distribution of charge, we consider that (q1 = q3) <q2. This models the situation where the three atoms form part of a chain, andthe charge density wave formation gives the central atom a higher average elec-tron density than the outer atoms. Furthermore, since spin is not included inthe microscopic model for Se and Te, also the spin degree of freedom will beneglected here. A further simplification is that we will disregard the issue ofanti-symmetrising the wave-function of the total system, and, like in the Hartreeapproximation, the many-body wave function will be given simply by the productof orthonormalised one-electron wave functions. Within these crude simplifica-tions, prior to interactions, the wave function of the total system is then theproduct of three hydrogen wave functions:

ψsystem = ψa(r1)ψb(r2)ψc(r3) , (A.14)

where

ψl(ri) =1√πa3

0

e−|ri−Rl|/a0 , with i = {1, 2, 3} and l = {a, b, c}. (A.15)

The previous equation implies that the electrons occupy s-orbitals. We arguethat choosing p-orbitals instead, which would grant a closer analogy to Se andTe, should not change the qualitative nature of the results. The anisotropy ofthe orbital shape should give rise to an anisotropy in the direction of the ionicdisplacements, but should not be the determinant factor causing the displace-ment in the first place. Thus, independent of the orbital content, it should stillhold that an anisotropic charge distribution should affect the optimal distancebetween the ions. Namely, we expect that if more charge is located near thecentral ion, its distance from the outermost ions should decrease.

We note that the labels {a, b, c} identify the three ions, while the labels {1, 2, 3}refer to the electrons. The distance between electron i and ion l is given by|ril| = |ri −Rl|, while the distance from the electron i to the origin of the co-ordinate system is given by |ri| (we consider the ions to be fixed). We will takethe origin to coincide with the location of ion b.The full Hamiltonian, including interaction terms, is given by:

H = T + H ionsCoulomb + He

Coulomb + H ions-eAttractive . (A.16)

To establish a parallel with the model for Se and Te, where, in each orbital sector,only nearest neighbours interact, we will also limit here the range of interactions.For example, as the three hydrogen atoms come closer together, electrons 1 and2 interact, as do 2 and 3, but not 1 and 3, and the same goes for the the ionic

51

Appendix A. Part I

and ionic-electronic interactions. With this approximation in mind, each termin H reads:

T = −3∑i=1

~2

2m∇2i (A.17)

H ionsCoulomb =

e2

4πε0

(1

|Ra −Rb|+

1

|Rb −Rc|

)(A.18)

HeCoulomb =

e2

4πε0

(q1q2

|r1 − r2|+

q2q3

|r2 − r3|

)(A.19)

H ions-eAttractive =− e2

4πε0

(q1

|r1a|+

q1

|r1b|+

q2

|r2a|+

q2

|r2b|+

q2

|r2c|+

q3

|r3b|+

q3

|r3c|

),

(A.20)

where Rl denotes the position of the ion. If the two outermost ions are con-strained to be equally distanced from the middle ion, i.e. |Ra−Rb| = |Ra−Rc| =R, our task becomes to determine the value of R that minimises the energyeigenvalue of H, up to first order in perturbation theory. The energy eigenvalue,computed from the unperturbed wave function in (A.14), is given by:

E =e2

2Rπε0+ (q2

1 + q22 + q2

3) EHydrogen0 + 〈ψsystem|HeCoulomb|ψsystem〉

− e2

4πε0

(q1〈ψsystem|

1

r1b|ψsystem〉+ q2〈ψsystem|

1

r2a|ψsystem〉

+ q2〈ψsystem|1

r2c|ψsystem〉+ q3〈ψsystem|

1

r3b|ψsystem〉

).

(A.21)

The two last terms in E can be evaluated using prolate spheroidal coordinates(ρi, µi, φi) [58]. For instance, if we take the ion a and b as foci, the coordinate

52


transformation is given by:

ρi = 1R

(|ri,a|+ |ri,b|

)1 ≤ ρi < +∞

µi = 1R

(|ri,a| − |ri,b|

)− 1 ≤ µi ≤ 1

0 ≤ φi ≤ 2π

dVi = R3

8

(ρ2 − µ2

)dρdµdφ .

(A.22)

In the last equation, dVi represents the volume element re-expressed in the pro-late spheroidal coordinate system. From the definition of (A.20), to evaluatethe attractive interaction between the ions and the electrons, we will need toevaluate a series of integrals similar in structure. Let us consider explicitly oneof such integrals, since the remaining can be obtained following the same stepsand yield the same result.

I =〈ψsystem|1

|r1b||ψsystem〉 =

1

πa30

∫dV1

e−2a0|r1−Ra|

|r1 −Rb|=

=R2

4πa30

∫dρ1dµ1dφ1(ρ1 + µ1)e−

Ra0

(ρ1+µ1) .

(A.23)

Defining ν = Ra0

, the previous integral can be integrated by parts and is equalto:

I =1

a0ν

(1− e−2ν − νe−2ν

). (A.24)

Using the fact that the energy of the ground state of the hydrogen atom is

given by EHydrogen0 = − e2

8πε0a0, the attractive interactions provide the following

contribution to the energy E:

− e2

4πε0

(q1〈ψsystem|

1

r1b|ψsystem〉+ ...+ q3〈ψsystem|

1

r3b|ψsystem〉

)=

= 2EHydrogen0 (q1 + 2q2 + q3)

(1− e−2ν

ν− e−2ν

).

(A.25)

We are now left with evaluating the electronic Coulomb repulsion contribution.As before, let us focus on one of the terms in 〈ψsystem|He

Coulomb|ψsystem〉, and use

53

Appendix A. Part I

it to highlight the common structure shared with all the remaining contributingterms:

Λ =1

(πa30)2

∫ ∫dV1dV2

e−2a0

(|r1−Ra|+|r2−Rb|)

|r1 − r2|

=( ν3

8π

)2∫ ∫

dρ2dµ2dφ2 dρ1dµ1dφ1 e−ν(ρ2−µ2+ρ1+µ1) (ρ2

2 − µ22)(ρ2

1 − µ21)

η12

=( ν3

8π

)2(Λ1 − Λ2 − Λ3 + Λ4

),

(A.26)

where the sum in Λi’s comes from the sum of integrals that originates from theproduct (ρ2

2 − µ22)(ρ2

1 − µ21), and η12 corresponds to the term |r1 − r2|, but now

written in prolate spheroidal coordinates instead. In this coordinate system, itis useful to express η−1

12 in terms of the Neumann expansion [57, 59]:

1

η12=

2

R

∞∑τ=0

τ∑λ=0

(Dλ,τP

λτ (ρ2)Qλτ (ρ1)

Pλτ (µ2)Pλτ (µ1) cos(λ(φ1 − φ2)

)), for ρ1 ≥ ρ2

exchange ρ1 with ρ2 in the previous expression, for ρ1 < ρ2 .

(A.27)

In (A.27), Pλτ (ρi) refers to the associated Legendre polynomials, while Qλτ (ρi) isthe associated Legendre function of the second kind. The term Dλ,τ is given by:

Dτ,λ = (−1)λ ελ

(2τ + 1

2

)( (τ − λ)!

(τ + λ)!

)2

, where ε0 = 1, εn = 2 (for n > 0) .

(A.28)

From (A.27), we can immediately see that the only term in (A.26) that willdepend on the angular variables φ1 and φ2 is the term cos

(λ(φ1 − φ2)

)in the

Neumann expansion. Thus, the integration in the angular components can beperformed independently of the others and gives rise to:

∫ 2π

0

∫ 2π

0

dφ1dφ2 cos(λ(φ1 − φ2)

)= 4π2

( sin(λπ)

λπ

)2

. (A.29)

54


Since by definition λ is an integer, the only way the previous integral can benon-zero is if λ = 0. Consequently, in Λ (in (A.26)), the associated Legendrepolynomials and the associated Legendre functions of the second kind becomethe Legendre polynomials (Pτ (ρi)) and the Legendre functions of the second kind(Qτ (ρi)).

In each Λi, the integration in the variables µ1 and µ2 can be performed sep-arately and independently of the rest, and in both cases the integrand functionis of the form µne±(νµi)Pτ (µi), where n is either zero or two. Let us then takethe case n = 0, and see that this integral can actually give us some reasonableguidance as to where the sum in τ can be truncated, without it being too far fromconvergence. To simplify notation, let us call the integrand function Fτ (µi, νi),i.e. Fτ (µi, νi) = e±(νµi)Pτ (µi). If the series expansion of the exponential func-tion is used, alongside with µni written as a series expansion in terms of Legendrepolynomials, Fτ (µi, νi) can be expressed as:

Fτ (µi, νi) =

+∞∑n=0

gn(ν)Pτ (µ)Pn(µ) , (A.30)

where gn(ν) is given as below,

gn(ν) = δ1,(−1)n

∑m={0,2,...}

νm2n+ 1

2(m−n)/2(1/2(m− n))!(m+ n+ 1)!!

± δ−1,(−1)n

∑m={1,3,...}

νm2n+ 1

2(m−n)/2(1/2(m− n))!(m+ n+ 1)!!

(A.31)

We note that the ± sign in the definition of gn(ν) depends directly on the signof the exponent of e±(νµi), and that δi,j is the Kronecker delta. Using theorthogonality property of the Legendre polynomials, over the range [−1, 1], theintegral of Fτ (µi, νi) can be immediately obtained:

∫ 1

−1

dµi Fτ (µi, νi) = gτ (ν)2

2τ + 1. (A.32)

In reference [57], for a symmetric wave function (regarding electron exchange),the optimal interatomic distance in the triatomic hydrogen ion molecule waspredicted to lie in between a0 and 2a0

1. This motivates our expectation thatν (recall that ν = R/a0) should not be above 3. We therefore restrain thenumerical study of the monotonic behaviour of gn(ν) to the special cases where1 ≤ ν ≤ 3. In figs.(A.1) and (A.2), the function gn(ν) is shown for some values

1In [60], the optimal iteratomic distance for the H3 molecule was found to be close to 2a0.

55

Appendix A. Part I

of ν. It is clear that the function oscillates rapidly towards zero, and we thusapproximate the infinite sum in τ with a finite one, running only from τ = 0 upto τ = 7.

Figure A.1: 22τ+1gτ (ν), as a function of the discrete variable τ , at fixed values

of ν. The plotted function assumes eνµi (and thus with a positive exponent) inthe definition of Fτ (µi, νi).

Note that if instead the integrand function is µ2e±(νµi)Pτ , the integration overµi ∈ [−1, 1] yields the same result as in (A.32), the only difference is that n →n+ 2 in the definition of gn(ν).There are now two integrations left to go to determine the Λi’s completely: theintegration over the variables ρ1 and ρ2. Since the expansion of η−1

12 (equation(A.27)) changes depending on whether ρ1 is above or below ρ2, the integrationin ρ1 needs to be split into two parts. Additionally, the dependency of η−1

12 onwhether ρ1 is bigger or smaller compared to ρ2 also requires us to perform theintegration in ρ1 first. The resulting expression will depend on ρ2, and thus thisshould be the last integral to be computed. Let us then consider first the casewhere ρ1 < ρ2, for which the Λi’s require computing integrals of the form:

∫ +∞

1

dρ2 ρm2 Qτ (ρ2)e−νρ2

I2(ρ2)︷︸︸︷∫ ρ2

1

dρ1 ρk1Pτ (ρ1)e−νρ1︸︷︷︸

I1

, with m, k = {0, 2} .

56


Figure A.2: 22τ+1gτ (ν), as a function of the discrete variable τ , at fixed values of

ν. The plotted function assumes e−νµi (and thus with a negative exponent) inthe definition of Fτ (µi, νi).

(A.33)

Computing the integral I2 first, we obtain:

I2(ρ2) =

∫ +∞

1

dρ1 ρk1Pτ (ρ1)e−νρ1 −

∫ +∞

ρ2

dρ1 ρk1Pτ (ρ1)e−νρ1

=

∑τl=0,2,... cl(τ)

(fl+k(ν, 1)− fl+k(ν, ρ2)

)∑τl=1,3,... cl(τ)

(fl+k(ν, 1)− fl+k(ν, ρ2)

),

(A.34)

for which the coefficients cl(τ) correspond to the coefficients composing the Leg-endre polynomial of order τ . It is for this reason that the sum in l runs exclusivelyeither through odd (if τ is odd) or even (if τ is even) integers. For instance, ifPτ=2(ρ1) is the Legendre polynomial of order 2, then c0 = −1/2 and c2 = 3/2.The function fl(ν, ρ2) is defined as:

fl(ν, ρ2) =

∫ +∞

ρ2

dρ1 (ρ1)le−νρ1 =e−νρ2

νl+1

l∑m=0

l!

m!(νρ2)m , (A.35)

where, for a given l, the integral can be solved using partial integration, and aclosed formula can then be obtained through the mathematical induction prin-

57

Appendix A. Part I

ciple. The expression for fl(ν, 1) can be readily obtained from the previous oneby fixing ρ2 = 1.The integral I1 is then obtained by adding two contributions: one that comesfrom integrating the function ρm2 Qτ (ρ2)e−νρ2 , and the other that requires inte-grating ρm2 Qτ (ρ2)fl+k(ν, ρ2)e−νρ2 :

I1 =

τ∑l

cl(τ)

(fl+k(ν, 1)

∫ +∞

1

dρ2 ρm2 Qτ (ρ2)e−νρ2+

−∫ +∞

1

dρ2 ρm2 Qτ (ρ2)fl+k(ν, ρ2)e−νρ2

).

(A.36)

The integral I1, and alike integrals that involve the Legendre functions of thesecond kind or the Legendre polynomials in the same range of integration, werecomputed individually for each τ . To evaluate the integrals involving the Leg-endre functions of the second kind, one is inevitably forced to evaluate integralswith the same structure as Hn(ρ2, ν, a) (defined below).

Hn(ρ2, ν, a) =

∫ +∞

ρ2

dρ e−νρ ρn ln(ρ+ a)

= ln(ρ2 + a)fn(ν, ρ2)− Ei(−ν(ρ2 + a))fn(ν,−a)

+1

νθ(n− 1)Bn(ν, ρ2, a)

+ θ(n− 2)

n−1∑j=1

(j∏l=1

(n− l + 1)

)Bn−j(ν, ρ2, a)

νj+1,

(A.37)

where fn(ν, ρ2) is the function given in (A.35), Ei(−ν(ρ2 +a)) is the exponentialintegral function, θ(m) is the Heaviside unit step function, and Bn(ν, ρ2, a) isthe function defined in (A.38). The lower bound in the integral defining Hn

does not need to be restricted to ρ2, and the validity of the closed formula in(A.37) remains, after replacing ρ2 for the new lower bound of integration inthe corresponding terms in (A.37). We note that the above closed formula wasobtained by combining partial integration with the definition of the exponentialintegral function, and the mathematical induction principle.

Bn(ν, ρ2, a) =e−νρ2

νδn,1

+ θ(n− 2)

(fn−1(ν, ρ2)

+

n∑j=2

(δa,−1 − (−1)nδa,1

)fn−j(ν, ρ2)

) (A.38)

58


Since some integrals were obtained by evaluating it for every individual value ofτ , and because to Λ (given in (A.26)) contributes a large number of integrals, wewill not list all the steps needed to construct Λ. The steps presented so far high-light the type of calculations that one encounters when evaluating the electronicCoulomb repulsion term. The final expressions are long, but can be built fromthe auxiliary functions listed above, such as fn (in (A.35)) and Hn (in (A.37)),along with the special function Ei(x).

The ground state energy (given by E in (A.21)) is obtained by collecting allcontributing terms, and the results are best illustrated by plotting E as a func-tion of ν. This is shown in figs.(A.3) and (A.4). In (A.3), q1 = q2 = q3 = 1 andthe value of ν that minimises E can be found numerically to be ν ≈ 1.667. Ifwe let q1 = q3 = 0.5, q2 = 2 (fig.(A.4)), the new value of ν that minimises E isfound numerically to be ν ≈ 1.286, and thus smaller, when compared to the casewhere the distribution of charge is uniform. We can therefore conclude that adisplacement of the outermost ions to a distance closer to the middle one is con-sistent with a higher distribution of charge around the central ion, as assumed

in the form of H(2)e-ph in the main text.

Figure A.3: Energy eigenvalue E, defined in (A.21), as a function of the dimen-sionless parameter ν = R/a0. In the figure, 1 ≤ ν ≤ 3 and q1 = q2 = q3 = 1.

59

Appendix A. Part I

Figure A.4: Energy eigenvalue E, defined in (A.21), as a function of the dimen-sionless parameter ν = R/a0. In the figure, 1 ≤ ν ≤ 3 and q1 = q3 = 0.5, q2 = 2.

60

Part II

Edge states in 1D crystals

61

CHAPTER 1

Introduction and motivation

1.1 Why should one care about edge states in1D crystals?

While previously the interactions between the main constituents of a crystal wereat the core of our focus, with lattice symmetries derived as a consequence of thefirst, we now take an almost inverted perspective, as we bring the lattice symme-tries to the spotlight and deal with non-interacting electronic systems only. Inparticular, we will be interested in the lattice symmetries connected to insulatingcrystals. The driving motivation behind our interest stems out of the recentlyproposed topological classification in [5]. The lattice point symmetries allow fordefining a set of integers that count, at high symmetry points in the Brillouinzone, how many bands transform according to a given irreducible representation.While this statement follows immediately from group theory, it turns out thatthese integers can be proved to characterise the topology of the bulk bands. Itthen follows that two insulators, with band structures leading to different sets ofthose integers, are topologically distinct.By now the familiarity with Chern insulators, where topologically non-trivialphases imply the existence of protected edge states at the boundary betweendistinct topological states, makes almost inevitable the question of whether ornot the same picture applies here: will edge states exist at the boundary be-tween two crystals that have a different set of integers? Since there is no apriori bulk-boundary correspondence in this context, an answer cannot be givenimmediately, and forces us to address the existence of edge states solutions, inconnection to the lattice symmetries, as a problem on its own.

Even if topology is overlooked for a minute, being able to predict the existenceof edge state solutions at the interface between two crystalline systems, can be

63


of interest on its own. For example, if the crystals are insulators, but there areedge states present (localised states at the boundary of crystals were known toexist already in 1930’s), then one can think about current transport across atwo-dimensional interface.It can be argued that these edge states, being entirely protected by the latticesymmetries, are of weaker interest than those in Chern insulators, since it isprecisely at the surface that the point symmetries, granting their existence, canbe broken. However, if, from a practical stand point, it is possible to manipulatethe interface such that the lattice symmetries of the bulk are maintained there,then as long as the symmetry remains, so do the edge states.The first task should then be a proof of concept, i.e. to obtain edge state solu-tions and clarify their connection with the bulk properties. As one usually startswith the easiest case, 1D crystals are the desirable starting point. Although thefocus here is given to these simpler crystalline systems, the goal is to use theknowledge and insight learned here to then be able to address similar questionsfor higher dimensional crystals, which carry more variety of point group symme-tries and thus become potentially more interesting.Even for the simpler 1D systems, we will prove that there exists a type of bulk-boundary relation, connecting the number of disappearing bulk states with thepresence of edge state solutions. We show this relation to be inseparable fromthe symmetry constraints of the lattice, and thus independent of the model cho-sen to represent the electronic states in the system. The presence of these edgestates will be shown to agree with the expectations of the recent topologicalclassification in [5] (which by itself makes no statements about what happens atthe crystalline boundaries), and a strategy will be laid out, at the end of chapter9, for how to generalise these results to higher dimensions.

1.2 A very brief overview of edge states through-out time

1.2.1 When edge states went by the name of surface states

Until the advent of the transistor in the 1940’s, the theoretical interest in edgestates was sparse [61, 62]. The latter discovery, and its technological applica-tions, spiked the enthusiasm in the field, with the 1950’s accommodating a lotof further studies on the initial theories of edge states, particularly concerningthe ones developed by Tamm [63] and Shockley [64]. This first burst of interestin edge states was completely disconnected from the concepts of band topology,which would only become a wide-spread concept in condensed matter physicslater on. There are, nevertheless, common traits that are shared between theolder theories of edge states and the more modern concepts of band topology, aswe will see later, even if they do not agree on a common name for referring tothe localised states at the boundary.

64

1.2. A very brief overview of edge states throughout time

The theory of edge states, or as called at the time: surface states, starts withTamm in 1932 [63, 65]. The ideal 1D crystal was described using the Kro-nig–Penney model, where the periodic potential is modelled by a series of deltafunctions of the form δ(x−ma), with maximum amplitude at every integer mul-tiple of the lattice constant a.The crystal was made semi-infinite by the imposition of a boundary at x = 0 (xbeing the spatial coordinate). This delimited the crystal to the region of positivex. Confined to the region of negative x, was the vacuum, which was modelledby a positive and constant potential. To avoid having one of the Kronig-Penneyδ–functions at the boundary, at x = 0 the potential is made equal to the samefinite constant as the vacuum (a detail which now might seem insignificant, butwill turn out to be relevant for the type of edge states obtained)[65].When the energy associated with the crystalline wavefunction was smaller thanthe strength of the vacuum potential, Tamm found that it was possible to obtainan exponentially damped solution at both sides of the interface, and hence alocalised crystalline state near the boundary [65]. He called these states surfacestates. Three years later, Maue addressed in essence the same problem, yet us-ing the nearly free electron model instead [66, 67]. In addition, the analysis wascarried out assuming that the potential, within the crystal, kept everywhere thesame structure as in the bulk, i.e. the crystal’s termination was assumed to oc-cur without any boundary effects. This distinction from Tamm’s approach, wasactually important for the type of edge state solutions obtained. By omitting thelast δ–function, Tamm effectively introduced a difference between the potentialin the bulk of the crystal, and the one in the region closest to the boundary atx = 0, but still inside the crystal. The change in potential effectively acted asan impurity potential at the edge of the crystal, and the localised state could beseen as an impurity mode. In Maue’s analysis, no difference existed between thecrystal’s bulk potential and the crystal’s potential closest to the boundary, andtherefore the edge state solutions could not be interpreted as impurity modes.As a result, the type of edge states that were obtained by Maue were not of thesame type as Tamm’s.The distinction was also apparent in the conditions needed for the existence ofthe two types of edge states: while Tamm could obtain them by requiring the vac-uum potential to be higher in energy, compared to the electronic energies in thecrystal; Maue would not only need the previous requirement, but also a particularsign of the lth Fourier component in the lattice potential (this criterion is goingto be explained further in the following chapters) [66, 67]. The second conditionimplies that, depending on the sign of the Fourier lattice potential components(a bulk property), the energy spectrum may or may not host edge states. Today,this observation would have been cause for suspecting that perhaps topologicalproperties were at stake, but in the 30’s the story of edge states continued byre-examining the problem using ever-more detailed tight-binding models [68, 62].

