Eur. Phys. J. B (2020) 93: 67https://doi.org/10.1140/epjb/e2020-100544-y THE EUROPEAN

PHYSICAL JOURNAL BRegular Article

A Hamiltonian model of the Fibonacci quasicrystal usingnon-local interactions: simulations and spectral analysis?,??

Amrik Sen1,a and Carlos Castro Perelman2,3

1 School of Mathematics, Thapar Institute of Engineering & Technology, Patiala, Punjab, India2 Centre for Theoretical and Physical Sciences, Clark Atlanta University, Atlanta, Georgia, USA3 Ronin Institute, 127 Haddon Pl., Montclair, NJ 07043, USA

Received 9 November 2019 / Received in final form 4 February 2020Published online 8 April 2020c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature,

2020

Abstract. This article presents a Hamiltonian architecture based on vertex types and empires for demon-strating the emergence of aperiodic order in one dimension by a suitable prescription for breaking translationsymmetry. At the outset, the paper presents different algorithmic, geometrical, and algebraic methods ofconstructing empires of vertex configurations of a given lattice. These empires have non-local scope andform the building blocks of the proposed lattice model. This model is tested via Monte Carlo simulationsbeginning with randomly arranged N tiles. The simulations clearly establish the Fibonacci configuration,which is a one-dimensional quasicrystal of length N , as the final relaxed state