The first of these was developed by Goodwin, in 1939, and addressed, sepa-

65


rately, a linear chain of atoms made of, respectively, s and p orbitals [68]. Thetype of edge state solutions obtained were, however, of Tamm’s type, since themodel prescribed different Coulomb and exchange integrals for the atoms at theend of the chain, as compared to those in the bulk. In contrast to Maue, noother constraints were found, and the article is finished by concluding: ”Thepresent paper indicates that even when the electrons are to be considered astightly bound surface states will in general occur” [68].In that same year, Schockley applies the cellular method to a finite chain ofatoms, terminated in the absence of any boundary effects [64]. As a result,insight is provided on the microscopic origin of edge state solutions from bulkstates. This is done by analysing the energy spectrum of a finite chain of atoms asa function of the lattice constant. The starting point is the large lattice constantlimit (or in more modern terms, the atomic insulator limit), where the overlapbetween atomic orbitals is negligible; by decreasing the lattice constant, edgestates appear as soon as the energy bands, which originated from the separateatomic levels, begin to overlap [64, 65]. Two energy levels are then present inthe energy gap, each originating from one of the two bands that provided theoverlap between distinct atomic levels [64]. In going from one limit to the other,the boundary curves delimiting the energy levels would have crossed, and onlyafter this crossing would the edge states appear in the gap. Thus, in modernterms: one cannot continuously go from the atomic limit to the final insulatingstate, without in between closing the gap, a concept that we would now recogniseas a topological phase transition.Schockley’s paper goes on to analyse where in the energy spectrum, after or be-fore the boundary curves have crossed, Tamm’s and Maue’s edge state solutionssit. He concludes that Maue’s edge states sit just after the boundary curves havecrossed, as opposed to Tamm’s states [64]. These belonged to the region beforethe crossing, and resulted from making the lattice potential near the boundary ofthe crystal distinct from the one in the bulk. Hence, Tamm’s type of edge statesis ”essentially different from the origin of the states occurring after the bandshave crossed” [64]. Interestingly, Schockley also notes that if the edge stateslevels, from Maue’s type, are not completely filled, one should expect these tobe conducting. Yet, at the time no clear experimental confirmation existed forthe presence of edge states in crystalline solids, and so the interest for theseelectronic states remained small [61].

In the late 1940’s, nevertheless, the study of metal-semiconductor interfaces leadthe way to an experimental proof demonstrating the existence of edge states [69].It started with Meyerhof’s observation that the contact potential difference insilicon crystal rectifiers did not match the difference in the work functions ofthe two solids, which was, in first approximation, the expected theoretical pre-diction [70]. Bardeen proposed that Meyerhof’s observation could be explainedif a relatively high density of edge states existed at the surface of the semi-conductor, which would then be responsible for compensating the difference in

66

1.2. A very brief overview of edge states throughout time

work functions between the two solids [71]. Bardeen’s theory was soon after ver-ified experimentally by Schockley, Brattain and Pearson, and with that a risinginterest in the theory of edge states emerged [71, 72, 73, 61].

Yet, as time passed and the theoretical models for studying edge states becameincreasingly more complex, so the definitions of what made Maue–Schockleystates distinct from Tamm’s states became ”hazy, and sometimes misleading”,as can be read from Davison and Levine’s review on the subject, in 1970 [62].This situation was addressed in 1985, by Zak [74]. In the article, a simple, yetpowerful group theory approach is used to show how the difference between thetwo types of edge states is one of symmetry. The existence of Maue–Schockleystates, at boundaries coinciding with the high symmetry points of the 1D lat-tice, is then formulated in terms of a general (i.e. model independent) symmetrycriterion that is inseparable from the point symmetries of the lattice [74]. Theprocedure elucidates how the existence of edge states at high symmetry surfacesis symmetry protected, offering yet another similarity to the edge states in thecurrent context of topological crystalline insulators. Of course these similaritiesare not coincidental, Maue–Schockley edge states were always topological edgestates, even if today one can still find them referred to by their old name: surfacestates.

1.2.2 The topology fever

Edge states can now be found as one of the features of topological phases ofmatter [75], an active field of research in condensed matter physics [76, 77, 78].Hence, to address the current interest in edge states, we are bound to talk aboutits connection to topology.

The first well-known example in which topology entered the realm of condensedmatter, is the case of the 2D integer quantum Hall effect. In very brief terms,at low temperatures and strong magnetic field, the appearance of a quantisedconductance is linked with the existence of edge states, propagating only along agiven direction at the boundary of the system [79]. The ability to propagate cur-rent at the edge persists, even in the presence of some disorder, a property thatcharacterises the edge states as robust [79]. On the other hand, using Kubo’sformula to compute the Hall conductance reveals that it can be expressed interms of the Chern number [80], which establishes a direct link with topology.The calculation employs the wavefunctions of the bulk, yet the propagation ofcurrent in the quantum Hall effect is mediated by the edge states. This phe-nomenon is known as the bulk-boundary correspondence [75]. In the context ofthe quantum Hall effect, an explicit proof relating the conductance computedfrom the bulk (in a square lattice) and the edge states at the boundary can befound in [81].

More generically edge states arise at the boundary between any two (topologi-

67


cally) distinct Chern insulators [82]: given two insulators characterised by dif-ferent Chern numbers, their inherently distinct topology means that no smoothtransformation exists that interpolates between the two Hamiltonians, withoutalong the way closing the energy gap, which is done precisely by the edge states.Thus, in the context of Chern insulators, the existence of edge states is insep-arable from the topological properties of the bulk. It must be said that, eventhough the quantum Hall effect was first observed in 1980, using strong externalmagnetic fields, it was later shown, in 1988, that the same phenomena could arisein a 2D honeycomb lattice model, without any need for an external magneticfield [83]. The model was proposed by Haldane, and highlights the importance ofbreaking time reversal symmetry for the integer quantum Hall effect to be realis-able. This introduces a central idea in the study of topological phases of matter:the presence (or absence) of certain symmetries is determinant for the existenceof non-trivial topological systems. In fact by now, an entire catalog of topologicalclasses can be found [84], based on the presence or/and absence of time-reversal,particle-hole symmetry and their combination (referred to as sublattice or chiralsymmetry). These phases host edge states at the boundary, protected by thesymmetries that grant the non-trivial topology of the bulk. The classificationis based on the most general symmetries of a non-interacting electronic system,but many insulators in nature are associated to a given crystal structure, andwith that to a set of point symmetries. Thus, the birth of ”topological crystallineinsulators” was perhaps almost inevitable [85].

The topological crystalline insulators were proposed by Liang Fu in 2011 as anextension to the previous classification of topological phases. What was intendedwith the term was to describe a class of systems that had a non-trivial topologydue to the simultaneous presence of time-reversal symmetry and “certain pointgroup symmetries” of the lattice [85]. In particular, the principle was illustratedwith a tight-binding model, characterised both by time-reversal symmetry as wellas C4 rotational invariance. Edge states, closing the gap, were shown to existat a high symmetry surface, i.e. one that preserved the C4 symmetry. Theywere linked to the presence of a non-trivial value for the Z2 bulk invariant. Infact, despite the term ”topological crystalline insulators” not being used there,the work in [86] had already proved the usefulness of recognizing the presence ofinversion symmetry, when addressing Z2 time-reversal invariant systems.

By now, other lattice symmetries, alongside time-reversal symmetry, have beenconsidered [87], but a general theory illustrating a topological character thatis inherited from the lattice symmetries alone was only recently proved in [5].This includes a general framework to evaluate the topological properties of non-interacting electronic systems, in the absence of both time-reversal, particle-holesymmetry and their combination (class A in [84]). The formalism is based onK-theory, yet to obtain the set of integers that classify the topology of a givencrystalline structure, one only needs to apply simple concepts of group theory.

68

1.3. A reader’s guide, part II

Not long after, other works followed in the same line of thought [88, 89]. For in-stance in [88], group theory and graph theory were employed to derive a generalmethod to understand the topology of“all 230 crystal symmetry groups”. How-ever, in a subsequent article [90], the focus returned once more to the crystallinelattice symmetries at the service of the more traditional symmetries: ”(...) weintroduced the generalization of this theory to the experimentally relevant situ-ation of spin-orbit coupled systems with time-reversal symmetry”.With the promise of finding newly topological materials [91], the excitement isnow high. The term“topological crystalline system” can now even be found underthe new name of “topological quantum chemistry” [88, 91], despite both refer-ring to the same concept: that lattice symmetries contribute to the topologicalnature of band insulators. But if lattice symmetries are included as importantsymmetries for topology, then, at least in 1D, where they were proved crucialfor the existence of edge states at high symmetry surfaces, it becomes clear thatthe study of surface states initiated in 1930’s, has always been connected to themodern story of topological band theory.

1.3 A reader’s guide, part II

To help the reader, a brief description on the structure of the second part of thethesis is provided. Here, we will be mainly devoted to:

a) Understanding the connection between edge states, the lattice symmetriesand topology in band theory. The case study will be 1D crystalline insulators,with inversion symmetry, although the ultimate goal is to generalise the conclu-sions drawn here also to 2D and 3D insulators.

b) Proving that the existence of edge state solutions, in 1D inversion-symmetricinsulators, is always linked with disappearing Bloch solutions from the bulk statesand both are signalled in terms of a bulk quantity. Thus a type of bulk-boundaryrelation also exists in these systems.

In chapter 2, the results in 2.2 can be found in references [66] and [67], andare included merely as a review to introduce the notation and the method, butalso to motivate why the existence of bulk disappearing states should be an ex-pectation whenever edge state solutions are present. For instance, in [66], to thebest of the author’s knowledge, it was the first time that the existence of edgestates was explicitly shown to be accompanied by an obstruction to smoothlyconnecting bulk Bloch solutions. Maue’s proof was obtained for the particularcase of a finite crystal with symmetric interfaces [66]. For a semi-infinite crys-tal, Maue studied the existence of edge state solutions, within the nearly freeelectron model, but gave no proof relating these to disappearing bulk states [66].We show here, in section 2.3, that the edge state solutions in the semi-infinitecrystal are also accompanied by the disappearance of bulk states. In fact, we

69


will see throughout chapter 2 that the presence of edge states is systematicallyaccompanied by disappearing bulk states, and prove it to be deeply rooted intothe lattice symmetries, just as the edge state solutions are; this is done in chap-ter 5. These are thus two sides of the same story, underpinning a bulk-boundaryrelation in topological 1D crystals, whose topology is intertwined with the pointsymmetries of the lattice.

To study if the symmetry-based topological classification proposed in [5] im-plies edge states at the interface between two topologically distinct crystals, it isof interest to analyse a setup where a crystal no longer shares a boundary withthe vacuum, but instead shares it with another crystal. We will often refer tothis setup as a heterostructure. The problem is then divided into steps: first weaddress how to determine if edge states are present at the interface, and onlyafter we examine their connection with the topological classification in [5]. Thefirst of these tasks can be found in chapter 3, where the problem of finding edgestates (and disappearing bulk states) is addressed in the context of the nearlyfree electron model (sections 3.1 and 3.2).All throughout these cases, it will become clear that, in the nearly free electronmodel, all aspects of the appearing edge states and disappearing bulk states de-pend on the occurrence of a single criterion, which is summarised in the lastsection of chapter 3.We use the nearly free electron model throughout chapters 2 and 3, because ofits simplicity. The model depends on only one parameter, the strength of theFourier transformed lattice potential, and the wave functions acquire a simpleform. The confidence that there is no crucial gain in insight from using morerealistic models is given by the symmetry arguments1 in chapters 4 and 5.As long as certain symmetry constraints are met, the existence of edge states isindependent of the detailed microscopics.We note that in the end of section 2.1, there are some important observationsthat are useful to keep in mind while reading the remaining sections in chapters2 and 3.

In chapters 4 and 5, we address the problem using symmetry arguments alone,and therefore in a model-independent way. The symmetry constraints come fromthe point symmetries of the unit cell and have important repercussions on thesymmetries of the Bloch states. Since solving the boundary problem requires amatching procedure that involves equating the states on both sides of the in-terface, and their corresponding first derivatives, these symmetries turn out toalso have important consequences for what happens at the boundary [74]. Thegeneral procedure to address the existence of edge states, based on symmetryalone, is due to Zak [74], and we review it in section 4.1. In Zak’s work, how-

1Which are also related to the topological invariants in [5]. This allows us to understandwhy the results do not depend on the choice of microscopic model, since only the topologicalnature of the model matters for the presence of edge states.

70

1.3. A reader’s guide, part II

ever, the relation between the edge states and the disappearing bulk states is leftanswered. We generalise Zak’s procedure to provide a proof for their interde-pendency (chapter 5). Applying Zak’s procedure to the interface between twosemi-infinite crystals can be found in section 4.2. This section is divided intotwo: the first (subsection 4.2.1) concerns the analysis of edge state solutionsonly from a symmetry band perspective, while in the second part (subsection4.2.2) we relate the predicted edge state solutions with changes in the topolog-ical invariants, across the interface [5].

The symmetry arguments used in this part of the thesis can be placed in thecontext of a broader framework: the band representation theory, which allowsfor identifying symmetry labels that characterise a full band, rather than onlyspecial points in the Brillouin zone [92]. Referring to this approach is crucialin the sense that it provides the tools to generalise the methods presented inchapters 4 and 5 to higher dimensional crystals. We briefly discuss how thismay be accomplished in chapter 6.

Before starting, it is important to point out the set of premises that will beshared among all the remaining chapters and sections. The first one concernsthe lattice potential, which will be taken to be the same in every unit cell, in-cluding the ones close to the boundary. The second one refers to the crystal’stermination, which will not to be located at an arbitrary position, but at a highsymmetry point. For 1D lattices, the only available point symmetry is inversionsymmetry, and in fact only two distinct inversion centres exist: if x is defined asthe spatial coordinate and a is the lattice constant, these are x = 0 or x = a/2.Consequently, interfaces separating the crystal from the vacuum or from a dif-ferent crystalline system, will always sit at either precisely these positions, or atrelated ones by pure translations. In other words, we neglect any edge effectsthat can cause distortions in the unit cells close to the boundary, and deal exclu-sively with perfect crystal terminations at high symmetry points in the lattice.The third and last remark is on the absence of electronic interactions. All crys-talline systems studied here are implicitly described by one-body, non-interactingelectronic Hamiltonians, subjected to a periodic lattice field, that is the sourceof all point symmetries.

71

CHAPTER 2

Edge state solutions in the nearly free electronmodel

2.1 An analogy with the ”finite square well prob-lem”

The central question of this chapter is simply: can there be edge state solutions,localised at the interfaces of a 1D crystal?An edge state solution is nothing more than a wave function that is bound to asurface, decaying exponentially as the distance from the boundary increases ineither direction. If this boundary separates the crystal from a different environ-ment, as for instance the vacuum, then we need to account for the discontinuityimplied in the lattice potential, when solving the Schrodinger equation for thefull system (crystal + vacuum). This is reminiscent to the well-known ”finitesquare well” problem in quantum mechanics: also there the discontinuities inthe potential, as a function of the spatial coordinate, lead to the appearance ofconfined electronic states. The analogy is of course not one-to-one, since thestates in the ”finite square well” were only exponentially suppressed solutionsoutside the well, but the concept of localisability is present. This is an instruc-tive comparison, because it turns out that the strategy needed to answer ourcentral question in the following sections (within chapters 2 and 3) is similarto the one used in the ”finite square well” problem. Thus, as learnt from ourfirst encounters with quantum mechanics, given a finite 1D crystal with lengthL = Na, where a is the lattice constant and N the number of atoms in thesystem, if we define x to be the spatial coordinate, in order to determine if edgestate solutions are present, we will need to:

1. Solve the Schrodinger equation, separately, for the left and right regions

73

Chapter 2. Edge state solutions in the nearly free electron model

outside the crystal, as well as inside the crystal.

2. Constrain the wave functions found in 1. to match the boundary condi-tions, namely, requiring the wave function and its first derivative to both becontinuous functions of x at the interface. Using the labels i and j to identifyneighbouring regions, ψ′ to represent the first derivative with respect to x, andxν an interface position, the boundary conditions are:

ψi(xν)− ψj(xν) = 0, i 6= j

ψ′i(xν)− ψ′j(xν) = 0, i 6= j .

(2.1)

Hence, if an exponentially decaying function in x is a solution of the Schrodingerequation, within the crystal, and satisfies all boundary conditions, this consti-tutes an edge state solution.

Before proceeding, we call attention to the following set of observations, per-meating the next sections. To the reader that finds them obvious, take them asreminders or feel free to skip them.

3. The boundary conditions need to be fulfilled by states ψi and ψj having thesame energy E. We are interested in the stationary solutions of the Schrodingerequation describing the global system, and thus characterised by a unique eigen-value.

4. The potential in the vacuum region is modelled by a constant potential(V (x) = cst). Taking, for simplicity, the potential in the vacuum to be zero andassuming the energies, within the crystal, to be negative, the allowed states inthe vacuum region always assume the form of a real single exponential (ψ(x) ∼

e±x

√2m~2 |E|), decaying as the distance from the boundary increases. This fol-

lows immediately from solving the time-independent Schrodinger equation witha negative eigenvalue E. A linear combination of single exponentials is excluded,because in chapter 2 and 3 the interface is always kept at xν = 0, and we alsowork under the assumption that states should be normalisable.

5. The set of boundary conditions in (2.1) can be very often reduced to asingle condition, without loss of generality:

ψ′i(xν)

ψi(xν)=ψ′j(xν)

ψj(xν), with i 6= j . (2.2)

To justify this statement, let us consider a simple example, where a step functionmodels the potential V (x) and the discontinuity occurs at xν . If ψI(x) = Af(x)and ψII(x) = Bg(x) are the solutions of the Schrodinger equation in the two

74

2.2. The finite crystal with symmetric interfaces: a review of Maue’s work

regions, (2.2) is immediately obtained from (2.1) by representing the system ofequations in matrix form and requiring that the determinant is zero. The onlyexceptions in which a solution of (2.2) might not be immediately useful is whenψ′(x)ψ(x) is either zero or infinity. For instance, in the first case, while the equal-

ity between ψ′i(xν) and ψ′j(xν) is satisfied, this is made without constraining thevalues of the wave functions themselves, which can lead to a solution that cannot

guarantee ψi(xν) = ψj(xν). The case where ψ′(x)ψ(x) goes to infinity is the reverse:

it imposes an equality between the wave functions, but can lead to solutions thatmight not provide an equality between the first derivatives. In these cases, oneshould then go back to the individual equations in (2.1) and explicitly verify thatthe obtained solution is a simultaneous solution for both equations.

The function ψ′(x)ψ(x) is the derivative of log(ψ(x)), and so it is referred to as the

logarithmic derivative.

6. If an edge state solution exists within the crystal, this solution occurs inthe gap. An edge state is, by definition, a solution whose probability densitydecays, as we move away from the interface. For the 1D Schrodinger equation,in which the Hamiltonian corresponds to non-interacting electrons, subjected toa periodic lattice field, the eigenvalues and wave functions can be determinedexactly [93]. The resulting wave functions are of two types: the Bloch wavefunctions, and the ones that are proportional to real exponentials [94]. More-over, it is possible to determine which range of energies give rise to which typeof solutions. Since physical solutions describing the bulk states are required tobe normalisable, the second type is usually disregarded, giving rise to forbiddenenergy ranges for the bulk states (the energy gaps) [94]. Thus, when edge statesolutions are allowed, they have energies in the gap.

2.2 The finite crystal with symmetric interfaces:a review of Maue’s work

In 1935, Maue proved that for a finite 1D crystal, bounded by equidistant inter-faces from its midpoint, two bulk electronic states disappear, one from the lthand the other from (l+ 1)th band [66] (l labels the energy bands in the Brillouinzone). To clarify, with bulk states what is meant is the electronic states thatare determined by the nearly free electron model, subjected to periodic bound-ary conditions. He then expected that these two disappearing solutions shouldbe linked with the appearance of two edge state solutions, with energies in thelth gap. Goodwin extended his method, rendering it applicable to any crystallattice, within the framework of the nearly free electron model [67]. In the samearticle, he also included a review on Maue’s work. The algebraic manipulationsneeded to follow Maue’s original paper are slightly different from the ones foundin Goodwin’s review section. To avoid any confusion that may result from this,and to lay the foundation for the application to interfaces with new types of

75


topology discussed afterwards, we will give our own brief review of Maue’s andGoodwin’s work here. The notation will be close to that used by Goodwin, butexplicit comparison to Maue’s notation is included whenever useful.

We consider a 1D crystal of length L = 2Na, where a is the lattice constantand 2N the number of atoms in the crystal. If x is the spatial coordinate, themidpoint of the crystal will be located at x = 0, while the interfaces sit diamet-rically opposite to it, at x = −Na and x = Na. The periodic lattice field isexpanded as:

V (x) =

+∞∑l=−∞

Vl e2πia lx , (2.3)

where the requirement that V (x) should be a real field demands Vl = V ∗−l. It isfurthered assumed that Vl is itself real, and consequently that Vl = V−l, whichrenders V (x) symmetric1 with respect to inversion in x. No constraints are madeon Vl’s sign, and both options are explored. Later on, it will become clear howVl’s sign is actually a determining factor for both the existence of edge statesolutions and of disappearing bulk states. It will also be its sign that providesthe link between these two phenomena and the symmetry-based topology in [5].In this section we will start by exploring whether any bulk states disappear dueto the open boundary conditions.

A disappearing solution is a solution that would otherwise be present in thebulk system, defined with periodic boundary conditions, but that is now incom-patible with the open boundary conditions. In accordance with the strategysketched in section 2.1, if a Bloch solution is a suitable solution within the crys-tal, it needs to satisfy equation (2.2). That is, the logarithmic derivative of theBloch state needs to match that of the vacuum state. The latter states, apartfrom a normalisation constant, are given by:

ψvac(x) ∼

ex√

2m~2 |E|, for x < −Na

e−x

√2m~2 |E|, for x > Na ,

(2.4)

where E coincides with the energy of the particular Bloch state we are consider-ing (point 3., in the last section). The simple form of ψvac(x) fixes the sign of the

logarithmic derivativeψ′vac(x)ψvac(x) to always be positive on the left (x < −Na) and

always negative on the right (x > Na). As a consequence, a Bloch solution can

1Remarkably, neither Maue nor Goodwin mentions the fact that the reality of Vl mustoriginate from a lattice symmetry.

76


only be a suitable solution within the crystal if its logarithmic derivative is pos-itive on the left and negative on the right. Otherwise it becomes incompatiblewith the boundary conditions, giving rise to a disappearing state. Addition-ally, given that the states in the vacuum are real valued functions, satisfyingthe boundary conditions requires, necessarily, also real valued wave functionsto exist within the crystal. In the nearly free electron model, the majorityof Bloch states are given by single plane waves, which will not provide for areal valued logarithmic derivative. Hence, ”proper” crystalline solutions becomelinear combinations of independent Bloch states that share the same energy:ψcrystal(x) = c1ψk(x) + c∗1ψ−k(x), where ψk(x) ∼ eikx. If ψk and ψ−k are twolinearly independent solutions of the bulk Schrodinger equation, there should beno constraints on what values the coefficient c1 can take. This allows us to alwaysfind a particular choice for it that assures the compliance with the boundary con-ditions. The origin of disappearing bulk states is then linked to the appearanceof constraints on the values of the coefficients. This can happen near the edges ofthe Brillouin zone, where the solutions of the nearly free electron model alreadyconsist of combinations of Bloch waves:

ψ(x) = αeikx + βei(k−2πla )x . (2.5)

This combination of Bloch states again needs to be combined with its complexconjugate partner in the presence of open boundary conditions, in order to yielda real-valued wave function. Note that, contrary to what happens for the otherstates in the interior of the band, there is now a restriction on the ”freedom ofchoice” regarding the coefficients, since, in equation (2.5), α and β cannot bechosen at will, but are actually related through the Schrodinger equation,

α

β

∣∣∣∣±

=λ2Vl

2(πla

)(k − πl

a

)∓√λ4V 2

l + 4(πla

)2(k − πl

a

)2(2.6)

λ2E∣∣± = −λ2V0 +

(πla

)2

+(k− πl

a

)2

±√λ4V 2

l + 4(πla

)2(k − πl

a

)2

, (2.7)

with λ2 = 2m~2 and V0 the average lattice field over a unit cell. The eigenvalues

in (2.7) are obtained from (2.5). This is done by inserting the Ansatz into theSchrodinger equation, from which we can obtain a set of two equations: oneresults from multiplying the equation by e−iqx, and integrating it over a unitcell, the other by multiplication with e−i(k−

2πla )x, followed also by integration.

Equation (2.7) is then the outcome of looking for a non-trivial solution to the setof equations, i.e. one where the coefficients α and β are not simultaneously zero.Using one of the equations of the set, (2.6) can be straightforwardly obtained.

77


The notation αβ

∣∣∣±

is meant to specify to which eigenvalues, λ2E∣∣±, the coeffi-

cients αβ

∣∣∣±

are associated2. Let us stress that equation (2.6) is dependent on Vl,

and therefore also on its sign, but also that α and β depend on the wavenumberk. All these factors combined act as constraints that might render the boundaryconditions (equation (2.2)) impossible to satisfy.To verify whether a solution consistent with the boundary equations exists oneshould write a real-valued wave function of the form ψcrystal(x) = 1

2

(clψ(x) +

c∗l ψ∗(x)

), and see if it can obey the boundary conditions. In its most generic

form, cl is complex, but because its amplitude will factor out in the logarithmicderivative, we can effectively write it only in terms of its phase cl = eiθ. Apriori, two interfaces would imply the need to evaluate the boundary conditionstwice, however, since the potential is symmetric (V (x) = V (−x)) and becausethe two interfaces sit diametrically opposed to each other, symmetry will renderthem equivalent, and the boundary conditions can be studied at only one of theinterfaces. Likewise, symmetry forces the crystalline states to either be even orodd with respect the midpoint of the chain.Looking ahead at applying this analysis to interfaces between topologically dis-tinct materials, it will be useful to phrase this as a group-theoretical argument.The only point symmetry of the 1D chain is inversion symmetry, thus the as-sociated point group contains only two elements: the identity and the inversionoperator, taking x→ −x. In particular, the inversion operator has two distinct1D irreducible representations, ±1. Since the Hamiltonian inherits the symme-tries of the lattice, it will commute with the inversion operator, and through the(time-independent) Schrodinger equation we will obtain ψ(x) = ±ψ(−x), allow-ing us to see that the solutions are indeed even or odd, with respect to x = 0(the midpoint of the chain).

The symmetry of the crystalline states requires the phase θ to be restrainedto either θ = 0 (mod 2π) or θ = π/2 (mod 2π). In particular, the symmetrisedsolutions become:

ψEvencrystal(x) = α cos(kx) + β cos

((2πl/a− k)x

)(e)

ψOddcrystal(x) = α sin(kx)− β sin

((2πl/a− k)x

)(o) ,

(2.8)

yielding, respectively, the following form for equation (2.2) at the right interface,x = Na:

tan(kNa)(k − 2πl

aβ

α+β

)= λ

√|E| (e)

cot(kNa)(k − 2πl

aβ

α+β

)= −λ

√|E| (o) .

(2.9)

2Here, our notation follows Goodwin, with V0 = −V Mau0 and Vl = −V Mau

l .

78


There are now two limiting cases that are instructive to consider: the limitk << πl

a , where we move away from the edges of the Brillouin zone and analyse

how states in the interior of the band satisfy (2.9); and the limit k = πla ± δ,

with δ << πla , where one explores the region close to the edges of the Brillouin

zone. Within the first limit, if the square root in (2.6) is rewritten as 2πla

√1 + y,

with y =λ4V 2

l

4(πl/a)2 , then, due to ~2

2m

(πa

)2

>> |Vl|, it is valid to approximate it

by its Taylor expansion, truncated in first order in y. Consequently, the relationbetween the coefficients α and β becomes:

α

β

∣∣∣∣+

→∞ ,α

β

∣∣∣∣−<< 1 (2.10)

which in turn implies that, in this limit, (2.5) is only given by a single planewave, as it should for states in the interior of the band. Furthermore, using(2.10) in (2.9) the boundary conditions are simplified in form. Associated tothe higher eigenvalue λ2E

∣∣+

, (e) goes to k tan(kNa) = λ√|E| and (o) becomes

k cot(kNa) = −λ√|E|. The graphical method provides proof that solutions ex-

ist for both equations, since an intersection can always be found between left andright terms, both in (e) and (o). The same holds true for the state with eigenvalueλ2E

∣∣−, where now the boundary conditions read (e)→ 2πl

a tan(qNa) = −λ√|E|

and (o)→ 2πla cot(qNa) = λ

√|E|, for which it is always possible to find a solu-

tion, since the trigonometric functions span all R domain. This gives strength tothe previous expectation that no disappearing solutions should occur for statesin the interior of the band.Going on to k = πl

a ± δ, equation (2.6) becomes a function of δ. For this rea-son, the problem of determining which states are compatible with the boundaryconditions in (2.9) is equivalent to finding a suitable value for δ. Let us re-call that, given the periodicity of the tangent and cotangent, if equations (2.9)allow for solutions, then within the range ]0, π/Na[ we should be able to findexactly one, and only one, solution. This allows δ to be taken as a small shift,around q = πl/a, within the region δ ∈]0, π/Na[. This introduces an importantassumption that has gone unmentioned so far3:

1 <<

(πa

)2λ2|Vl|

<< N , (2.11)

which, combined with the range of allowed values for δ, means δ π/aλ2|Vl| << 1.

The square root can then again be approximated, up to first order in y, by

3As Maue points out, this assumption is important if we want the two edge states, localisedat the two ends of the crystal, to be well separated from each other.

79


λ2|Vl|√

1 + y, with y = 4(πl/a)2δ2

λ4|Vl|2 , so that:

α

β

∣∣∣∣±≈ sign(Vl)

1/√y

±1∓ (1/√y)

. (2.12)

Since, in the nearly free electron model, the weak lattice field induces gaps toopen up near the edges of the Brillouin zone, a meaning can be ascribed to the±δ shift: in the reduced Brillouin zone scheme, the ”+” sign should be seen asprobing the region near the edge, from the upper band side, while the ”−” sign,corresponds to inspecting the neighbourhood of q = πl/a, in the lower band,

below the lth gap. Thus, αβ

∣∣∣+

is associated to a ”+δ” shift, and αβ

∣∣∣−

to a ”−δ”shift.

For q = πla ± δ the set of equations in (2.9) can be conveniently rewritten as:

± tan(δNa)

(± δ + πl

a

αβ

∣∣∣±

+ 1

αβ

∣∣∣±− 1

)= λ

√|E| (e)

± cot(δNa)

(± δ + πl

a

αβ

∣∣∣±

+ 1

αβ

∣∣∣±− 1

)= −λ

√|E| (o) .

(2.13)

We are now ready to clarify why precisely close to the edges of the Brillouin zonethere might exist states, built from linear combinations of single Bloch solutions,that cannot satisfy the boundary conditions. Starting with an even state, withenergy λ2E

∣∣+

, and using (2.12), the boundary condition reads

λ√|E| =

tan(δNa)

(δ + πl

a

(2√y − 1

)), if Vl > 0

tan(δNa)(δ + πl

a

(1

(2/√y)−1

)), if Vl < 0 .

(2.14)

Note that given the definition of y and the fact that δ π/aλ2|Vl| << 1, we have

1√y >> 1, for finite δ, and 1√

y → ∞ when δ → 0. Therefore, when Vl < 0,

the right hand side is a function of δ behaving (qualitatively) in the same wayas tan(δNa). The latter is a positive and continuous function of δ in the re-gion δ ∈]0, π/2Na[, where a unique solution to the boundary condition equationcan be found (see fig.(2.1) (D)). For Vl > 0, this is no longer true. Over the

80


range δ ∈]0, π/2Na[, the left side is still a positive, monotonically increasing,function of δ; however, its minimum value is obtained from l’Hopital’s rule to beλ2|Vl|Na. Due to (2.11), and given that the energy is fixed by energy conserva-tion to be close to 1

λ2 (πla )2 − λ2V0, the vacuum’s logarithmic derivative, λ√|E|,

is always below λ2|Vl|Na. Therefore the boundary condition cannot be satisfied(see fig.(2.1) (C)). Let us recall that V0 is the average lattice potential, over aunit cell, and given that the system size is assumed to be the largest scale in the

problem, also

(πa

)2λ2V0

<< N . Hence, an even electronic state, with energy λ2E∣∣+

,

disappears from the bottom of the (l+1)th band, for Vl > 0, but is present whenVl < 0.An odd state, again with energy λ2E

∣∣+

, is subjected to same boundary condi-

tions as in (2.14), if tan(δNa) is replaced by − cot(δNa). Now, no indeterminateform exists for Vl > 0, and no obstructions to finding a solution, over the rangeδ ∈]π/2Na, π/Na[, are present (see fig.(2.2) (G)). When Vl < 0, a 0 × ∞ in-determinate form arises close to δ → 0, limiting the possible values taken bythe crystal’s logarithmic derivative over the interval δ ∈]0, π/2Na[, but leavingunaffected the region δ ∈]π/2Na, π/Na[, where the logarithmic derivative spansR+. Since there is now an extra minus sign, if a suitable solution exists it needsnecessarily to occur within δ ∈]π/2Na, π/Na[, and therefore, for Vl < 0, a finitecrystal, with symmetric interfaces located with respect to x = 0, has no (bulk)odd disappearing states from the upper band (above the lth gap), as can be seenfrom fig.(2.2) (H). Thus, as Maue concluded, from the upper band (relative tothe lth gap), there is one bulk state disappearing: the lowest even electronicstate.Continuing with the lower eigenvalue λ2E

∣∣−, an even state is subjected to the

boundary condition equations (simplified by 2.12):

λ√|E| =

− tan(δNa)

(− δ + πl

a

(1

(2/√y)−1

)), if Vl > 0

− tan(δNa)(− δ + πl

a

(2√y − 1

)), if Vl < 0 .

(2.15)

Comparing (2.15) with (2.14) we see that the terms in brackets multiplying πla

have interchanged for the different signs of Vl, and that these terms carry an extraminus sign. These two effects combine to secure compliance with the boundaryconditions for the even state in the lower band, at the edge of the Brillouin zone.The first one removes the obstruction in fulfilling the boundary condition whenVl is positive, since now the logarithmic derivative essentially behaves qualita-tively as − tan(δNa), for all allowed values of δ (see fig.(2.1) (A)). The secondeffect assures the indeterminate form, originating when δ → 0 for Vl < 0, is of the”inoffensive” type, already encountered previously for the odd solutions in theupper band: it restricts the range of values the logarithmic derivative can take,but only in the region where it is a negative valued function (see fig.(2.1) (B)). It

81


Figure 2.1: Solving (2.14) and (2.15) for the even states through the graphicalmethod, for the different sigs of Vl. When the indeterminate forms cause nointersections, the value at δ → 0 is indicated, otherwise it is omitted.

allows the logarithmic derivative’s codomain, in the interval δ ∈]π/2Na, π/Na[,to be R+, therefore guaranteeing that a solution to the boundary condition equa-tion exists.

For the odd state in the lower band, (2.15) has − tan(δNa) replaced by cot(δNa).This poses no problems when Vl is negative, since the resulting logarithmicderivative behaves, as a function of δ, similarly to cot(δNa) (see fig.(2.2) (F)),but it makes it impossible to satisfy the boundary condition when Vl > 0 (seefig.(2.2) (E)). This is due to the 0×∞ indeterminate form arising in the regionwhere the crystal’s logarithmic derivative is a positive valued function. Notethat cot(δNa) is a positive and monotonically decreasing function in the regionδ ∈]0, π/2Na[, and therefore the effect of the indeterminate form comes as aconstraint on the maximum value taken by the crystal’s logarithmic derivative

in this region. This value is set by l’Hopital’s rule to be (πl/a)2

λ2|Vl|Na , which given

(2.11) fixes the maximum value to be much smaller compared to λ√| E|− |.

82


Hence, when Vl > 0, from the lower band (relative to the lth gap), there is onebulk state disappearing: the highest odd electronic state.These results are summarised in table (2.1). Maue was then lead to the conclu-

Figure 2.2: Solving (2.14) and (2.15) for the odd states, through the graphicalmethod, for the different sigs of Vl. When the indeterminate forms cause nointersections, the value at δ → 0 is indicated, otherwise it is omitted.

sion that a single criterion for disappearing states can be identified in the nearlyfree electron model. In particular, the criterion resides in the sign of Vl: positiveVl leads to disappearing states (at the edges of the Brillouin zone), while negativeVl leads to no disappearing states at all.Maue concluded that two edge states must exist for the finite crystal, but neverconsidered them explicitly. Instead he showed that edge states exist in the semi-infinite crystal when the same criterion Vl > 0 is satisfied. In the latter casehowever, he never explicitly showed that bulk states disappear. In the followingsection, we will make the correspondence between the disappearing bulk statesand the presence of edge states explicit within a single crystal. This shows theemergence of edge states to be part of a broader theme: that of a bulk-boundarycorrespondence. The disappearance of bulk states and the emergence of edgestates have a common cause, which in the case of the nearly-free electron model

83


Sates at the edge of the BZ Vl > 0 Vl < 0 Fig.Even state: top of the lth band X X fig.(2.1) A and B

Even state: bottom ofthe (l + 1)th band

× X fig.(2.1) C and D

Odd state: top of the lth band × X fig.(2.2) E and FOdd state: bottom of the

(l + 1)th bandX X fig.(2.2) G and H

Table 2.1: Disappearing states at the edges of the Brillouin zone, depending onthe sign of Vl. The X indicates that the correspondent crystalline state satisfiesthe boundary conditions, while the × signals that it cannot, thus representing adisappearing state.

is Vl being positive.

Maue’s results are instructive in providing a clear example of how edge statesolutions can stem out of bulk properties (Vl is solely set by the lattice, andis calculated in the system with periodic boundary conditions), having conse-quences even for the bulk density of states, but one can still hesitate on howgeneral these results might be: how do they cary through beyond the nearly freeelectron model? Does the presence of indeterminate forms in the crystal’s loga-rithmic derivative always signal the disappearance of bulk states or were thesesomehow only artefacts coming from the particular form wave functions have inthe nearly free electron model? Why is the sign of Vl important for the emer-gence of edge state solutions, and how relevant is the fact that the lattice fieldhas inversion symmetry? Can there be any relation between these edge statesolutions and the topology of the bulk spectrum? These are all valid questionsthat are left unanswered, but whose explanations will become clearer in the fol-lowing chapters. In particular, at least for 1D crystals, we will be able to providea general answer, connecting all these questions to a single criterion: symmetry.

2.3 The semi-infinite crystal

In this section, we will consider a semi-infinite crystal in contact with the vac-uum, and show explicitly that the disappearance of bulk states is necessarilyaccompanied by the emergence of edge states. This property identifies a bulk-boundary relation for 1D inversion symmetric crystals. The task might seem atfirst a small complement to Maue’s work, but it is, nevertheless, a necessary one.Not merely for completeness, but because it hides an example where the origin ofedge state solutions is actually arising from the bulk properties, a feature whichis now very common in the context of topological band theory.In topological insulators, the proof for the bulk-boundary correspondence, relat-

84

2.3. The semi-infinite crystal

ing the topological invariant of the bulk with the number of edge states at theboundary, is based on Chern–Simons theories and applies to topological invari-ants that depend on the existence of a Berry curvature [82]. This covers, forexample, the Chern number and the Z2 invariant [95], but without an explicitproof, it is not obvious that all topological invariants must obey the same prin-ciple. It is then relevant that one can define a relation between the bulk andthe edge states at the boundary, for other topological phases, such as the onesthat depend on crystal symmetries. This is especially the case, given that thepresence of edge states provides signatures to experimentally identify non-trivialtopologies in band insulators.Maue showed that the condition which both edge state solutions (in the semi-infinite geometry) and disappearing electronic states (in the finite geometry, withsymmetric interfaces) relied on was one and the same: the sign of the Fouriertransformed lattice potential Vl (defined in equation (1.3), defined in the firstpart of the thesis). As will become clearer later on, the sign of Vl is in realityrelated to the symmetry properties of the bands, and therefore provides a linkto the topological classification in [5]. Thus, being able to establish a one-to-one relation between the existence of edge states and the disappearance of bulkstates, becomes a route to establishing a bulk-boundary correspondence for 1Dcrystalline systems, whose symmetries are limited to the point symmetries of thelattice.

2.3.1 From finite to semi-infinite

It is useful, for the purpose of counting the number of disappearing states, towork with a finite density of states. For this reason, we start with a crystal withlength L = Na, where N is the total number of atoms in the crystal, assumedto be finite, even if a large integer (large in the sense of (2.11)). The crystal isbounded by two interfaces, where, if we define x to be the spatial coordinate,one is located at x = 0, and the other at x = Na. To obtain the semi-infinitecrystal, we set N → ∞ as the very last step, and examine how the edge statesolutions and the disappearing states are affected by it. In order to simplify theproblem, we consider the potential field in the region x > Na to be infinitelylarge, which will force the wave functions inside the crystal to be zero at x = Na.In the region x < 0, the potential is left finite, making x = 0 the interface withthe vacuum and the region where localised states might occur (see fig.(2.3)).

2.3.2 Disappearing Bloch states

As established in section 2.2, if a disappearing state exists, it will necessarilyoccur close to the edges of the Brillouin zone. Thus, given that we are interestedin probing their existence, we will constrain the analysis to wave functions of theform

ψcrystal(x) = α cos(kx+ θ) + β cos((2πl

a− k)x− θ

), (2.16)

85


Figure 2.3: Finite crystal with length L = Na, with interfaces located at x = 0and x = Na. At x > Na the potential is infinite, while at x < 0 the potential isfinite.

with k = πla ± δ, and δ being the small shift within the range ]0, π/Na[. This

shift allows us to probe the states close to the edges of the Brillouin zone, whoseenergies either belong to the top of the lth band (k = πl

a − δ) or to the bottom

of the (l + 1)th band (k = πla + δ). Note that ψcrystal(x) is already the result

of taking a real linear combination of single Bloch states (defined in equation(2.5)), making θ the phase associated to the complex valued coefficients. Sincewe impose the potential field in the region x > Na to be infinite, the requirementthat the wave function should be zero at x = Na fixes the phase θ to be:

θ =π

2−N(π ± δa) . (2.17)

Using (2.17) in (2.16), the boundary condition at x = 0 (interface separating thecrystal from the vacuum) becomes equivalent to (2.13), written for the odd bulkstates in the finite crystal with symmetric boundaries. There, the attributionof the symmetry label, even or odd, was motivated by the symmetrical locationof the boundaries, which separated the crystal from the same finite potential[67]. Here, however, this motivation is lost and even direct substitution of (2.17)in (2.16) will not allow for a clear symmetry identification, at least before asuitable δ is defined, which can only be done by solving the boundary conditionequation. On the other hand, the fact that we do get an identical boundarycondition equation is, at the very least, a cause for scepticism towards neglectingall together the role of symmetry in the problem. In reality, the particular formψcrystal(x) adopts, within our particular choice of model, conceals the actual roleinversion symmetry plays on both the emergence of disappearing states and onthe appearance of edge state solutions.To determine the existence of disappearing bulk states, we should now solvethe boundary condition equation. This is simplified since the equation is thesame as (2.13) for the odd states, which lets us preempt the results (see table(2.1)): one disappearing state at the top of the lth band, if Vl is positive, andno disappearing states if Vl is negative. Also note that, even if we take N →∞,

86


none of these results change. This can be seen by considering the cases where thegraphical method allowed no solution in the last section: if we let N → ∞, theobstruction to an intersection in fig.(2.2) still remains. Thus, as in the previoussection, the sign of Vl is our immediate criterion for the origin of disappearingstates.

2.3.3 Edge state solutions

According to Maue’s claim, the number of bulk disappearing states should bethe same as the number of appearing edge states [67]. For the semi-infinite crys-tal, this would then imply finding one edge state solution, in the lth gap, for Vlpositive, and no edge state solutions if Vl is negative. To prove this statement,we start by specifying the wave function that represents an edge state solutionfor the crystal sketched in fig.(2.3). Given that we start with a finite N , ourAnsatz should have the form:

ψedge(x) = c1ψ1(x) + c2ψ2(x) , (2.18)

for which

ψ1(x) = e−kIx

(α1e

iπla x + β1e−iπla x

)ψ2(x) = ekIx

(α2e

iπla x + β2e−iπla x

).

(2.19)

In the previous equation, kI is defined to be the amplitude of the imaginarycomponent of the wavenumber k and is fixed to be positive. The real part of k isset to πl/a. One can check that writing k as a complex number will only allowfor real eigenvalues in (2.7), if indeed the real part is fixed to be πl/a.Just as in (2.6), the coefficients αj and βj , with j = {1, 2}, are constrained bythe Schrodinger equation

αjβj

∣∣∣∣±

=λ2Vl

−2i(−1)j kIπla ∓

√λ4|Vl|2 − 4

(πla

)2

k2I

, (2.20)

with the exception that now the ratio αβ becomes a complex number. In the semi-

infinite crystal extending over the region x > 0, an edge state solution shouldhave essentially the same form as ψ1. Additionally, a suitable solution needs toprovide a real-valued logarithmic derivative at x = 0. The two remarks motivateletting βj=α

∗j [66, 67]. We set the phase associated to αj to be given by θj .

87


Using the fact that (2.20) is the ratio between two complex conjugate numbers,allows us to obtain θj :

θj =(−1)j

2arcsin

(2kI sign(Vl)

λ2|Vl|πl

a

). (2.21)

It then becomes clear that θ1 = −θ2. With (2.21), and the requirement thatthe wave function should go to zero at x = Na, a relation can be found for thecoefficients c1 and c2:

c2 = −c1 e−kINa . (2.22)

Hence, given a suitable solution for kI , ψedge(x) → ψ1(x) when N → ∞, andthe semi-infinite crystal will have an edge state localised near the interface, atx = 0. The question then becomes whether or not a finite kI exists, such thatψedge(x) can satisfy the boundary condition at x = 0 (equation (2.23)),

λ√|E| = − coth(kINa)

(kI −

πl

atan(θ2)

). (2.23)

Given that kI is defined to be a positive and real number, (2.21) implies thatθ2 ∈ ]0, π/2[, if Vl > 0, and θ2 ∈ ] − π/2, 0[ for Vl < 0. The importance ofthis statement comes in when examining the solutions of (2.23): − coth(qINa)is always negative, and therefore, if a solution exists, the other term on the rightside of (2.23) has to also be negative, which is only possible when tan(θ2) ispositive, and thus when Vl is positive. This can be proven by using (2.21) toexpress qI in terms of θ2 and using it to re-write qI − πl

a tan(θ2) as an explicitfunction of θ2. Hence, if Vl < 0, just as there are no bulk disappearing states,there are also no edge states possible. On the other hand, if Vl is positive, onecan check, using the graphical method, that a unique solution exists, within theinterval θ2 ∈ ]0, π/2[. In the region θ2 → 0, the right side in (2.23) holds a0 ×∞ indeterminate form, but the constraint that it imposes on the function’scodomain is negligible, as can be checked using L’Hopital’s rule (see fig.(2.4)).Consequently, the left side spans almost all R+, and all of R+ in the limitN →∞.It is thus always possible to find a solution to (2.23) (see fig.(2.4)).We can then conclude that, for the semi-infnite crystal in the nearly free electron

model, the disappearance of bulk states is a direct signal of the existence ofedge state solutions in the system, and that, once more, the sign of Vl is thedeterminant factor for which both occur.

88


Figure 2.4: Solving (2.23) for Vl > 0, using the graphical method. An intersectioncan be found, even in the limit N →∞.

89

CHAPTER 3

At the interface between two crystals

So far we focused on the existence of edge state solutions in crystals boundedby interfaces that separate them from the vacuum. In these setups, a singlefeature was identified as the predictive factor behind the appearance of allowededge states and forbidden bulk states. This setup can be generalised to the caseof joint materials bounded by a common interface, and thus requires us to findout what the generalisation of the criteria for getting edge states is. Beyond theimmediate interest on probing the generality of Maue’s criterion, the study ofjoint materials bounded by interfaces can be seen as an interesting problem on itsown. Exploring the electronic transport features of heterostructures has lead toimportant technological innovations, in which the bipolar junction transistor isperhaps the most notorious example. In this context, since the existence of edgestate solutions has direct consequences on transport properties at the interface,being able to predict them becomes not just of theoretical interest, but holdsalso practical repercussions [71, 72]. Additionally, preempting the connectionbetween the edge state solutions obtained so far and the topological propertiesof the bulk, heterostructures become also of interest as probes to study bandtopology, with richer possibilities for edge state solutions than the simpler cases,where interfaces separated only single crystals from the vacuum.

3.1 Disappearing Bloch states

Let us then consider an interface between two crystals, as schematically depictedin fig.(3.1). We define as usual x to be the spatial coordinate. The left crystal isbounded by two interfaces, one at x = −Na and the other at x = 0. In the regionx < −Na, the potential is considered to be infinitely large. The second interface,at x = 0, separates the left crystal from the other crystalline system, which wewill refer to as the right crystal, extending over the region x ∈]0, Na]. Beyondx = Na, the potential is also taken to be infinite, making the right crystal too,

91

Chapter 3. At the interface between two crystals

Figure 3.1: Two crystals, left (L) and right (R), sharing an interface at x = 0.At x < −Na and x > Na, the potential V (x) is infinite.

in the limit N →∞, a semi-infinite crystal.

Close to the edge of the Brillouin zone, the wave functions for each crystal aregiven by:

ψ

(L)crystal(x) = α(L) cos(kx+ θ(L)) + β(L) cos

((2πl′

a − k)x− θ(L)

)ψ

(R)crystal(x) = α(R) cos(qx+ θ(R)) + β(R) cos

((2πla − q

)x− θ(R)

),

(3.1)

where the upper indices, (L) and (R), identify to which crystal region the wavefunctions belong. These are obtained by taking a linear combination of thecorresponding single Bloch state ψ(x) (given in 2.5) with its complex conjugate.We consider that the two crystalline systems can have different average lattice

fields, i.e λ2(V(R)0 − V (L)

0 ) 6= 0. In this way our setup is closely related to theprevious examples, except now, on one side, the vacuum is replaced by another

crystal. For simplicity, we take l = l′ and |V (R)l |=|V (L)

l′ |. The sign of Vl can,nevertheless, still differ between the two crystals. Since k and q correspond tomomenta close to the edges of the Brillouin zone, we proceed as usual and expressthese in terms of small shifts around πl/a: k = πl

a ±∆ and q = πla ± δ, where the

shifts δ and ∆ are not necessarily the same. They are, however, related throughthe energy eigenvalues (see (2.7))

λ2(V(R)0 −V (L)

0 ) = δ2−∆2±√λ4V 2

l + 4(πl/a)2δ2∓√λ4V 2

l + 4(πl/a)2∆2 , (3.2)

as well as through the boundary condition equations. These sets of equations

guarantee that for a given value of λ2(V(R)0 −V (L)

0 ), the pair of shifts (δ,∆), andthe energy, are completely specified. Given that the potential becomes infiniteat both x = −Na and x = Na, the requirement that the corresponding wavefunction should vanish at these points determines, as in (2.17), the values of θ(L)

92

3.1. Disappearing Bloch states

and θ(R):

θ(L) = −π2 ±∆Na

θ(R) = π2 ∓ δNa ,

(3.3)

which after direct substitution in (3.1) leads to the boundary condition at theinterface between the two crystals (x = 0):

± cot(∆Na)

(±∆+

πl

a

α(L)

β(L)

∣∣∣±− 1

α(L)

β(L)

∣∣∣±

+ 1

)= ∓ cot(δNa)

(±δ+

πl

a

α(R)

β(R)

∣∣∣±− 1

α(R)

β(R)

∣∣∣±

+ 1

).

(3.4)

The ratios α(L)

β(L) and α(R)

β(R) can be independently obtained using (2.12). As a result,

for states with the same energy, they will only differ in the two crystals if the

sign of Vl is different on both sides, i.e. if V(L)l = −V (R)

l . To simplify notation,let us define the functions:

F(±)αβ,R(δ) = ±δ +

πl

a

(sign(V

(R)l )± 1

)∓√y(

sign(V(R)l )∓ 1

)±√y

with y =4(πl/a)2δ2

λ4|Vl|2, (3.5)

F(±)αβ,L(∆) = ±∆+

πl

a

(sign(V

(L)l )± 1

)∓√w(

sign(V(L)l )∓ 1

)±√w

with w =4(πl/a)2∆2

λ4|Vl|2, (3.6)

which we can use to re-express the ratios α(L)

β(L) and α(R)

β(R) , in (3.4). Note that the

indices ± indicate whether the corresponding state has energy λ2E+ or λ2E−.

Let us then start by considering the case where V(L)l = V

(R)l = |Vl|. For this

setup, (3.4) reads:

± cot(∆Na) F(±)αβ,L(∆) = ∓ cot(δNa)F

(±)αβ,R(δ) , (3.7)

which we solve using the graphical method. The functions F(±)αβ,L and F

(±)αβ,R are

finite everywhere, except when δ → 0 (or ∆→ 0), in which case one can check,using (3.5) and (3.6), that:

limη→0 F

(+)αβ,J(η)→∞

limη→0 F(−)αβ,J(η)→ 0 ,

(3.8)

93


where η can either be set to δ or ∆ and J to the R or L labels. For states whoseenergy is λ2E+, the logarithmic derivatives in the left and right side of (3.7)behave qualitatively as 1

∆ cot(∆Na) (see fig.(3.2) (A)) and − 1δ cot(δNa) (see

fig.(3.2) (B)), respectively. From fig.(3.2), it is immediate that if δ is a solutionto (3.7) in the region ]0, π

2Na [, which we will refer to as the first quadrant, thecorresponding solution for ∆ is within the range ] π

2Na ,πNa [, the second quadrant.

Furthermore, we can always find a value of ∆ that provides a matching valueto the logarithmic derivative on the right, and vice-versa. For example, takeδ = π

3Na , a solution for ∆ in the second quadrant exists with a finite value ofthe logarithmic derivative, as illustrated in fig.(3.3). Depicted in fig.(3.4) are thesolutions for the shifts ∆ and δ, obtained with a particular set of parametersin the nearly free electron model. Note that changing the strength of |Vl| onlyaffects the particular numerical value solutions can have (through (3.5), (3.6)and (3.2)), but does not compromise the possibility of finding a pair (δ,∆) sat-isfying (3.7). As N increases, the shifts δ and ∆ become smaller and smaller inamplitude, but the product δNa and ∆Na stays finite, which guarantees thatthe boundary conditions at x = ±Na are always satisfied.

Figure 3.2: Sketch for the functions 1∆ cot(∆Na) (A) and 1

δ cot(δNa) (B), overthe region ]0, π

Na [. As seen from (A) and (B), in order for the logarithmic deriva-tives at x = 0 to agree in sign, a solution of (3.7) with δ, for example, in ]0, π

2Na [,requires a ∆ in ] π

2Na ,πNa [, if the two logarithmic derivatives are to be negative.

For states with energies corresponding to λ2E−, the left hand side in (3.7) be-haves qualitatively as −∆ cot(∆Na), while the right side as δ cot(δNa). Follow-ing a similar argument, solutions can be found for the shifts δ and ∆, in oppositequadrants, that solve (3.7) (see fig.(3.5)). Consequently, we must conclude that

given V(L)l = V

(R)l = |Vl|, bulk states are always allowed, even close to the edges

of the Brillouin zone.

The question now remains what will happen when the two crystals have different

94


Figure 3.3: Numerical evaluation of a cot(∆Na) F(+)αβ,L(∆) (blue curve), for the

fixed set of parameters: N = 104, l = 1 and (π/a)2

λ2|Vl| = 98.696. The yellow curve

represents the right hand side of (3.7), with δ = π3Na . As seen from the figure,

the logarithmic derivatives match with a finite value. By multiplying (3.4) by thelattice constant a, the numerical evaluation of the boundary condition equationcan be done in terms of the variable ∆a, which simply goes from zero to π

N , fora fixed value of a, as seen in the horizontal axis in the figure.

signs for the Fourier transformed lattice potential. Let us then consider V(L)l =

−V (R)l , with V

(R)l = |Vl|. The boundary condition (3.4) in this case can be

rewritten as:

± cot(∆Na)(± 2∆ + F

(∓)αβ,L(∆)

)= ∓ cot(δNa)F

(±)αβ,R(δ) . (3.9)

States with energy λ2E+, give rise to a logarithmic derivative behaving similarto ∆ cot(∆Na), in the left side of (3.9), and to − 1

δ cot(δNa) at the right side (seefig.(3.6)). Over the region ]0, π

Na [, given generic values for δ and ∆, the right handside in (3.9) will be generally much larger compared to the left, which constrainsthe number of (δ,∆) pairs that can satisfy the boundary condition equation. Theonly possibilities for δ and ∆ that can lead to two logarithmic derivatives withthe same value are if: ∆ → π

Na , and δ is within the first quadrant, but awayfrom δ = π

2Na , or if δ → π2Na , for which we expect multiple values of ∆ to be

able to grant a matching logarithmic derivative, except those close to ∆ = πNa

(see fig.(3.7)).This means that, in the first case, a matching occurs with the logarithmic deriva-tive in the left diverging, while, in the second case, a matching requires the log-

95


Figure 3.4: Set of δ and ∆ values that satisfy (3.7), for states with eigenvalue

λ2E+. The data points where obtained by fixing N = 1012, l = 1 and (π/a)2

λ2|Vl| =

98.696. Taking the lattice constant a as a fixed parameter, the vertical andhorizontal axes span the entire range of values that δ and ∆ can take on.

arithmic derivative on the right to go to zero. As previously mentioned, in caseswhere the logarithmic derivative either goes to zero or diverges, the conditionsfor the matching of the wave functions, and first derivatives, at the interfaceshould be individually examined. Let us then first start by considering the casewhere ∆→ π

Na . The first order derivatives of the wave functions in (3.1) at theinterface read:

ψ′(R)(0) ≈ −πla (α

(R)+ − β(R)

+ ) cos(δNa)

ψ′(L)(0) ≈ −∆ (α

(L)− + β

(L)− ) ,

(3.10)

where we have used that α± ± β± << α± ∓ β±, which is true for both (L) and(R) coefficients. This follows from (2.12). To have ψ

′(R)(0) = ψ′(L)(0), we need

96


Figure 3.5: Set of δ and ∆ values that satisfy (3.7), for states with eigenvalue

λ2E−. The data points where obtained by fixing N = 1012, l = 1 and (π/a)2

λ2|Vl| =

98.696. Taking the lattice constant a as a fixed parameter, the vertical andhorizontal axes span the entire range of values that δ and ∆ can take on.

Figure 3.6: Sketch for the functions ∆ cot(∆Na) (A) and − 1δ cot(δNa) (B), over

the region ]0, Na[.

then

β(R)+

β(L)−

cos(δNa) ≈ 1

lN

α(L)−

β(L)−

+ 1

α(R)+

β(R)+

− 1

∝ −1

N. (3.11)

97


Figure 3.7: Numerical solutions for δ and ∆ that satisfy (3.9), for states withenergy λ2E+. The data points were obtained by fixing N = 109, l = 1 and(π/a)2

λ2|Vl| = 98.696. Taking the lattice constant a as a fixed parameter, the vertical

and horizontal axes span the entire range of values that δ and ∆ can take on.

Consequently, eitherβ(R)+

β(L)−

is finite, but then δ ≈ π2Na , which is incompatible with

our initial argument for the matching of the logarithmic derivatives, orβ(R)+

β(L)−∝ 1

N .

This second statement would then imply β− ∝ N , with β+ finite, or β+ ∝ 1N ,

with β− finite. The first option would violate the constraint of normalisation ofthe wave function when N → ∞, and we therefore neglect it. The second case,would imply that also α+ ∝ 1

N , since the ratio α+/β+ is set by (2.12) to movecloser and closer to −1, as N becomes larger and larger. However, this wouldmean that as N → ∞, both α+ and β+ would vanish, a contradicting result towhat is assumed in the nearly free electron model. Recall that the eigenvaluesnear the edges of the Brillouin zone are constructed under the assumption thatboth coefficients are not simultaneously zero. Thus, we must conclude thatthere is no physically consistent solution that can solve the boundary conditionequations in this case. If now consider the regime of solutions in which δ → π

2Na ,

98


the wave functions in (3.1), at x = 0, are:

ψ(R)(0) ≈ (α

(R)+ + β

(R)+ )

ψ(L)(0) ≈ sin(∆Na) (α(L)− + β

(L)− ) ,

(3.12)

for which an equality is possible if

β(L)−

β(R)+

sin(∆Na) ≈

α(R)+

β(R)+

+ 1

α(L)−

β(L)−

+ 1

∝ − 1

N. (3.13)

The situation is therefore analogous to the previous case, and the same reasoningapplies. Having exhausted all potential solutions of (3.9) with eigenvalue λ2E+,we must conclude that the corresponding bulk states are no longer physically

acceptable solutions, when V(L)l = −V (R)

l .When states have energies corresponding to λ2E−, the left hand side in (3.9)behaves qualitatively as − 1

∆ cot(∆Na), while the right hand side as δ cot(δNa).The boundary condition equation is then identical to the previous one, if oneinterchanges δ and ∆. As a result, matching logarithmic derivatives are confined,this time, to: δ → π

Na and ∆ within the first quadrant, but away from ∆ = π2Na ,

or ∆ → π2Na , with δ away from π

Na (see fig.(3.8)). The first case leads to adivergent logarithmic derivative on the right side, while the second produces alogarithmic derivative that goes to zero on the left.

99


Figure 3.8: Numerical solutions for δ and ∆ that satisfy (3.9), for states withenergy λ2E−. The data points were obtained by fixing N = 109, l = 1 and(π/a)2

λ2|Vl| = 98.696. Taking the lattice constant a as a fixed parameter, the vertical

and horizontal axes span the entire range of values that δ and ∆ can take on.

Not surprisingly, as expected from the similarity between the boundary conditionequations, the same results as in (3.11) and (3.13) will follow (just interchangethe labels (L) and (R)).

We have now seen that at the interface between two crystals, bulk states areno longer suitable solutions at the edges of the Brillouin zone, if Vl changes signacross the interface. This generalises Maue’s result of the bulk–vacuum interface.As in that case, we now expect to find localised states at the interface betweenthe two crystals.

3.2 Edge state solutions

Let us now explicitly prove that, just as for the interfaces separating crystalsfrom the vacuum, the existence of disappearing bulk states in the crystallineheterostructure comes hand-in-hand with the presence of edge state solutions.

We start by writing down the Ansatz for these states in the left (ψ(L)edge(x)) and

100

3.2. Edge state solutions

right (ψ(R)edge(x)) crystals:

ψ

(R)edge(x) = c

(R)1 ψ

(R)1 (x) + c

(R)2 ψ

(R)2 (x)

ψ(L)edge(x) = c

(L)1 ψ

(L)1 (x) + c

(L)2 ψ

(L)2 (x) ,

(3.14)

for which

ψ

(R)1 (x) = e−q

(R)I x cos

(πla − θR

)ψ

(R)2 (x) = eq

(R)I x cos

(πla + θR

) (3.15)

and

ψ

(L)1 (x) = eq

(L)I x cos

(πl′

a + θL

)ψ

(L)2 (x) = e−q

(L)I x cos

(πl′

a − θL).

(3.16)

In the previous equation, q(J)I , with J = {L,R}, identifies the imaginary part of

the momenta in the left or right crystal. The states in (3.14) result from taking a

linear combination of (equal energy) single Bloch states, denoted by ψ(J)1 (x) and

ψ(J)2 (x), whose momenta is πl

a (or πl′

a ) plus an imaginary part, given by ±i q(J)I ,

making q(J)I , by definition, a positive number. The possible energy eigenvalues

for the states ψ(J)1 (x) and ψ

(J)2 (x) are given by:

λ2E

(L)± =

(πl′

a

)2

−(q

(L)I

)2 ±√λ4V 2l′ −

(q

(L)I

)2(πl′/a)2 − λ2V

(L)0

λ2E(R)± =

(πla

)2

−(q

(R)I

)2 ±√λ4V 2l −

(q

(R)I

)2(πl/a)2 − λ2V

(R)0 ,

(3.17)

as a consequence of replacing q by a complex number in (2.7). Strictly speaking

then, ψ(J)1 (x) and ψ

(J)2 (x) should also carry ± lower index specifying the energy

eigenvalue to which they correspond, but to simplify notation they are omitted.

As before, we will consider l = l′, |V (L)l | = |V (R)

l | = |Vl|, but maintain V(L)0

and V(R)0 as potentially distinct. The imaginary components q

(L)I and q

(R)I are

related through the eigenvalues in (3.17) and the boundary condition equations.

Thus, given a fixed V(L)0 and V

(R)0 , the states in (3.14) and their corresponding

energies become completely determined:

101


λ2(V(R)0 − V (L)

0 ) =(q

(L)I

)2 − (q(R)I

)2 ±√λ4V 2l −

(q

(R)I

)2(πl/a)2

∓√λ4V 2

l −(q

(L)I

)2(πl/a)2 (3.18)

The phases θL and θR are determined in a similar way as for the case of thesingle semi-infinite crystal, namely through the ratio α

β and the demand thatβ = α∗. They are given by:

θL = 1

2 arcsin(

sign(Vl′) q(L)I

2(πl′/a)λ2|Vl′ |

)θR = 1

2 arcsin(

sign(Vl) q(R)I

2(πl/a)λ2|Vl|

),

(3.19)

where the principal value of arcsin(...) is taken to be in the usual range [−π2 ,π2 ].

From (3.19) it is clear that the sign of the Fourier transformed potential is decisivein constraining θL and θR to a particular quadrant, which will be important whenaddressing the boundary conditions at x = 0.Next we consider the effect the infinite potentials at |x| > Na have on the wave

functions. The constraint is that ψ(L)edge(−Na) = 0 and ψ

(R)edge(Na) = 0, which

imposes a particular relation between the coefficients c(J)1 and c

(J)2 , just as with

the single semi-infinite crystal in contact with the vacuum:

c(J)2 = −c(J)

1 e−2q(J)I Na . (3.20)

Thus, if a finite q(J)I exists that allows for (3.14) to satisfy the boundary conditions

at x = 0, |c(J)2 | << |c

(J)1 |, as N becomes larger and larger, leading to ψ

(L)edge ≈

c(L)1 ψ

(L)1 (x) and ψ

(R)edge ≈ c

(R)1 ψ

(R)1 (x), the expected edge state solutions in ideal

semi-infinite crystals, which will be maximally localised near the interface. The

possibility of finding a finite q(J)I should then be rooted in having a different sign

for Vl across the interface. Taking into account (3.20) in (3.14), the boundarycondition for the logarithmic derivatives at x = 0 reads:

coth(q

(L)I Na

) [q

(L)I −

πl

atan(θL)

]= − coth

(q

(R)I Na

) [q

(R)I − πl

atan(θR)

].

(3.21)

Let us start analysing equation (3.21) by looking at the expressions in square

brackets. Since by definition q(J)I is positive, it follows from (3.19) that θJ must

102

3.2. Edge state solutions

be in the first quadrant, if V(J)l is positive, and in the fourth quadrant, if V

(J)l is

negative. We can then always write q(J)I from (3.19) as

q(J)I =

λ2|Vl|πl/a sin(θJ) cos(θJ) if V

(J)l = |Vl|

−λ2|Vl|πl/a sin(θJ) cos(θJ) if V

(J)l = −|Vl| ,

(3.22)

allowing us to re-express the terms in square brackets:

q(J)I −

πl

atan(θJ) =

−πla tan(θJ)

(1− λ2|Vl|

(πl/a)2 cos2(θJ))

if V(J)l = |Vl|

−πla tan(θJ)(

1 + λ2|Vl|(πl/a)2 cos2(θJ)

)if V

(J)l = −|Vl| .

(3.23)

Figure 3.9: Numerical solutions for θL and θR that satisfy (3.21). The data points

were obtained by fixing N = 1012, l = 1 and (π/a)2

λ2|Vl| = 98.696. The solutions θL

and θR are only slightly shifted from the line θL = θR, making q(L)I and q

(R)I also

close in value.

103


Given that in the nearly free electron model λ2|Vl|(πl/a)2 << 1, the expressions in

(3.23) are either positive or negative solely depending on whether θJ is in the

first or the fourth quadrant. In (3.21), coth(q

(J)I Na

)converges quickly to 1 for

large N , as long as q(J)I is finite, and we can therefore factor it out from the

equation. We are left with the square brackets, but due to the extra minus signin the right side of (3.21), these can never agree in sign, if θL and θR are inthe same quadrant. Consequently, a solution can only exist if Vl changes sign

across the interface. Furthermore, because λ2|Vl|(πl/a)2 << 1, the boundary condition

equation is satisfied for q(L)I ≈ q(R)

I , as confirmed also by numerical evaluation of(3.21) (see fig.(3.9)).

For edge state solutions to be present in heterostructures, it is then imperativethat the two crystals harbour opposite signs for their Fourier transformed po-tential, the same criterion that has lead to the existence of disappearing bulkstates from in-band states, near the edges of the Brillouin zone. Thus, oncemore, the two come hand-in-hand, as a natural consequence of their shared ori-gin, establishing in this way a bulk–boundary relation also for the case of aninterface between two crystals. Since both edge states and bulk disappearingstates can be inferred from the knowledge of Vl’s sign across the interface, whichfor both left and right crystals is computed using periodic boundary conditions(and thus refers to the bulk system), the sign of Vl becomes the bulk propertythat determines what happens at the boundary.

3.3 Searching for a physical meaning in Vl’s sign

The multiple examples in this chapter have been consistent in one unique trait:the emergence of edge state solutions and disappearance of bulk states was sig-nalled by the sign of Vl. The question remains as to why this parameter, whichin the nearly free electron model is even only a small perturbation to the freemotion of electrons, has such a determinant influence for what happens at theboundary of crystals. To begin understanding what physical meaning hides be-hind the sign of Vl, let us look more carefully at the single Bloch states ψ(x) and

the coefficient ratio αβ

∣∣∣±

. For convenience, these equations are repeated bellow:

ψ(x) = αeikx + βei(k−2πla )x ,

α

β

∣∣∣∣±

=λ2Vl

2(πla

)(k − πl

a

)∓√λ4V 2

l + 4(πla

)2(k − πl

a

)2.

104

3.3. Searching for a physical meaning in Vl’s sign

At precisely the Brillouin zone edge, k = πla , the ratio α

β

∣∣∣±

simplifies to

α± = ∓ β± sign(Vl) , (3.24)

and therefore

ψ+(x) ∼ sin

(πla x), ψ−(x) ∼ cos

(πla x), if Vl > 0

ψ+(x) ∼ cos(πla x), ψ−(x) ∼ sin

(πla x), if Vl < 0 ,

(3.25)

Figure 3.10: Two possible energy bands in the nearly free electron model, rep-resented in the reduced Brillouin zone scheme. The symmetry of the states atk = 0 and k = π

a are indicated for the two possible signs of Vl, and are obtainedby systematically choosing either Vl positive or Vl negative. The different coloursindicate bands of distinct symmetry types.

meaning that, the sign of Vl determines if the state, corresponding to the high-est (ψ+(x)) or the lowest (ψ−(x)) energy level, is even or odd (see fig.(3.10))with respect to x = 0. The lattice potential is then altering the bands in thenearly free electron model, not only because it leads to the opening of energygaps, but also by determining the symmetry labels for the states at the edges ofthe Brillouin zone. One clear effect is that, from a symmetry point of view, thesign of Vl throughout the energy spectrum determines which different types ofbands appear in the crystal. We note that there is nothing a priori forbiddingVl to have different signs for different values of l. Moreover, in a group theoryperspective, the bands can be characterised by the symmetry labels occurringat the high symmetry points in the Brillouin zone, which in 1D are only two,k = 0 and k = πl

a . Since Vl = V−l, the potential has inversion symmetry, and

105


therefore the labels that can be ascribed to these points are precisely the even orodd trait under inversion, determining that in 1D inversion symmetric crystalsonly four distinct types of bands can potentially exist in the reduced Brillouinzone scheme. These four different types of bands correspond then to: bands thatare odd for k = 0 and for k = π

a , even for both k = 0 and k = πa , even at k = 0

but odd at k = πa , or the reverse. This establishes a direct link between the

sign of Vl and the band symmetry type. What remains to be determined is howthe symmetry properties of the bands determine the presence or absence of edgestates.

There is, indeed, a link between the band symmetry types and the origin ofboth disappearing bulk states and edge states at the boundary, even indepen-dent of the nearly free electron model. A detailed explanation is postponed tothe next chapter. For now, let us motivate the interplay between symmetry andthe compliance with the boundary conditions, written for the bulk states, bylooking at the logarithmic derivative of ψ(x) (as given in the beginning of thissection), at x = 0, with k = πl

a ± δ:

ψ′±(0)

ψ±(0)= i

(πl

a

αβ

∣∣∣±− 1

αβ

∣∣∣±

+ 1± δ

). (3.26)

This is a familiar object and, in fact, in all previous examples, direct inspection

enables us to systematically find the term −i ψ′±(0)

ψ±(0) in the boundary conditions

for the real linear combination of Bloch states. This means that the logarith-mic derivative of real wave functions contains information, in quite a direct way,about the logarithmic derivative of individual Bloch states, and thus about theirsymmetry properties. In the nearly free electron model, these come into playthrough how the coefficients α and β are related to each other (which followsfrom (3.24) and (3.25)).Consider, for example, the boundary condition for the odd bulk states in thesingle crystal with two equidistance interfaces about its midpoint. If we take itsenergy to lay close to the top of the l = 1 band: when α

β → 1 as δ → 0, which is

the same as approaching k = πa closer and closer, the right hand side in (3.26)

goes to zero, and thus is proportional to δ. At the same time, the boundary

condition equation for the odd states is given by the product of −i ψ′±(0)

ψ±(0) with

cot(δNa), and so the effective logarithmic derivative behaves as −δ cot(δNa).Hence, the boundary condition becomes impossible to satisfy between the corre-sponding bulk state in the crystal and the vacuum. However, if instead α

β → −1,the impossibility would not be present. The two limits for the ratio α

β are directlylinked with symmetry and the sign of V1 since α

β → 1, as δ → 0, is consistent

with an even (single) Bloch state and requires V1 > 0, while a state with thesame energy, but with α

β → −1, corresponds to an odd single Bloch state and

106

3.3. Searching for a physical meaning in Vl’s sign

nkj Fig.(3.10) (A) Fig.(3.10) (B)

n0even 2 1n0

odd 0 1

nπ/aeven 1 1

nπ/aodd 1 1

Table 3.1: Number of bands that transform according to the irreducible repre-sentation j at the high symmetry point k, for both energy bands depicted infig.(3.10).

exists only if V1 < 0. The two possible signs for V1 lead then to the two distinctsymmetry types of bands depicted in fig.(3.10), with only one of them support-ing edge state solutions at the interface with the vacuum. Note that, within thenearly free electron model, the states in the first band, away from k = π

a , arewell approximated by single plane waves, which consequently makes the Blochstate at k = 0 a constant, and thus even under inversion.The fact that there exists a bulk property, in this case the sign of Vl, that deter-mines if Bloch states are able to smoothly connect to the vacuum is reminiscentto what happens at the boundary of topological insulators. Indeed, there is a re-lation between the symmetry labels given to the bulk states and the topology of1D crystals (which will be addressed in more detail in the next chapters). To ex-emplify it, imagine then a setup where the two crystals have the band structuresdepicted in fig.(3.10): (A) corresponds to the left crystal, while (B) to the rightone. From our previous analysis, we know edge states must be present, since thesign of Vl changes across the interface (fixed at x = 0). Let us further say that,for instance, the two bands in the crystals correspond to occupied states. If wehad no knowledge regarding the sign of Vl, we can still find a topological distinc-tion between these two crystals, using the methods of reference [5]. Note that fora 1D inversion symmetric crystal, at k = 0 and k = π

a , the point group is formedby the identity and the inversion operation. There are then two irreducible rep-resentations, one associated to states that are even under inversion and the othercorresponding to the odd ones. Hence, using a similar notation as in [5], nkj isthe number of bands that transform according to the irreducible representationj at the high symmetry point k. The list of integers nkj for the crystal on the leftis necessarily different from that for the crystal on the right (see table (3.1)). Inaccordance with [5], since left and right crystals are characterised by a differentset of integers, the two crystals are topologically distinct. Thus, given that thetopological nature of the crystals stems out of the lattice symmetries, and these,in turn, affect the boundary conditions through the constraints imposed on thewave functions, the existence of edge state solutions at the interface between thetwo crystals is necessarily linked to their distinct topology. Furthermore, sincethe existence of edge state solutions is always accompanied by an equal number

107


of disappearing bulk states, this points to the topological character of this bulk-boundary relation.

Before proceeding into the next chapter, we note that the location of the in-terface throughout chapter 2 has been systematically set at x = 0, one of theinversion centres in the crystal. As shows in [74], only when the boundary isfixed at a high symmetry point are the edge state solutions a consequence of thepoint symmetries of the crystal. This is consistent with Maue’s criterion, which,as we have seen, is rooted in the crystal’s lattice symmetries. For the 1D sys-tems under consideration, the interface should then be fixed at a high symmetryposition, if the edge state solutions are to be protected by the point symmetriesof the lattice.

108

CHAPTER 4

Symmetry as the predictor of edge statesolutions

So far, the only foolproof indicator of whether or not edge states exist, is givenby the sign of Vl. But, as discussed in the end of the last chapter, a sign changein Vl is directly linked to symmetry changes in the energy spectrum. Thus, wemight as well look for symmetry indicators that can tell us about the existenceof edge states. This becomes particularly instructive when one is interested inmaking statements outside the scope of the nearly free electron model.One hint on how we might succeed in this task was already uncovered in the endof the last chapter, and concerns the logarithmic derivative of the real Bloch so-lutions. This object carries information about the symmetry labels at the edgesof the Brillouin zone (even or odd under inversion). In the last chapter, this wasestablished with the interface fixed at x = 0, an inversion centre of the lattice.We can, therefore, identify at least two instances in which symmetry permeatesthrough the boundary condition equations: fixing the interface at a high sym-metry point of the lattice, and through the role of Bloch states close to highsymmetry points in the Brillouin zone.Indeed, a general framework can be built combining these two features to devise asymmetry criterion predicting the existence of edge state solutions at the bound-ary of 1D inversion symmetric crystals, as shown by Zak in 1985 [74]. Preciselybecause this criterion is built only on symmetry arguments, it offers a generalframework, independent of the particular detailed form wave functions adopt inthe crystal. This formalism will allow us to settle the essence of the emergenceof edge states in heterostructures as deeply rooted in symmetry. Additionally,it lays the foundation for us to prove that, independently of the chosen model,disappearing bulk states are inherently a result of symmetry constraints. Thisclarifies their origin as unavoidably entwined with the appearance of edge states.All these aspects combined, plus the anticipation that the formalism can be

109

Chapter 4. Symmetry as the predictor of edge state solutions

generalised to higher dimensions, with lattice symmetries other than inversionsymmetry, justifies starting this chapter with a review of Zak’s symmetry crite-rion. The band symmetry labels used in this framework are directly connectedto the quantised Zak phase [96], with its quantisation stemming from the latticesymmetries. This signals the presence of a non-trivial topology in the system, aproperty which was not a priori obvious from the nearly free electron model.In analogy to Chern insulators, it becomes then almost unavoidable to wonderhow a bulk-boundary correspondence can be established, rendering the edge statesolutions found so far a direct consequence of a particular value assumed by theZak phase. This has been addressed in the literature, which often refers to thetask as ambiguous because, as it stands in its original form, the Zak phase isdependent on the choice for the unit cell’s origin [97]. This has led some authorsin the endeavour of extracting an origin-independent part from the original Zakphase, so that the standard bulk-boundary correspondence can be recovered incrystalline systems sharing a boundary with the vacuum [97, 98].Let us note that, even without prior knowledge about the band topology, thesame issue could have been raised for whether or not edge states exist in ourdiscussion so far. Maue’s criterion regarding a positive sign for the Fourier trans-formed potential was unambiguous only because the interface with the vacuumwas always chosen to be at x = 0. As pointed out in [74], had he fixed theinterface at x = a/2, the criterion for edge state solutions would have been theopposite. Likewise, the same problem exists for Zak’s symmetry criterion. Andyet, this will not be the case for crystalline heterostructures. As we will see,as long as the interface is fixed at a high symmetry point of the lattice, theprediction of edge state solutions is indifferent to which of the two is picked.Hence, this highlights the heterostructure setup as an ideal system to study thetopological properties rooted on the lattice discrete symmetries.We will also discuss the equivalence between Zak’s, Maue’s and the topologi-cal classification in [5] for the identification of topological phase transitions inheterostructures. Although the number of edge states is given by the num-ber of disappearing bulk states from the occupied electron states, in the threeformalisms, a single bulk criterion can be identified that signals the change intopology at the interface, and hence the presence of edge states.

4.1 A review on Zak’s symmetry criterion foredge state solutions

Suppose a 1D semi-inifinite crystal, defined over the region x < xν , has an in-terface at x = xν , separating it from the vacuum. Is it possible to predict theexistence of edge state solutions? Up until now, the route to answer this questionhas been: given a Bloch solution ψk(x), with k complex, and the wave function

in the vacuum, ψvac(x) ∼ e−√λ2|E|x, compute the corresponding logarithmic

derivatives, and analyse the possibility of matching them up at the interface.The simple form of ψvac(x) leads to a logarithmic derivative that is merely a

110

4.1. A review on Zak’s symmetry criterion for edge state solutions

negative constant on the vacuum side. On the other hand, for the crystal’s log-arithmic derivative, we relied on a specific model to attain the wave functions,and no other statements could be made beforehand.What we are, nevertheless, able to say a priori is that the crystal’s logarithmicderivative, at x = xν , should also be negative, if the boundary conditions are tobe satisfied at the interface. The question should then be whether we can predictthe sign of the crystal’s logarithmic derivative, for states in the gap. Using Zak’sframework we can answer this question, without any knowledge of the detailedform of the Bloch states.

The starting point is then to consider a bulk 1D crystal, with a set of wellseparated energy bands, and in particular, one whose underlying lattice has in-version symmetry. The Bloch states, ψk(x), are, therefore, the solutions of the1D Schrodinger equation, with a periodic and inversion symmetric potential.The analytic properties of these states, as a function of a complex variable k, werestudied by Kohn [99]. He showed that, by appropriately choosing the phases ofthe Bloch states, these can be analytically continued into the complex k-plane1.Recall that, in the end, we are interested in the logarithmic derivative of in-gapstates, which have complex valued wave numbers, making these results particu-larly relevant in this context.Along with the Bloch states, Kohn also addressed the properties of the associatedWannier functions. It was proven that there exists a unique Wannier function2

for each band that is of one of the following four forms [99]:

1. symmetric (m = +) with respect to x = 0 (q = 0), so that a(0,+)(−x) =a(0,+)(x);

2. anti-symmetric (m = −) with respect to x = 0 (q = 0), so that a(0,−)(−x) =−a(0,−)(x);

3. symmetric (m = +) with respect to x = a/2 (q = a/2), so that a(a/2,+)(−x+a) = a(a/2,+)(x);

4. anti-symmetric (m = −) with respect to x = a/2 (q = a/2), so thata(a/2,−)(−x+ a) = −a(a/2,−)(x),

As a result, a band acquires the symmetry labels (q,m), and only four distinctsymmetry types of bands are possible. The relation between these symmetrytypes and the symmetry labels at high symmetry points, in the reduced Brillouinzone, is given in appendix B. The single Bloch state, in a given band, can thenbe written as follows:

ψ(q,m)k (x) =

( a2π

)1/2 +∞∑n=−∞

eikan a(q,m)(x− na) , (4.1)

1Over a strip of finite width, enclosing the real axis.2Kohn also proves that the resulting Wannier function ”falls off exponentially” [99].

111


which is associated to the corresponding logarithmic derivative

ρ(k, xν) =ddxψ

(q,m)k (xν)

ψ(q,m)k (xν)

. (4.2)

By using the symmetry properties of the Wannier functions, Bloch’s theorem,and the definition (4.1), it is possible to determine the values of k and xν for

which either ψ(q,m)k (x) or d

dxψ(q,m)k (x) vanishes, and, consequently, also the points

where the logarithmic derivative, respectively, diverges or goes to zero (see ta-ble (4.1)). Since the wave number can now also be complex, k = kR + ikI , itbecomes desirable to examine the properties of (4.2) for complex k values. Zakproves that ρ(πa + ikI , x) and ρ(ikI , x) are real valued functions, for any x, butthat ρ(kR, 0) and ρ(kR, a/2) are purely imaginary. This has important conse-quences for the sign of the logarithmic derivative: that ρ(kR, 0) and ρ(kR, a/2)are purely imaginary, means that, by fixing xν to coincide with either one of thetwo inversion centres, the logarithmic derivative is imaginary for any single Blochstate that is an energy state belonging to any band; however, since ρ(πa + ikI , x)and ρ(ikI , x) are real for any x, the logarithmic derivative, with xν = {0, a/2},goes from being purely imaginary for states with energies belonging to any givenband, to a real-valued function for states with energies in the gap.

The zeros of ψ(q,m)k (x), and of its derivative that are constrained by symmetry

are listed in table (4.1). Note that the sign of the logarithmic derivative switches

only if ψ(q,m)k (x) or d

dxψ(q,m)k (xν) changes sign, which can only happen when one

of the two goes through zero.Thus, by fixing the interface at either xν = 0 or xν = a/2, the behaviour ofρ(k, xν), as a function of k, is determined: for any kR describing states alonga given energy band, the logarithmic derivative has a fixed and common sign;only when the edges of the reduced Brillouin zone are approached, i.e kR → 0 or

kR → πa , where either ψ

(q,m)k (x) or d

dxψ(q,m)k (x) goes to zero, can the logarithmic

derivative go through a sign change, allowing ρ(−iε, xν), or ρ(πa − iε, xν), to pos-sibly differ in sign from ρ(ε, xν), or ρ(π/a− ε, xν), (ε ∈ R and ε << 1). The signof the logarithmic derivative, for any given set of bands, can then be predictedby establishing how the sign(ρ(k, 0)) and sign(ρ(k, a/2)) behave when traversinga band with a given (q,m) symmetry.

To do so, we can use the known relation between the signs of the logarithmicderivatives at the two inversion centres:

sign(ρ(−iε, 0)) = sign(ρ(−iε, a/2))

sign(ρ(π/a− iε, 0)) = −sign(ρ(π/a− iε, a/2)) .

(4.3)

This is proven in [74] and follows almost immediately from the fact that theWronskian, W (x), computed from two independent solutions of the Schrodinger

112

4.1. A review on Zak’s symmetry criterion for edge state solutions

(q, l) (k, xν) : (0, 0) (0, a/2) (π/a, 0) (π/a, a/2)

(0,+) ψ′ = 0 ψ′ = 0 ψ′ = 0 ψ = 0(0, -) ψ = 0 ψ = 0 ψ = 0 ψ′ = 0(a2 ,+) ψ′ = 0 ψ′ = 0 ψ = 0 ψ′ = 0(a2 , -) ψ = 0 ψ = 0 ψ′ = 0 ψ = 0

(0,+) ρ = 0 ρ = 0 ρ = 0 ρ =∞(0, -) ρ =∞ ρ =∞ ρ =∞ ρ = 0(a2 ,+) ρ = 0 ρ = 0 ρ =∞ ρ = 0(a2 , -) ρ =∞ ρ =∞ ρ = 0 ρ =∞

Table 4.1: Zeros of ψ and ψ′, and the corresponding effects for the logarithmicderivative ρ(k, xν) at high symmetry points in the reduced Brillouin zone (table1 in [74]).

equation is independent of the coordinate x, and therefore W (0) = W (a/2).Thus, as soon as one knows how the sign of ρ(k, 0) changes for all the (q,m)symmetry bands, the sign behaviour of ρ(k, a/2) becomes fully determined.Let us then fix xν = 0 and see how Zak’s method yields rules for the sign(ρ(k, 0)),as we go through the set of bands in fig.(4.1), representing a possible energy spec-trum of a left semi-infinite crystal. We will proceed as Zak, and assume that thereare no edge state solutions below the first band. Then, under this assumption,ρ(−iε, 0) > 0, with ε << 1. Because the Bloch states are guaranteed to be ana-lytic for small ε [99] and ρ(k, 0) vanishes at k = 0 (see table (4.1)), ρ(−iε, 0) ∼ εand consequently ρ(ε, 0) ∼ iε. As we approach k = π/a, ρ starts to diverge (table4.1) and hence ρ(π/a − ε, 0) ∼ i

ε , implying ρ(π/a − iε, 0) ∼ 1ε . Therefore, the

sign of ρ(k, 0) stays unchanged by traversing a (a/2,+) band while, from (4.3),ρ(k, a/2) will change sign after going through the same band.

Immediately below the next band (a (0,+) symmetry type), since ρ(k, 0) can-not change sign in the gap and will be forced to go to zero at k = π/a,ρ(π/a − iε, 0) ∼ ε. This makes ρ(π/a − ε, 0) ∼ −iε, and sets the logarithmicderivative for states in the (0,+) to be a negative imaginary number.At k = 0, ρ(k, 0) will again go to zero, and so ρ(ε, 0) ∼ −iε, but as a resultρ(−iε, 0) ∼ −ε. Given that we started with a positive logarithmic derivative justbelow the (0,+) band and end up with a negative one, we need to conclude thatρ(k, 0) switches sign by going through a (0,+) band.Had we considered ρ(k, a/2) instead, we would have started with a negative signimmediately below the (0,+) band, and would have ended up with a negativesign immediately above, as a direct consequence of (4.3). Thus, ρ(k, a/2) main-tains its sign while traversing a band of (0,+) symmetry.

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Figure 4.1: Four distinct symmetry types of bands, as given by their the indices(q,m). The energy bands are represented in the reduced Brillouin zone scheme.The ± signs indicate the sign of ρ(k, 0) for in-gap states. It is assumed in thefigure that xν = 0.

The next band is a (0,−) type, for which we know the sign of ρ(k, 0) imme-diately below to be negative. Since ρ(k, 0) will diverge at k = 0, ρ(−iε, 0) ∼ − 1

ε ,

making ρ(ε, 0) ∼ iε . As k → π/a, ρ(k, 0) will again diverge and so ρ(π/a−ε) ∼ i

ε ,which determines ρ(π/a− iε) ∼ 1

ε . Hence, ρ(k, 0) changes sign as it goes througha (0,−) band.For ρ(k, a/2), we would start and end with a negative sign: ρ(k, a/2) maintainsits sign while traversing a (0,−) band.

The last band is a (a/2,−) type, for which ρ(k, 0) is positive immediately be-low. At k = π/a, ρ(k, 0) will go to zero, and thus ρ(π/a − ε) ∼ −iε. As weapproach k = 0, ρ(k, 0) will start diverging, and so ρ(ε, 0) ∼ − i

ε . Consequently,ρ(−iε) ∼ 1

ε , a positive logarithmic derivative. Thus, we see that by traversinga (a/2,−) band, ρ(k, 0) keeps its sign, while from (4.3), ρ(k, a/2) would switchsign after going through the same band. These results are summarised in table(4.2).If we now return to our initial question regarding the existence of edge statessolutions for the left semi-infinite crystal, we can conclude at once that for asemi-infinite crystal with the bulk bands in fig.(4.1), edge state solutions are

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4.2. At the interface between two semi-infinite crystals

(q,m) xν = 0 xν = a/2(0,+) ρ(k, xν) changes sign ρ(k, xν) keeps sign(0,−) ρ(k, xν) changes sign ρ(k, xν) keeps sign(a/2,+) ρ(k, xν) keeps sign ρ(k, xν) changes sign(a/2,−) ρ(k, xν) keeps sign ρ(k, xν) changes sign

Table 4.2: Rules for the sign change of ρ(k, 0) and ρ(k, a/2), when a band of agiven symmetry type (q,m) is traversed.

possible around k = 0, in the energy gap enclosed by the (0,±) bands.In general, given an interface fixed at xν = {0, a/2}, and the assumption thatbelow the first band ρ > 0, we need at least one symmetry type band that hasq = xν to have edge state solutions.We note that a comparison between Zak’s symmetry criterion and Maue’s rulefor the sign of Vl can be found in appendix B and that they both agree in theirpredictions.

4.2 At the interface between two semi-infinitecrystals

In semi-infinite crystals bounded by the vacuum, symmetry has been the hiddencriterion for edge state solutions. It becomes then natural to expect that thesame holds true for crystalline heterostructures.To prove it, let us then consider a left and right semi-infinite crystals, sharingan interface at x = xν . At both sides of the interface, if edge state solutions arepresent, the logarithmic derivatives from left and right crystals have to matchone another for fixed xν . Since edge states are in-gap states, this can only happenwhen they share the same sign in gaps defined in the same energy range,

sign(ρleft(kleft, xν)) = sign(ρright(kright, xν)) . (4.4)

From the last section, it is now clear that the sign of the logarithmic derivative isdependent on the set of (q,m) bands characterising the crystal. For this reasonthe right and left hand side of (4.4) can be determined independently throughZak’s symmetry criterion, reducing the problem to a matter of then searchingfor a common sign at the energy equivalent gaps.We will assume that there are no edge state solutions in the gap below the firstband, and take the same starting signs that guaranteed no matching with thedecaying states in the vacuum, i.e. ρleft(kleft, xν) begins with a positive sign,but ρright(kright, xν) is initially negative. Taking this assumption will allow us tomake a direct comparison with the nearly free electron model approach.It is instructive to start with a specific example, where a particular set of bands isspecified for both left and right crystals, and use it to infer the general properties

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necessary for the existence of edge state solutions in heterostructures. This willbe our next task.

4.2.1 Band symmetry arguments

Let us then consider a left and a right semi-infinite crystal with the particular ar-rangement of bands in fig.(4.2). We will only look at direct hybridization gaps, sothat a parallel with the nearly free electron model can be immediately obtained.Other ordering of bands, although in theory possible, will not be discussed here.The signs of ρleft(kleft, xν) and ρright(kright, xν) can be readily determined fromZak’s criterion at each gap, once xν is fixed at one of the inversion centres in thelattice. We start by setting xν = 0 first, addressing the other choice, xν = a/2,later. It follows immediately from table (4.2) that the logarithmic derivative’ssign changes only by traversing a band with q = xν . This means that, aftertraversing the first band, the sign in the lowest energy gap is reversed on theright, but stays the same on the left crystal. Recall that the assumption of noedge state solutions below the first band requires starting with different signs forthe left and right logarithmic derivatives. As a consequence then, we end up witha matching sign in the first energy gap. Since the second band in both systemsis characterised by q = 0, the sign is inverted from the first to the second energygap, but now on both crystals, leading also to a sign match in the second energygap. By applying the rules in (4.2) to the remaining two bands, we will concludethat no other matching signs are possible in the gaps, as illustrated in fig.(4.2).

If instead we fix xν = a/2, traversing the first band will change the sign of thelogarithmic derivative on the left crystal, but not on the right. The situation isnow reversed from when the same set of bands were transversed with xν = 0,yet, despite that, the matching signs are maintained in the two energy gaps.This is in clear opposition to what happened for the case of a single semi-infinitecrystal with the vacuum: there, changing xν meant going from having allowededge state solutions, in a given gap, to none at all.The disparity is, nonetheless, a natural consequence of the symmetry propertiesof the (q,m) bands. Given the different signs in the gap below the first energyband, a first matching sign can only occur if one of the logarithmic derivativeschanges sign, but not both. Hence, what is needed in order to have edge statesis that both left and right crystals have energy equivalent bands with differentsymmetry q labels. Referring to such energy bands as the pair (qL, qR), where qLand qR indicate the q label for the left and right crystal bands, having qL 6= qRguarantees that by specifying either xν = a/2 or xν = 0, what changes is merelywhich of the two logarithmic derivatives changes sign and which one keeps thesign unchanged.If a (qL, qR) pair, with qL 6= qR, appears twice in the energy spectrum of the leftand right crystals, it will undo the matching sign the second time it occurs. If inbetween these two pairs, there exists energy equivalent bands sharing the same

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Figure 4.2: Example of a particular set of (q,m) energy bands, characterisingthe left (L) and right (R) crystals, in the reduced Brillouin zone. The ± signs, ineach crystal, indicate the sign of the logarithmic derivative for to in-gap states.In red are the signs obtained by fixing xν at zero, while in green are the signscorresponding to xν = a/2. The rectangle highlights the cases where a signmatch occurs in energy equivalent gaps.

q, on both left and right crystals, the sign match is propagated through the gaps,until the second pair with qL 6= qR is reached. Thus, edge state solutions willdepend on having (qL, qR) energy pairs with qL 6= qR, occurring at least oncethroughout the energy spectrum of the crystals.

4.2.2 Connection with topological invariants

In [5], a topological phase transition is defined by an exchange of the symmetryproperties between the valence and conduction bands. The initial and final stageof the process are separated by the closing of the gap at the Fermi level. Theinformation concerning the band symmetry properties is given by the set of nkiintegers: the number of occupied bands, at the high symmetry point k in theBrillouin zone, that transform according to the irreducible representation labeledby i. An exchange in the symmetry properties of the valence and conductionbands is then signalled by changes in these numbers. As a consequence, if weimagine building a composite structure out of two crystals, characterised by adistinct set of nki integers, at the interface, a topological phase transition mustoccur. The question then becomes whether or not this transition implies theappearance of edge states at the surface. Let us note that if we could establish

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a clear correspondence between the edge state solutions predicted by Maue andZak, and the change in the nki numbers across the interface, then not only theanswer to the previous question would be positive, but also the topological char-acter of these states would be settled. This sets the aim of this section, which willfocus entirely on the topological phase transitions between different crystallinesystems.To make a direct comparison between the classification in [5] and Maue’s andZak’s criterions for edge state solutions, it is instructive to consider an orderingof bands that is consistent with the nearly free electron model. What this meansis that, at a given high symmetry point in the reduced Brillouin zone, there isalways associated to it an even and odd Bloch wave function, whose energy dif-ference corresponds to the size of the nearby energy gap. For this band ordering,it then follows immediately that if the gap between the last occupied band andthe conduction band (called valence–conduction gap, from now on) occurs atk = 0, at k = π/a, there are exactly the same number of even states, as there

are odd ones: nπ/aeven = n

π/aodd. This holds true for both left and right crystals.

As usual, we will assume that the number of occupied bands is the same in both

crystals. Consequently, when the valence–conduction gap is fixed at k = 0, nπ/aeven

takes the same integer value for any crystals that share the same number of oc-cupied bands. As a result, a change in the topology across the interface has tooccur by having a different set of integers n0

i that characterise the left and rightcrystals.When the valence–conduction gap occurs at k = π/a instead, because the lowestenergy state is fixed to be even under inversion (to be consistent with the nearlyfree electron model), it will always hold true that n0

even = n0odd + 1. As soon as

the number of occupied bands is specified, n0even is fixed, determining completely

the symmetry labels at k = 0. But more importantly, given the same number ofoccupied bands and the valence–conduction gap at k = π/a, n0

even will remainthe same for any crystal. Thus, in this case, a topological phase transition across

the interface is present, whenever the set of integers nπ/ai , characterising the left

and right crystals, are distinct.Let us further specify how the relevant set of integers can differ across the in-terface. Since in the nearly free electron model each energy gap separates aneven from an odd state, the set of integers can only be different between the leftand right crystals, if at the high symmetry point, where the valence–conductiongap sits, a different (inversion) symmetry label characterises the two crystallinesystems (see fig.(4.3)). This automatically implies that a topological phase tran-sition can be inferred from:

valence–conduction gap at k = 0 : |n0even,L − n0

even, R| = 1

valence–conduction gap at k = πa : |nπ/aeven,L − n

π/aeven,R| = 1

(4.5)

where the L and R subindices refer to the corresponding nki integers for the left

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Figure 4.3: Two cases where the valence–conduction gap sits either at k = π/a(upper half) or k = 0 (lower half). For both left and right crystals, the symmetrylabels, even (e) or odd (o), at the high symmetry points in the reduced Brillouinzone are indicated. The symmetry label at which the valence–conduction gap islocated is highlighted in red. Note that if we exclude this label, there is exactlythe same set of nki integers across the interface.

and right crystals. Indeed, for the equations in (4.5) to hold true, the two crystalsneed to have a different sign for VN , where N refers to the last occupied band.When this is not the case, the relevant set of integers is the same for both crystals,and the equations in (4.5) become equal to zero. Thus, when VN changes signacross the interface, the composite structure goes through a topological phasetransition, with edge states appearing at the interface as a consequence.The previous argument sets the equivalence between the formalism in [5] andthe extension of Maue’s criterion introduced in chapter 3. Nevertheless, we stillwish to analyse how the set of nki labels can be related to Zak’s band labels.To do this, let us first note that, in appendix B, we have established the equiv-alence between Maue’s and Zak’s criterion, both for single semi-infinite crystalsand for composite crystalline structures. This ensures that both methods pre-dict edge state solutions in the same energy gap, and consequently that bothare connected to a topological phase transition. However, by itself this does notprovide a clear criterion that allows for a quick identification of a topologicalphase transition, at least based exclusively in terms of the (q,m) band symme-try labels. To devise such criterion, it is useful to know yet another consequenceof having each energy gap separating an even from an odd state. The result

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is that the (q,m) symmetry labels cannot be assigned randomly to the set ofoccupied bands. This provides restrictions to which band symmetry types canfollow after a particular (q,m) band occurs in the energy spectrum (this is dis-cussed in more detailed in appendix B). These constraints, along with the fixedsymmetry of the lowest energy state, allows us to have information about thesum of the q labels over all occupied bands. In particular, given the k position ofthe valence–conduction gap and a fixed (q,m) symmetry label for the last occu-pied band, this sum turns out to be either 0 (mod a) or a/2 (mod a) (discussedin appendix B), as summarised in table (4.3). To see how this can clarify the

Last (lth) occupied band l even l odd(0,+)

∑l ql = a/2 (mod a)

∑l ql = 0 (mod a)

(0,−)∑l ql = 0 (mod a)


(a/2,+)∑l ql = a/2 (mod a)


(a/2,−)∑l ql = 0 (mod a)

∑l ql = 0 (mod a)

Table 4.3: Summing the symmetry label q of all occupied bands can either beequal to an integer multiple of a (as represented by na, n ∈ N0) or an integermultiple of a/2, depending on the symmetry type of the last occupied band, andwhether it occurs at an l even or l odd position in the energy spectrum. Notethat l even corresponds to having the valence-conduction gap sitting at k = 0,while l odd corresponds to the valence-conduction gap at k = π

a .

relation between the change in the set of integers nki and the band symmetrylabels (q,m), let us consider the cases (A) and (B) in fig.(4.4). (A) and (B)represent, respectively, the last valence band in the left and right crystal, for thetwo possible positions of the valence–conduction gap. In both cases, the left andright crystals are characterised by different sets of nki integers. The different setsoriginate from the distinct symmetry labels that are present at the high symme-try k point, where the valence–conduction gap sits. In fig.(4.4), the compatible(q,m) bands are also indicated. Let us stress that all those choices lead exactlyto the same symmetry label at the corresponding high symmetry k point, wherethe valence-conduction gap sits. For any of the possible choices for the symmetrytype of the valence band, the difference (in absolute value) between the q labelsacross the interface, summed over all occupied bands, is always given by:

∣∣∣∣∣∑l

(q

(L)l − q(R)

l

)∣∣∣∣∣ =a

2(mod a) . (4.6)

In (4.6), the (L) and (R) upper indices refer, respectively, to the q labels forthe left and right crystals. Note that this equation holds true, independently ofwhich of the lattice inversion centers is chosen for the position of the interface,and follows immediately from table (4.3). In the case where Vl stays the same

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Figure 4.4: Possible (q,m) bands that lead to a change in VN across the interface,for the two locations of the valence–conduction gap in the reduced Brillouin zone.Recall that if VN is positive, the even state has lower energy compared to theodd state, while if VN is negative the reverse is true.

across the interface, one can check (using table (4.3)) that (4.6) becomes equalto zero (mod a).We can therefore conclude that, at the interface between crystalline systems,Maue’s and Zak’s criterion predict the existence of topological edge states, as adirect result of the distinct bulk band topologies in the left and right crystals.The equivalence between the formalism in [5], Maue’s and Zak’s criterion is givenin tables (4.4) and (4.5).

Choice of notation T.P.T.

Maue’s notation sign(V(L)N ) = −sign(V

(R)N )

Zak’s notation Abs(∑

l

(q

(L)l − q(R)

l

))= a

2 (mod a)

[5]’s notation Abs (nkeven,L − nkeven, R) = 1

Table 4.4: Summary of all equivalent bulk properties that indicate a topologicalphase transition (T.P.T.) at the interface between two crystalline systems.

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Choice of notation No T.P.T.

Maue’s notation sign(V(L)N ) = sign(V

(R)N )

Zak’s notation Abs(∑

l

(q

(L)l − q(R)

l

))= 0 (mod a)

[5]’s notation Abs (nkeven,L − nkeven, R) = 0

Table 4.5: Summary of all equivalent bulk properties that lead to no topologicalphase transition (No T.P.T.) at the interface between two crystalline systems.

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CHAPTER 5

Full symmetry criterion for 1D crystals

In the previous chapter, the emphasis was placed on the existence of edge states.Zak’s original symmetry criterion made only statements about edge state so-lutions, and we used this criterion to examine their presence at the interfacebetween two crystals. Nevertheless, one of the main conclusions of chapters 2and 3 was that the edge states and the disappearing bulk states always cametogether.To prove the generality of this relation, we will now devise a symmetry crite-rion for disappearing bulk states in single crystals, sharing interfaces with thevacuum. Moreover, we will prove their presence to always be linked to Zak’ssymmetry criterion for edge states.To prove that this is precisely the case in 1D crystals, we need real-valued wavefunctions as the representatives of the bulk states. These can be obtained di-rectly by considering linear combinations of equal energy, but independent, singleBloch states. They are given by ψk(x) and its complex conjugate. From (4.1),it follows immediately that ψ−k(x) = ψ∗k(x). Thus, crystalline bulk states, in agiven (q,m) band, are given by:

φ(q,m)(x) ∼ eiδψ(q,m)k (x) + e−iδψ

(q,m)−k (x) . (5.1)

To guarantee a real wave function, the coefficients involved in (5.1) are necessarilythe complex conjugate of one another and are also, in general, complex-valuednumbers, hence the presence of the phase factor δ. Because their magnitudecan always be fixed a posteriori, and does not contribute to the logarithmicderivative, it is omitted in (5.1).Let us, for instance, consider, as in chapter 2, a crystal bounded by the samefinite potential at x = −Na and at x = Na. If the crystalline wave functionscannot provide for a negative logarithmic derivative at x = Na, then it follows

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Chapter 5. Full symmetry criterion for 1D crystals

that disappearing bulk states must be present.

γ(q,m)(k, xν) =ddxφ

(q,m)(xν)

φ(q,m)(xν)(5.2)

Thus, for predicting the disappearance of bulk states, it is useful to identify acriterion that allows us to preempt the sign of (5.2). Similarly to Zak’s criterion,how the logarithmic derivative behaves will depend on the type of symmetry bandthat is traversed. To clarify how the procedure works, we will revisit two exam-ples from chapter 2: the finite crystal, with interfaces located at diametricallyopposite sides from the middle point of the crystal, and the semi-infinite crystal,emulated from the infinite potential on one side and an interface separating itfrom the vacuum on the other.

5.1 The finite crystal with symmetric interfaces

Let us then consider a finite crystal, with two interfaces located at x = Na andx = −Na. As before, we are only interested in interfaces located at high sym-metry points within the unit cell. This motivates placing the interfaces either atx = ±Na, related to x = 0 by pure translational vectors, or at x = ±Na+ a/2,equally related by pure translations to x = a/2. The periodic potential is as-sumed to be kept the same up to the very end of the last unit cells, as schemat-ically depicted in fig.(5.1). The symmetric location of the interfaces, along with

Figure 5.1: Lattice potential V (x) terminated at the symmetric integer latticepositions x = ±ma. Not only the lattice potential for the infinite crystal hasinversion symmetry, V (x) = V (−x), but now also the finite crystal potential issymmetric with respect to x = 0.

the inversion symmetric potential, allows us to preempt that even and odd (withrespect to the midpoint of the crystal) linear combinations of states exist for(5.1). This restricts the allowed values for δ, just as in chapter 2. There are thentwo possible cases:

1) The midpoint of the crystal is located at x = 0. This means linear com-binations of Bloch states that are either even or odd with respect to x = 0.

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5.1. The finite crystal with symmetric interfaces

2) The midpoint of the crystal is located at x = a/2. This means linear combi-nations of Bloch states that are either even or odd with respect to x = a/2.

Thus, for any (q,m) symmetry band, there are both available even and oddlinear combinations of Bloch states. These are listed in tables (5.1) to (5.4),along with the resulting logarithmic derivative, defined in (5.2).

Band (q,m) φ(Even)xmid=0(k, x) ≡ φ(e)

0 (k, x) γ(q,m)(e,xmid=0)(k, x = Na)

(0,+) 12{ψ

(0,+)k (x) + ψ

(0,+)−k (x)} i ρ(0,+)(k, 0) tan(kNa)

(0,−) 12{ψ

(0,−)k (x)− ψ(0,−)

−k (x)} i ρ(0,−)(k, 0) tan(kNa)

(a/2,+) 12{e

ik a2ψ(a/2,+)k (x) + e−ik

a2ψ

(a/2,+)−k (x)} i ρ(a/2,+)(k, 0) tan(kNa)

(a/2,−) 12{e

ik a2ψ(a/2,−)k (x)− e−ik a2ψ(a/2,−)

−k (x)} i ρ(a/2,−)(k, 0) tan(kNa)

Table 5.1: Linear combination of ψ(q,m)k (x) states, even with respect to the

midpoint x = 0, and corresponding logarithmic derivative γ, calculated at theright interface.

We note that since φxmid(k, x) is either even or odd, the corresponding first

derivative in x is, respectively, also either odd or even, making γ always oddwith respect to the midpoint of the crystal. At the same time, to satisfy theboundary conditions, at the right most interface γ should be negative, while,for the one placed at the left of the crystal’s midpoint, γ is required to be pos-itive. Hence, the problem can be restricted to solving the boundary conditionsfor only one of the interfaces. It is also important to note that now, if there areedge state solutions, they will also be a linear combination of real exponentials,φedge(x) ∼ aekx±be−kx. This is a consequence of having the same finite potentialbounding the crystal at both ends, and because the crystal is finite, extendingover both positive and negative regions of x. As a result, also the existence ofedge state solutions should now be inferred from the sign of γ.Direct inspection of tables (5.1) to (5.4) allows the identification of a commonstructure in γ: being even with respect to xmid implies γ(k, xmid + Na) =iρ(k, xmid) tan(kNa), while being odd leads to γ(k, xmid +Na) = −iρ(k, xmid)cot(kNa). This is a joint consequence of Bloch’s theorem, which allows us toexpress the states φ at x = xmid + Na in terms of the states at x = xmid, and

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Band (q,m) φ(Odd)xmid=0(k, x) ≡ φ(o)

0 (k, x) γ(q,m)(o,xmid=0)(k, x = Na)

(0,+) 12{ψ

(0,+)k (x)− ψ(0,+)

−k (x)} −i ρ(0,+)(k, 0) cot(kNa)

(0,−) 12{ψ

(0,−)k (x) + ψ

(0,−)−k (x)} −i ρ(0,−)(k, 0) cot(kNa)

(a/2,+) 12{e

ik a2ψ(a/2,+)k (x)− e−ik a2ψ(a/2,+)

−k (x)} −i ρ(a/2,+)(k, 0) cot(kNa)

(a/2,−) 12{e

ik a2ψ(a/2,−)k (x) + e−ik

a2ψ

(a/2,−)−k (x)} −i ρ(a/2,−)(k, 0) cot(kNa)

Table 5.2: Linear combination of ψ(q,m)k (x) states, odd with respect to the mid-

point x = 0, and correspnding logarithmic derivative γ, calculated at the rightinterface.

the symmetry properties of the single Bloch states ψ(q,m)k (x) found in [74]. The

logarithmic derivative γ, unlike ρ (the logarithmic derivative for single Blochstates, defined in (4.2)), is now always a real valued function at fixed xν = Na,or xν = Na + a/2, for both bulk states and at k = −iε and k = π/a− iε. Thisis easily seen by recalling that ρ is a purely imaginary quantity, when fixed atthe inversion centres of the lattice and for real k, but it becomes real for k setto either k = −iε or k = π/a − iε, while at these k values the trigonometricfunctions give rise to purely imaginary quantities, which are then compensatedby the extra i in the expressions for γ.From tables (5.1) to (5.4), it is clear that γ can only go to zero or diverge eitherthrough ρ, and thus at the edges of the reduced Brillouin zone (table (4.1)), orthrough the zeros and infinities of the trigonometric functions. In the reducedBrillouin zone 0 < |k| < π/a, and so the range of positive k values accommo-dates N periods of the functions tan(kNa) and cot(kNa). To avoid the zerosand infinities associated to the trigonometric functions, k is replaced by km + ε,with km = π

Nam, m ∈ Z, where ε is then a small shift, in the interval ε ∈]0, πNa [

(see fig.(5.2)), determined by the boundary condition equations. This is a neces-sary procedure, if the boundary conditions are to be satisfied, given that in thevacuum region the logarithmic derivative is always finite.Because now γ is non-zero and non-divergent everywhere, except at the edges ofthe reduced Brillouin zone, it will always be possible to find an ε that allows theboundary conditions to be fulfilled, for any k belonging to the bulk states in theinterior of band. Our previous experience with addressing the same problem inthe framework of the nearly free electron model, allows us to anticipate why the

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Band (q,m) φ(Even)xmid=a/2

(k, x) ≡ φ(e)a/2(k, x) γ

(q,m)(e,xmid=a/2)(k, x = Na+ a/2)

(0,+) 12{e−ik a2ψ

(0,+)k (x) + eik

a2ψ

(0,+)−k (x)} i ρ(0,+)(k, a/2) tan(kNa)

(0,−) 12{e−ik a2ψ

(0,−)k (x)− eik a2ψ(0,−)

−k (x)} i ρ(0,−)(k, a/2) tan(kNa)

(a/2,+) 12{ψ

(a/2,+)k (x) + ψ

(a/2,+)−k (x)} i ρ(a/2,+)(k, a/2) tan(kNa)

(a/2,−) 12{ψ

(a/2,−)k (x)− ψ(a/2,−)

−k (x)} i ρ(a.2,−)(k, a/2) tan(kNa)

Table 5.3: Linear combination of ψ(q,m)k (x) states, even with respect to the

midpoint x = a/2, and correspnding logarithmic derivative γ, calculated at theright interface.

Figure 5.2: Example of an ε shift moving all the k points away from the zerosof tan(kNa), so that the new values form a discrete set for which tan(kNa) isalways finite.

previous statement might not apply for k points near the edges of the Brillouinzone. Let us recall that when ρ→ 0 or ρ→∞, it sometimes happened that thequadrant in which ε would grant the correct sign to γ could no longer harboursolutions due to the presence of indeterminate forms.To illustrate that this is also the case here, let us consider a specific example.Consider a finite crystal, formed by the four well separated bands in fig.(5.3).Let us further assume that the midpoint of the crystal is located at x = 0.The assumption that there are no edge state solutions below the first band,means that, in the gap below the first band, we start with ρ(−iε, 0) > 0. The

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Band (q,m) φ(Odd)xmid=a/2

(k, x) ≡ φ(o)a/2(k, x) γ

(q,m)(o,xmid=a/2)(k, x = Na+ a/2)

(0,+) 12{e−ik a2ψ

(0,+)k (x)− eik a2ψ(0,+)

−k (x)} −i ρ(0,+)(k, a/2) cot(kNa)

(0,−) 12{e−ik a2ψ

(0,−)k (x) + eik

a2ψ

(0,−)−k (x)} −i ρ(0,−)(k, a/2) cot(kNa)

(a/2,+) 12{ψ

(a/2,+)k (x)− ψ(a/2,+)

−k (x)} −i ρ(a/2,+)(k, a/2) cot(kNa)

(a/2,−) 12{ψ

(a/2,−)k (x) + ψ

(a/2,−)−k (x)} −i ρ(a/2,−)(k, a/2) cot(kNa)

Table 5.4: Linear combination of ψ(q,m)k (x) states, odd with respect to the mid-

point x = a/2, and correspnding logarithmic derivative γ, calculated at the rightinterface.

analycity of ψ(q,m)k (x) and the fact that ρ(0,+)(0, 0) = 0 (table (4.1)) provides

ρ(0,+)(ε, 0) ∼ iε⇒ iρ(0,+)(ε, 0) < 0. As a consequence, the even state in the (0,+)

band, close to k = 0, has at the right interface γ(0,+)(e,xmid=0)(ε,Na) ∼ −ε tan(εNa).

Given the negative sign of the logarithmic derivative in the region extendingover x > Na, the boundary condition can only be met if tan(εNa) > 0. Thisrequires choosing ε in the first quadrant, i.e. ε ∈ ]0, π

2Na [. Near k = π/a,

ρ(0,+)(π/a − ε, 0) ∼ iε ⇒ iρ(0,+)(ε, 0) < 0, and thus γ(0,+)(e,xmid=0)(π/a − ε,Na) ∼

ε tan(εNa), which means that the boundary condition can be satisfied by takingε to be in the second quadrant. We can, therefore, conclude that for an even

state φ(e)0 (k, x), in a (0,+) band, there is always a solution for ε guaranteeing

that the boundary conditions are satisfied. The conclusion is general, and holdseven if (0,+) would not be the lowest energy band.For the odd states in this same band, the corresponding trigonometric functionis multiplied by the term −iρ(0,+)(ε, 0) > 0, and hence we need cot(εNa) < 0,

seeing that γ(0,+)(o,xmid=0)(ε,Na) ∼ ε cot(εNa) (table (5.2)). This requires choosing

a shift ε in the second quadrant. As we move close to k = π/a, the sign of

−iρ(0,+)(k, 0) is fixed and thus γ(0,+)(o,xmid=0)(π/a− ε,Na) ∼ −ε cot(εNa). In order

for γ to be negative, we would need to restrain the domain of suitable solutionsfor ε to the first quadrant, but this is precisely where we now run into trouble:when ε→ 0, we find ε cot(εNa)→ 0×∞, our expected cause behind the inabilityto satisfy the boundary conditions. The indeterminate form can be lifted usingL’Hopital’s rule, after which we obtain limε→0 ε cot(εNa) = 1

Na . Because N is to

128


Figure 5.3: Particular choice of bands in a crystal. The red crosses indicate thatnear k = π/a, two propagating Bloch solutions disappear: an odd state withrespect to x = 0 in the (0,+) and an even state in the (0,−) band. In the gapenclosed by these bands, γ can satisfy the boundary conditions, and hence edgesate solutions are present.

be taken in the macroscopic limit, it is impossible to match γ with the vacuum’s

logarithmic derivative. Consequently φ(o)0 (π/a − ε, x) is not a suitable solution

in the (0,+) band, and cannot feature among the allowed states in the crystal.If the (0,+) band was not the lowest energy band, there would be no reason toconstrain the sign of ρ(0,+)(ε, 0) to be positive. Taking it to be negative, wouldonly change the location of the indeterminate form, which would now occur forstates near k = 0, leading to the disappearing state to arise there.

In the gap, ρ(π/a− iε) ∼ −ε, while cot(πN − iεNa) = i coth(εNa), resulting inγ(o,xmid=0)(π/a− iε,Na) ∼ −ε coth(εNa), for which the boundary condition canbe satisfied: the gap near k = π/a can harbor edge state solutions, in accordancewith the existence of an (odd) disappearing state.

Proceeding in this way, one can prove that, at the bottom of the (0,−) band,one even bulk state disappears, while all odd states in that same band satisfy theboundary conditions. Likewise, in the gap above k = 0, near the (0,−) band,

129


γ is a positive valued function (for both even and odd solutions), and hence noedge states will occur in this gap. For the remaining (q,m) bands, the sameprocedure will allow us to conclude that there are no more disappearing Blochsolutions or edge states (the main conclusions are summarised in fig.(5.3)).These results are in agreement with the ones obtained from the nearly free elec-tron model: the single Bloch state in the top of the (0,+) band is even underinversion (see appendix B), while the one at the bottom of the (0,−) is odd; thismeans that the gap was opened with V1 > 0, for which Maue predicts two disap-pearing bulk states [66, 67]. These are precisely the even state (with respect toxmid = 0), at the bottom of the upper band (here the (0,−) band), and the oddstate (with respect to xmid = 0), at the top of the lower band (here the (0,+)band). The remaining gaps are open by V2 and V3 negative. This is impliedby the presence of a single Bloch state at the top of the lower band that is oddunder inversion, and a single Bloch state at the bottom of the upper band that iseven under inversion, in the neighbourhood of both energy gaps. It then followsfrom Maue’s criterion that no edge states, as well as no disappearing bulk states,are present.

5.2 The more general proofWe now prove, in more general terms, that bulk disappearing states are insepa-rable from the appearance of edge states. Strictly speaking the proof is immedi-ately applicable to the finite crystal with symmetric interfaces, but also carriesthrough to the left semi-infinite crystal (using the same setup as in chapter 2),when only direct hybridisation gaps are considered. We note that, under thesame condition, if one does not demand a negative logarithmic derivative in thegap, similar arguments can also be applied to the right semi-infinite crystal.

Our general argument relies on proving the following:

1. First we show that if γ hosts a crucial indeterminate form at the edges ofthe Brillouin zone, ρ is forced to be negative in the gap.

2. Second, we verify that having γ negative in the gap is equivalent to ρ beingnegative in-gap.

3. Third, we prove that ρ cannot be negative in the gap (and, hence, alsoγ), if there are no crucial indeterminate forms present for the bulk states,either coming from the even or the odd states.

This will settle the emergence of edge states to always be linked to the disap-pearing bulk states, and will explicitly reveal the nature of this bulk-boundaryrelation to be deeply rooted to the band symmetry labels. Since these are con-nected to the classification in [5], this bulk-boundary relation can be seen as aconsequence of the (symmetry-based) topology of the bulk bands.

130

5.2. The more general proof

We start by identifying how the crucial (sign wise) indeterminate forms appearin γ. From tables (5.1) to (5.4), these occur for even and odd states when:

Even states: γ ∼ −1

εtan(εNa), occurs for ρ→∞ (5.3)

Odd states: γ ∼ −ε cot(εNa), occurs for ρ→ 0. (5.4)

These are the only sign-wise relevant indeterminate forms, since they are theonly possible ways to constrain the range of negative images to a sub-intervalthat excludes the values taken by the vacuum’s logarithmic derivative. Preciselybecause there is a limited number of ways to achieve this, not all (q,m) bandscan host them. For instance, if xmid = 0, indeterminate forms, linked to evenbulk states, cannot occur at a (0,+) band, since ρ never diverges there.

We now show that crucial indeterminate forms imply a negative sign for ρ inthe nearby gap (step 1). We will prove this explicitly for odd states, since theproof for even states follows very similar steps and gives rise to the same conclu-sion. For the bulk odd states, γ = −iρ cot(kNa),

near k = 0: γ ∼ −ε cot(εNa) =⇒ ρ(ε, xmid) ∼ −iε (5.5)

near k = π/a: γ ∼ −ε cot(εNa) =⇒ ρ(π/a− ε, xmid) ∼ iε. (5.6)

But this means that in the nearby gap, ρ is given by:

near k = 0: ρ(−iε, xmid) ∼ −ε (5.7)

near k = π/a: ρ(π/a− iε, xmid) ∼ −ε (5.8)

Hence, we find ρ < 0. Thus, if there exists indeterminate forms for which theboundary conditions cannot be satisfied (cases (5.3) and (5.4)), these automat-ically imply a negative logarithmic derivative in the gap, for the single Blochstates. In fact, this also means that γ is negative in the gap. This follows fromthe definition of γ in tables (5.1) to (5.4), and the fact that, in 1D crystals, thegaps are always located near k = 0 and k = π/a (in the reduced Brillouin zonescheme). This can be seen immediately by looking at the explicit expressions forγ in the gap, without a priori assuming any particular sign for ρ:

131


iρ×

{tan(−iεNa)

tan((π/a− iε)Na)= ρ tanh(εNa)

−iρ×

{cot(−iεNa)

cot((π/a− iε)Na)= ρ coth(εNa) .

Thus, ρ negative is equivalent to also having a negative sign for γ in the gap(this concludes step 2). The only remaining thing to check is if it is possible forρ to be negative in the gap, without there being any crucial indeterminate formsoccurring at the edges of the Brillouin zone. In the gap where ρ < 0:

ρ(−iε, xmid) < 0︸︷︷︸in the gap close to k = 0

⇒

ρ(ε, xmid) ∼ i

ε =⇒ ρ(0, xmid)→∞

ρ(ε, xmid) ∼ −iε =⇒ ρ(0, xmid)→ 0

(5.9)

ρ(π/a− iε, xmid) < 0︸︷︷︸in the gap close to k = π/a

⇒

ρ(π/a− ε, xmid) ∼ −iε =⇒ ρ(π/a, xmid)→∞

ρ(π/a− ε, xmid) ∼ iε =⇒ ρ(π/a, xmid)→ 0

(5.10)

If then ρ ∼ ± iε , no crucial indeterminate forms exist for odd bulk states, since

these require ρ → 0 (see eq.(5.4)). However, for even bulk states (where γ =iρ tan(kNa)),

ρ(ε, xmid) ∼ i

ε=⇒ γ ∼ −1

εtan(εNa) (5.11)

ρ(π/a− ε, xmid) ∼ − iε

=⇒ γ ∼ 1

εtan(Nπ− εNa) = −1

εtan(εNa) , (5.12)

we are left with the crucial indeterminate forms in (5.3). On the other hand,when ρ ∼ ±iε, no indeterminate forms are possible for even bulk states, becausethey are associate with ρ→∞ (see eq.(5.3)), but these are present for the bulkodd states (for which γ = −iρ cot(kNa)):

ρ(ε, xmid) ∼ −iε =⇒ γ ∼ −ε cot(εNa) (5.13)

132

5.3. The left semi-infinite crystal

ρ(π/a− ε, xmid) ∼ −iε =⇒ γ ∼ ε cot(Nπ − εNa) = −ε cot(εNa) . (5.14)

If ρ is then negative in the gap, this implies that a nearby even or odd bulk statewill hold a crucial indeterminate form. Recall that, for example, the existence ofedge state solutions in the finite crystal with symmetric interfaces is dependenton γ < 0 in the gap, which, as we have seen, coincides with ρ itself being negativethere. Yet, ρ cannot be negative without that implying either the even or oddbulk states to hold a crucial indeterminate form in the nearby k region. There-fore, the existence of edge states is inseparable from the presence of disappearingbulk states. Furthermore, as we have now established, the criterion predictingthese two phenomena can be restricted to inspecting the sign of ρ, which can

be directly probed from the single Bloch states ψ(q,l)k (x) and can be examined

even in a crystal with periodic boundary conditions. Thus, the sign of ρ providesa bulk-boundary correspondence for the 1D crystalline systems with inversionsymmetry.We note that this bulk-boundary relation is dependent only on general require-ments, in particularly the symmetry and the analyticity of the Bloch wave func-tions, and is thus model-independent.

5.3 The left semi-infinite crystal

In chapter 2, we addressed the case of a left semi-infinite crystal starting froma finite left crystal, bounded at x = Na by an infinite potential, and at xν bya finite and constant potential. The finite potential represented the vacuum,extending over the region x < xν . The spectrum of the ideal semi-infinite crystalwas then examined by analysing how the solutions of the boundary conditionequations changed as N increased.In the nearly free electron model, one edge state and one bulk disappearingstate were predicted. We now revisit this same problem using our full symmetrycriterion, and show that the same result is obtained. For this aim, we will makeuse of the conclusions drawn in the previous sections, which will allows us toreach our goal with fewer steps.Let us then start by setting xν = 0 (and the other boundary at x = −Na) andcomment later on the effect of choosing xν = a/2 instead. We consider thatfor both xν = 0 and xν = a/2, the crystal’s length (with N finite) stays thesame, meaning that when xν = a/2, the infinite potential bounding the crystalis shifted to x = −Na+a/2. The asymmetry between the potential at x = −Na(x = −Na + a/2) and the one at x = 0 (x = a/2) dictates also an asymmetry

133


between the boundary conditions at these two points:

Boundary conditions:

at x = −Na, φ(x) = 0 (a)

at x = 0, φ′(x)φ(x) = −

√2m~2 |E| (b)

(5.15)

with φ(x) given by (5.1). The boundary condition (a) is responsible for fixing thephase factor δ in (5.1), for each (q,m) band separately. The resulting crystallinewave functions are listed in table (5.5).

Band (q,m) φ(q,m)xν=0(k, x)

(0,+) i2{e

ikNaψ(0,+)k (x)− e−ikNaψ(0,+)

−k (x)}

(0,−) 12{e

ikNaψ(0,−)k (x) + e−ikNaψ

(0,−)−k (x)}

(a/2,+) i2{e

ika(N+1/2)ψ(a/2,+)k (x)− e−ika(N+1/2)ψ

(a/2,+)−k (x)}

(a/2,−) 12{e

ika(N+1/2)ψ(a/2,−)k (x) + e−ika(N+1/2)ψ

(a/2,−)−k (x)}

Table 5.5: Bloch solutions belonging to a given (q,m) symmetric band.

From tables (5.5) and (5.6) it is immediately obvious that for all (q,m) bands,the logarithmic derivative at x = 0 has the common structure:

γ(q,m)(xν=0)(k, 0) = −iρ(q,m)(k, 0) cot(kNa) , (5.16)

and is thus identical to the logarithmic derivative for odd bulk states in the finitecrystal bounded by symmetric interfaces. This can be understood in the follow-

ing way: the phase factors e±ikNa shift ψ(q,m)±k (x) to ψ

(q,m)±k (x + Na), through

Bloch’s theorem. This means that at x = 0, apart from a constant pre-factor,

the wave functions in (5.5) are equivalent to φ(Odd)xmid=0(k,Na), where x = Na was

precisely the location of the right interface that separated the crystal from thevacuum in the previous problem. Because multiplying the wave function by anyconstant pre-factor does not change γ, the two logarithmic derivatives, at theircorresponding interfaces, are exactly the same.

134

5.3. The left semi-infinite crystal

Band (q,m) γ(q,m)(xν=0)(k, 0)

(0,+) −i ρ(0,+)(k, 0) cot(kNa)

(0,−) −i ρ(0,−)(k, 0) cot(kNa)

(a/2,+) −i ρ(a/2,+)(k, 0) cot(kNa)

(a/2,−) −i ρ(a.2,−)(k, 0) cot(kNa)

Table 5.6: Logarithmic derivative γ(q,m)(xν=0)(k, x = 0) belonging to a given (q,m)

symmetric band.

The similarity between the two logarithmic derivatives is an instructive obser-vation, in that it immediately allows us to use the results from the previoussections. For example, if the crystal is characterised by the same set of bandsas in fig.(5.3), using our results from section 5.1, we automatically know thatthere will be only one disappearing state, occurring near k = π/a, in the (0,−)band, with an edge state solution in the nearby gap, and nothing else. This isin agreement with Maue’s criterion, since, within the nearly free electron model,only the first gap, near k = π/a, would be opened with Vl > 0 (the remainingimply Vl < 0, see appendix B), and thus only there would both edge states andbulk disappearing solutions be expected.Choosing the interface to be at xν = a/2 instead, implies evaluating the bound-ary condition (a) at x = −Na + a/2. The resulting equation can, nevertheless,be obtained from the previous one (at x = −Na) by taking N → N − 1/2. Thecorresponding crystalline wave functions follow then from table (5.5), by makingthe same substitution. This leads us to wave functions that are the same asφ

(Odd)xmid=a/2

(k,Na), and thus (5.16) can even more generally be written as:

γ(q,m)(k, xν) = −iρ(q,m)(k, xν) cot(kNa) . (5.17)

Given now the set of bands in fig.(5.3), the bulk disappearing state would occurnear k = π/a, in the (a/2,+) band (from (5.4) and (5.10)), with the edge statesolution in the nearby gap, enclosed by the (a/2,+) and (a/2,−) bands. Thus,just as for the finite crystal with symmetric interfaces, it is clear from (5.17)that the existence of disappearing bulk states and of edge state solutions reliessolely in the sign of ρ(q,m)(k, xν), making the two inherently dependent on thesymmetry constraints imposed by the lattice symmetries.

135

CHAPTER 6

Conclusions and outlook

6.1 Conclusions

In examining the origin of edge states at the interface of 1D crystals, symmetryhas shown to be the determinant factor. This is due to the constraints thatit imposes onto the logarithmic derivative, and thus to the ability of smoothlymatching the wave functions at the interface. Additionally, it is also symmetry,or better yet a change of the symmetry representations at the interface, thatallows the link between the presence of edge states and band topology. Indeed,we have shown that these two concepts, on the one hand the existence of edgestates, on the other the topological classification in [5], are connected for 1Dcrystals. The presence of edge states is signalled by the change at the boundaryof the number of occupied states that transform according to a given irreduciblerepresentation. This can equivalently be restated in terms of the q symmetryband label. In inversion symmetric systems, it has been explicitly shown thatthe Berry phase associated to a particular band is directly linked to its q label[96]. This immediately provides a relation between the existence of edge statesand having two sets of occupied bands that are characterised by distinct Berryphases. In the context of the nearly free electron model, this can also be restatedas a change in the sign of Vl across the interface. Thus, we see that in hindsightMaue’s and Schockley’s surface states have always been topological edge states.

The observation that no (symmetry preserving) transformation can exist betweenstates that transform according to different irreducible representations justifiesthe systematic encounter of disappearing Bloch solutions, whenever symmetrylabels changed across the interface. We have proven that the same symmetryconstraints that force the logarithmic derivative to render edge state solutionsare also responsible for forbidding Bloch states at the edges of the Brillouin zone.

137

Chapter 6. Conclusions and outlook

Therefore, if the edge states are always a cause of disrupting symmetry repre-sentations at the interface, it follows that these and bulk disappearing states arenecessarily tied together. Moreover, the symmetry conditions can be verified ina truly bulk crystal, with periodic boundary conditions. We thus find a bulk-boundary relation for 1D crystals characterised only by symmetry labels, eventhough the existence of such a correspondence is not a priori guaranteed by anytype of effective description in terms of a topological field theory.

6.2 Outlook: towards higher dimensions

In the study of edge states and bulk disappearing states, predicting the sign ofthe logarithmic derivative was highly simplified by the knowledge of how theWannier functions behaved under inversion. For the 1D crystals, Konh’s workon the analytic properties of the Bloch wave functions allowed us to use theknowledge that the Bloch states, belonging to a given energy band, all shared aunique Wannier function a(q,l)(x), with a precise symmetry under inversion [99].This symmetry was specified with respect to one of the two distinct inversioncentres of the crystal, q = 0 or q = a

2 , and allowed one to make statementsabout the sign of the logarithmic derivative for entire sets of bands of given (q, l)symmetry. Moreover, this was accomplished without ever having to specify theparticular form of the Bloch wave functions.

The previous approach contrasts with the one taken with the nearly free elec-tron model. There, although the procedure was much more straightforward, thecomputational effort required was considerably bigger. Thus, if one intends toexplore the interplay between the existence of edge states and the lattice sym-metries for crystals in 2D and 3D, looking for an extension of Zak’s symmetrycriterion is preferred. This choice of method can be further motivated by ar-guing that if the goal is to find edge state solutions that are protected by thelattice symmetries, then the behaviour of the logarithmic derivative should beconstrained by symmetry. If this is not the case, then the edge states need to beindependent of the symmetry properties of the bulk, and having, for example,an interface across which the band symmetry labels are distinct cannot have anyimpact on the conditions of existence of the edge states.

To move beyond 1D crystals, one then needs first to find all the distinct (q, l)-likelabels that correspond to a crystal with a given space group symmetry. This canbe done using the formalism of band representations for space groups developedby Zak [92]. One of the main features of this formalism is that, given a spacegroup G, it provides a procedure to obtain all its irreducible representationsexpressed in the basis of real-space localised functions. If in addition, the re-quirement is imposed that such basis functions form an orthonormal basis, thenthey are the Wannier functions. These sets of functions, and consequently therepresentations that are built from them, carry precisely the higher-dimensional

138

6.2. Outlook: towards higher dimensions

symmetry labels akin to the (q, l) labels, and hence determine how many bandswith different symmetry properties there can be in a crystal. These labels areintimately related to the type of point symmetries that characterise the spacegroup of a crystal. To highlight precisely this, let us consider the definition ofa lattice in accordance to Des Cloizeaux [100]: after choosing an origin in realspace, the lattice is obtained from it by applying all the transformation elements,belonging to the space group G. The lattice is then the collection of all pointsgenerated in this way. For example, for the 1D inversion-symmetric crystal, pick-ing the origin to be at x = ma, with m an integer, the infinite chain of atomscan be obtained by translating the point over any integer times a. Yet the samepattern could as well be obtained by picking the origin at, for example, x = a/2.The two choices of origin are distinct from a symmetry point of view, since thereis no element of the group G that can connect the two. From the above definition,they form two distinct symmetry lattices, and they will be generated by differ-ent sets of Wannier functions. If there is only one atom per unit cell, the first(corresponding to the lattice with origin at x = 0) has Wannier functions thatare atomic orbitals, while the second lattice is described by Wannier functionsthat result from orbital hybridisation [100]. Not incidentally, x = 0 and x = a/2are the two Wyckoff positions for the 1D crystal with inversion symmetry. Thismotivates why in Zak’s band representation formalism one starts by identifyingthe symmetry centres (the Wyckoff positions) in the conventional cell [92].

For every symmetry centre, labeled by q, the group Gq is constructed. Thisis the group formed by all the elements of G that leave the point q invariant, upto translations by a Bravais lattice vector. Consequently, for each of these sym-metry centres, the irreducible representations carry not only the q label, whichidentifies the corresponding Wyckoff position, but also a label l that distinguishesthe irreducible representations of Gq from one another. From the knowledge ofall the irreducible representations of the subgroups Gq’s, the band representa-tions of the full space group can be obtained by the induction method. It isfor this reason that, in the band representation formalism, the space group canbe though of as a collection of symmetry lattices, constructed from the finitenumber of symmetry centres [92] in the conventional cell.

After this procedure, one has not only figured out how many distinct symmetrytypes of bands there are, but also how the Wannier functions transform underthe symmetry operations of G, for every (q, l) representation. What symmetryconstraints this imposes onto the Bloch wave functions, along a given band, canbe obtained from the usual relation between Bloch and Wannier functions. Fromthis knowledge we can proceed in a similar way as done in the 1D case, and in-spect what these constraints imply for the sign of the logarithmic derivative. Inthis way, we hope to devise the symmetry criterion that allows for predictingedge states in 2D and 3D crystal structures and simultaneously relate it to thetopological classification in [5].

139

APPENDIX B

Part II

B.1 Dictionary between Maue’s and Zak’s crite-rion for edge state solutions

As discussed before, a direct link exists between the sign of Vl and the symmetrylabels attributed to the high symmetry points in the reduced Brillouin zone. Wewish to find what band symmetry types are compatible with the two possiblechoices for the sign of Vl in a given energy gap, and check that Maue’s criterionagrees systematically with Zak’s procedure for determining the presence of edgestate solutions. To refer to the band symmetry type, we will use Zak’s notation.

(q,m) k = 0 k = π/a(0,+) Even Even(0,−) Odd Odd

(a/2,+) Even Odd(a/2,−) Odd Even

Table B.1: (q,m) labels the band symmetry type: q indicates the inversioncentre for which the corresponding Wannier function is either even, label +, orodd, label −. The symmetry character adopted by the four types of bands atthe high symmetry points in the reduced Brillouin zone, k = 0 and k = π

a , isalso given.

To label the energy bands in ascending order in energy, in the reduced Brillouinzone scheme, the index l is used. In this way, larger values of l refer to higherenergy bands, and l = 1 corresponds to the lowest energy band. In the nearly freeelectron model, a (l− 1)th and a lth band are separated from one another by an

141

Appendix B. Part II

energy gap. We label these energy gaps with the same index as the band below,which means that in the previous case the two bands would be separated by the(l− 1)th gap, whose size depends on the value of |Vl−1|. As seen previously, thesign of Vl establishes whether the symmetry labels characterising the states atthe top of the (l− 1)th band, and the bottom of the lth band, are either even orodd (see fig.(B.1)).

Figure B.1: Even or odd symmetry labels characterising the bands, at the highsymmetry points in the reduced Brillouin zone, as imposed by the sign of Vl.The cases where l is an odd number, l = 2n−1, or an even number, l = 2n, withn = {1, 2, ...}, are represented separately.

Since the gaps are labeled with the same index l as the band below, from fig.(B.1)it is clear that Vl−1 induces the gap at k = 0, if l is even, and at k = π/a, ifl is odd. With this in mind, we can then determine what combination of signsfor Vl−1 and Vl are needed in order for the lth band to be characterised by aparticular (q,m) symmetry label (see table (B.2)). We note that the case l = 1 isexcluded from table (B.2). This is because in the nearly free electron model thestate at k = 0 in the lowest energy band is well approximated by a single planewave, which then fixes the symmetry label there to be even under inversion.Consequently, the band label (q,m), when l = 1, is fully determined by the signof V1: a (0,+) type, for V1 > 0, or (a/2,+), for V1 < 0. That these are the onlytwo possibilities in this case, follows immediately from table (B.1).

Direct inspection of table (B.2) allows us to conclude that bands cannot followeach other in a random order, since the sign of Vl, corresponding to a given (q,m)band, fixes the sign of Vl−1 for the next band (and so on). Consequently, given

142

B.1. Dictionary between Maue’s and Zak’s criterion for edge statesolutions

(q,m) l 6= 1 and odd l even(0,+) k = 0 : Vl−1 < 0 & k = π

a : Vl > 0 k = πa : Vl−1 < 0 & k = 0 : Vl > 0

(0,−) k = 0 : Vl−1 > 0 & k = πa : Vl < 0 k = π

a : Vl−1 > 0 & k = 0 : Vl < 0(a/2,+) k = 0 : Vl−1 < 0 & k = π

a : Vl < 0 k = πa : Vl−1 > 0 & k = 0 : Vl > 0

(a/2,−) k = 0 : Vl−1 > 0 & k = πa : Vl > 0 k = π

a : Vl−1 < 0 & k = 0 : Vl < 0

Table B.2: A given lth band, characterised by the symmetry label (q,m), isdefined by a precise combination of signs for Vl−1 and Vl, and the distinctionbetween l even or odd, is just a matter of deciding at which of the two highsymmetry k points are the (l − 1)th and the lth gap located.

a particular (q,m) band, only two symmetry types can follow. In particular,depending on whether the index l is even or odd, a given (q,m) band is followedby:{

l even: (qi,ml) =⇒ l + 1 : (qi,ml′) or (qj ,ml′), with i 6= j, l′ 6= l

l odd: (qi,ml) =⇒ l + 1 : (qi,ml′) or (qj ,ml), with i 6= j, l′ 6= l

(B.1)

This is schematically depicted in figs.(B.2) and (B.3). In fig.(B.2) the linesindicate which bands can proceed a (q,m) band in a l odd (and l 6= 1) position,while in fig.(B.3) the same is represented, but now l corresponds to an evenposition in the energy spectrum.

Figure B.2: Schematic representation for the symmetry type of bands that areallowed to follow immediately after a particular (q,m) band occurs in an l oddposition in the energy spectrum. For example, after a (0,+), the next band canonly be a (0,−) or a (a/2,+) symmetry type.

The fact that the freedom in ordering the symmetry types of bands through theenergy spectrum is constrained turns out to be an important point when it comesto showing the equivalence between Maue’s and Zak’s criterion. In fact, for aleft semi-infinite crystal, with an interface set at x = 0, the only possible waysfor which, in principle, Zak’s symmetry procedure could contradict Maue’s rulewould be if the cases in fig.(B.4) occurred. Recall that the label (q,m) is fixed,once we know the sign of both Vl−1 and Vl (table (B.2)); however, to predict the

143

Appendix B. Part II

Figure B.3: Schematic representation for the symmetry type of bands that areallowed to follow immediately after a particular (q,m) band occurs in an l evenposition in the energy spectrum. For example, after a (0,+), the next band canonly be a (0,−) or a (a/2,−) symmetry type.

Figure B.4: The possible mismatches between Maue’s and Zak’s criterion areillustrated by the cases (A1) to (D2), for a left semi-infinite crystal with theboundary fixed at x = 0. The logarithmic derivative’s sign in the gap, whichshould be negative if edge state solutions are to occur, is highlighted in red foreach case. In all these cases, a + sign in the lth gap is associated to Vl > 0,while a − sign in the lth gap occurs with Vl < 0. The lth gap is located, in (A1)to (D2), near the identified k point in the figure.

logarithmic’s derivative sign in the gap with Zak’s criterion, we need to knowthe symmetry type of the previous bands. Hence, if more than one band isfilled, we could imagine arriving to the last occupied band with the logarithmicderivative’s signs as given in fig.(B.4). In all the illustrated cases, the sign ofthe logarithmic derivative and the sign of Vl lead to contradicting conclusions:

144


when one predicts the existence of edge state solutions, the other does not. Ouraim is now to show that these configurations are not allowed in the nearly freeelectron model. Before we do that, we note that the analogous situation for aright crystal can be obtained from fig.(B.4) by reversing the signs bellow eachband. In addition, since what makes these cases ”problematic” is precisely thestarting sign, our conclusions will also hold in the case where the interface sitsat x = a/2.Let us then consider for example case (B2) in fig.(B.4), where the last occupiedband is a (0,−) type and sits in a l even position.

Figure B.5: Fixing the band in lth position (l even) to be a (0,−) symmetrytype, rule (B.1) allows us to deduce which band symmetry types are allowed tocome before. In the figure, four iterations towards the first band are shown. Byproceeding in this way, one can also deduce whether the sum of all q labels, overall occupied bands, is an integer multiple of a or of a/2.

By applying consecutively the rules in (B.1), we can deduce what are the sym-metry types of bands that allow for a (0,−) band to occupy the lth position.This is illustrated, up to four iterations, in fig.(B.5). Since the interface is fixedat x = 0, every band label with q = 0 reverses the sign of the logarithmic deriva-tive in the gap, while the bands labeled by q = a/2 leave the sign unchanged.Furthermore, the starting sign bellow the first band (l = 1) is fixed to be pos-itive in the nearly free electron model, with the first band itself constrained tobe a (0,+) or an (a/2,+) symmetry type. Thus, in order to arrive to the last(0,−) band with a positive sign for the logarithmic derivative in the (l − 1)thgap, we would need the bands before to contain an even number of symmetry

145

Appendix B. Part II

types labeled by q = 0. If we now go back to fig.(B.5), due to (B.1), by thetime we are in (l − 4)th iteration, all the available choices for the bands thereare exactly the same as the ones already present in the (l− 2) iteration, and thesame would be true (for both even and odd iterations) if we kept iterating back.Thus, even if the number of occupied bands (call it N) is bigger than four, therepetition pattern allows us to conclude that we cannot arrive to the last (0,−)band, without going through an odd number of bands labeled by q = 0, which,regardless of exactly how many they are, guarantees that in the gap below thislast band the sign is forced to be negative.As an example, consider the case where the last (0,−) band sits at l = 4 in theenergy spectrum. Since by our definition, the lowest energy band is identified byl = 1, iteration l− 3 in fig.(B.5) is our starting point. There we find all possiblepaths that allow us to reach the (0,−) band at the l = 4 position. Because in thenearly free electron model the first band can only be a (0,+) or a (a/2,+) type,this narrows down our possible paths to: (0,+) → (0,−) → (0,+) → (0,−),(a/2,+) → (a/2,−) → (0,+) → (0,−), (a/2,+) → (0,+) → (a/2,−) → (0,−)and (0,+) → (a/2,+) → (a/2,−) → (0,−). In all these paths, just before wereach the last (0,−) band, we can see that we always have an odd number ofbands with q = 0, forcing the sign of the logarithmic derivative in the gap be-low the (0,−) band to always be negative there. Case (B2) cannot, therefore,occur, and so the agreement between Zak’s and Maue’s criterion is preserved.Although, we will not work out the remaining cases in fig.(B.3), one can deducethat they will not be allowed in the nearly free electron model by proceedingin the exact same way as we did for case (B2). In more general terms, whatprevents the ”problematic” cases to exist is the combination of rules (B.1) withthe fact that the first band is restricted to either be a (0,+) or a (a/2,+).

A similar analysis is also what allows us to obtain the results in table (B.3).For instance, in fig.(B.5) because we cannot arrive at the (0,−) band in the lthposition without before having an odd number of occupied bands labelled byq = 0, if we sum the q labels of all occupied bands (which now includes the last(0,−) band) we will necessarily obtain as a result an integer multiple of a. Thisinformation is particularly useful when comparing the change in symmetry labelsacross an interface, separating two different crystalline systems. When the aimis to analyse the presence of edge state solutions in these composite structures,the knowledge of the incompatible sign configurations in fig.(B.4) is also usefulto establish a one-to-one correspondence between Zak’s symmetry criterion andthe rule that Vl should change sign across the interface. The correspondence canbe easily seen by listing all the compatible sign configurations for each distinctpair of bands. This is done in fig.(B.6), for the case where the lth gap is fixedat k = 0 (at k = π/a the procedure and conclusions are exactly the same). Asone can immediately check, only when the sign of Vl is different in both crystals,will Zak’s symmetry criterion predict the existence of edge state solutions at theinterface. We note that these results, including the information in table (B.3),

146


Figure B.6: All possible (q,m) pair of bands (excluding permutations of theleft and right crystals) that respect the rules in (B.1) and the initial constraintsimposed by the nearly free electron model, i.e. the sign in the gap bellow thefirst band is fixed as well as the symmetry of the lowest energy state. In the lthgap, the logarithmic derivatives of left and right crystals have matching signsonly when the sign of Vl changes across the interface.

have been used in chapter 4, in the section regarding the connection with topo-logical invariants.

Our previous analysis establishes the equivalence between Zak’s and Maue’s cri-terion, for all possible band structures in the nearly free electron model. We notethat these conclusions carry immediately through to any other model where noedge states occur below the first band, only direct hybridisation gaps are present,and the lowest energy state is even under inversion.To provide further clarity regarding the equivalence between Zak’s and Maue’scriterion, let us now consider three specific examples. As a first example, con-sider the sign of Vl to consecutively alternate between positive and negative, suchthat Vl−1 = −Vl. Starting with the case where Vl < 0, for l even, and Vl > 0, forl odd, the only allowed bands are a (0,+) type, every time l is an odd number,and a (0,−) type for every even l. This can be understood as follows: given

147

Appendix B. Part II

Last (lth) occupied band l even l odd(0,+)


∑l ql = 0 (mod a)

(0,−)∑l ql = 0 (mod a)


(a/2,+)∑l ql = a/2 (mod a)


(a/2,−)∑l ql = 0 (mod a)

∑l ql = 0 (mod a)

Table B.3: Summing the symmetry label q of all occupied bands can either beequal to an integer multiple of a or an integer multiple of a/2, depending on thesymmetry type of the last occupied band, and whether it occurs at an l even orl odd position in the energy spectrum.

V1 > 0, the first band is automatically (0,+) type, while for other bands labeledby l odd, since Vl−1 will be negative and Vl will be positive, the symmetry labelis forced to also be a (0,+) type (see (B.2)); on the other hand, bands labeled byl even are preceded by Vl−1 > 0 and followed by Vl < 0, and thus, according to(B.2), they are bound to be a (0,−) type. If these set of bands characterise theenergy spectrum of a left semi-infinite crystal (defined over the region x < 0),with the interface at x = 0, then Maue’s criterion predicts edge state solutionsat every gap where l corresponds to an odd number. In fig.(B.7), we compareit with Zak’s symmetry procedure, and conclude that the two are in agreement.Recall that, in order for an edge state solution to satisfy the boundary conditionswith the vacuum, a negative sign for the logarithmic derivative is needed for ingap states. If instead Vl > 0, for l even, and Vl < 0, for l odd, the first bandwill be a (a/2,+) type, due to V1 < 0, while the remaining bands will alternatebetween (0,+) (l even) and a (0,−) (l odd) type (directly from (B.2)). Maue’scriterion would predict now edge state solutions at every second gap, which is inagreement with Zak’s predictions (see fig.(B.8)).We can also compare the case of a heterostructure, formed by joining together aleft semi-infinite crystal, with the band structure given in fig.(B.7), and a rightsemi-infinite crystal, characterised by the energy bands in fig.(B.8). The interfaceis assumed to be at x = 0. We proved previously that edge state solutions areexpected to exist every time the sign of Vl changes across the interface, i.e.

V(L)l = −V (R)

l . In fig.(B.9), Zak’s procedure is applied to the heterostructure:at every gap above the first band, the logarithmic derivatives for states in thegap, share the same sign in both crystals. We can, therefore, conclude thatMaue’s and Zak’s criterion provide systematically the same results in all threeexamples, as anticipated.

148


Figure B.7: (q,m) bands compatible with V2l < 0 and V2l−1 > 0. Note that inthe nearly free electron model the lowest energy state at k = 0 is necessarily evenunder inversion. The sign of the logarithmic derivative in the gap is indicatedby the + and − sign. The − signs are highlighted in red, given that theseare the signs that imply the existence of edge state solutions for the left semi-infinite crystal in Zak’s procedure. In the gap below the first band, an edgestate solution would be proportional to ekx, leading to a positive logarithmicderivative, as indicated in the figure.

149

Appendix B. Part II

Figure B.8: (q,m) bands compatible with V2l > 0 and V2l−1 < 0. The sign of thelogarithmic derivative in the gap is indicated by the + and − sign. The − signsare highlighted in red, given that these are the signs that imply the existence ofedge state solutions for the left semi-infinite crystal in Zak’s procedure.

150


Figure B.9: The left image represents the energy spectrum for the left semi-infinite crystal, while on the right the energy bands represent the right semi-infinite crystal. On the right, an edge state solution, below the first band, isproportional to e−kx, therefore providing a negative valued logarithmic deriva-tive. The signs of the logarithmic derivatives on both left and right crystalswould agree in every other gap above the first, implying the presence of edgestate solutions in the heterostructure. This particular example assumes the in-terface to be set at x = 0, but the conclusion that edge state solutions exist isstill true even if x = a/2.

151

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Contributions to publications

[1] The idea of the project was formulated by J. van Wezel. With the exceptionof the construction of the Landau free energy with applied pressure, all the re-sults were obtained by me. The interpretation of the results was a joint effortbetween me and J. van Wezel, as well as for the writing of the manuscript.

[2] The idea of the project was formulated by J. van Wezel. All results, in-cluding adaptation of the Hamiltonian model to account for spin-orbit couplingand inclusion of the effect of temperature, were obtained by me. The interpre-tation of the results was a joint effort between me and J. van Wezel, as well asfor the writing of the manuscript.

[3] The idea of looking into edge states connected to the topological classifi-cation in [5] was proposed by J. van Wezel. All the results from there on wereobtained by me, including formulating the research plan to solve the problem.

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Summary

Crystal Symmetries in Charge-Orbital Order andTopological Band Theory

The present thesis has focused in the study of two particular problems: chargeand orbital ordered states, and later the study of edge states in 1D crystals.The first of these two themes is encountered in the first part of the thesis. In thefirst chapters, emphasis is given to the study of the crystal structure of selenium(Se) and tellurium (Te). In the crystal structure of these elements, along certainspatial directions, the positively charged ions sit at a smaller distance from eachother, forming shorter chemical bonds in a geometrical pattern that resembles aspiral. These shorter bonds are accompanied by an increase in electronic density,making the spiral pattern also a spiral arrangement of electronic charge.To distil the crucial ingredients necessary for the emergence of an ordered spiralarrangement of charge, a minimal model was constructed and discussed through-out the first part of the thesis. Although meant as a minimal model, and thusconstructed out of a number of simplifications, it allows insight on the crucialinteractions that are needed to render the chiral trigonal crystal structure ofSe and Te. In particular, it was discussed under which circumstances is theless symmetric crystal structure of Se and Te energetically more favourable thanadopting a simple cubic lattice structure instead. The reason behind comparingthese two crystal structures is that polonium, which sits just below Te in the pe-riodic table, crystallises at ambient temperature and normal pressure in a simplecubic lattice, the simplest possible crystal structure available in nature. Withinour minimal model, the evolution from one crystal structure into another can bebriefly summarised in terms of the competition, co-existence, and cooperationamong spin, charge, orbital, and lattice degrees of freedom.The second part of the thesis is devoted to the study of electronic states thatare localised near the physical terminations of the crystal, i.e. the probabilityof finding these electronic states decreases exponentially as we move away fromthe boundaries of the crystal. As in the first part of the thesis, the approachfollowed is one in which simplicity is employed. For this reason 1D crystallinesystems are analysed.The interest was to understand the link between the existence of these localisedstates and the modern concepts of topological band theory, involving the pointgroup symmetries of the crystal. The existence of these edge states depend cru-cially on the constraints imposed by point symmetries, which in turn depend onthe geometrical arrangements of atoms in the crystal. Thus, through this thesisthe underlying protagonists have always been crystal symmetries.

165

Samenvatting

Crystal Symmetries in Charge-Orbital Order andTopological Band Theory

Dit proefschrift bestudeert twee specifieke problemen: lading- en baangeordendetoestanden, en daarna het onderzoek naar randtoestanden in 1D kristallen.Het eerste van deze twee problemen wordt behandeld in het eerste deel van hetproefschrift. In de eerste hoofdstukken wordt nadruk gelegd op de bestuderingvan de kristalstructuur van selenium (Se) en tellurium (Te). In de kristalstruc-tuur van deze elementen, in bepaalde ruimtelijke richtingen, bevinden de positiefgeladen ionen zich op een kortere afstand van elkaar, waarbij kortere chemis-che verbindingen in een meetkundig patroon dat lijkt op een spiraal wordengevormd. Deze kortere verbindingen gaan gepaard met een hogere elektronen-dichtheid, waardoor de spiraalpatronen tevens een spiraalvormige elektronenlad-ing vormen. Om de cruciale ingredienten die noodzakelijk zijn voor het ontstaanvan geordende spiraalvormige lading te identificeren, wordt in het eerste deel vanhet proefschrift een minimaal model geconstrueerd en bediscussieerd. Hoewelbedoeld als een minimaal model, en dus gebruikmakend van een aantal vereen-voudigingen, biedt het inzicht in de cruciale interacties die nodig zijn voor devorming van de chirale driehoekige kristalstructuur van Se en Te. In het bij-zonder wordt besproken onder welke omstandigheden de minder symmetrischekristalstructuur van Se en Te energetisch voordeliger is dan een simpele kubischeroosterstructuur. De reden voor de vergelijking tussen deze twee kristalstruc-turen is dat polonium, het element dat zich net onder Te in het periodiek systeembevindt, bij kamertemperatuur en normale druk kristalliseert in een eenvoudigkubisch rooster, de meest eenvoudige kristalstructuur die voorkomt in de natuur.Binnen ons minimale model kan de evolutie van de ene kristalstructuur in de an-dere kort worden samengevat als een competitie, co-existentie en samenwerkingvan spin-, lading-, baan- en roostervrijheidsgraden.Het tweede deel van het proefschrift is gewijd aan de bestudering van elektrontoe-standen die gelokaliseerd zijn aan de fysische uiteinden van het kristal, waarmeewordt bedoeld dat de kans om deze elektrontoestanden waar te nemen exponen-tieel afneemt met de afstand tot de rand van het kristal. Net als in het eerstedeel van het proefschrift is voor een zo eenvoudig mogelijke aanpak gekozen. Omdeze reden worden 1D kristalachtige systemen geanalyseerd.Het onderzoek richt zich op het begrijpen van de relatie tussen het bestaan vandeze gelokaliseerde toestanden en moderne concepten van topologische banden-theorie die verband houden met de puntgroepsymmetrie van het kristal. Hetbestaan van deze randtoestanden hangt op cruciale wijze af van de restrictiesopgelegd door puntsymmetrieen, die op hun beurt afhangen van de meetkundigeordening van de atomen in het kristal. Dus, door dit proefschrift zijn de on-derliggende protagonisten altijd kristalsymmetrieen geweest.

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Documents

UvA-DARE (Digital Academic Repository) Crystal symmetries ... · Contents What is this thesis about9 I Combined Charge and Orbital Order in the Chalcogen Crystals11 1 Introduction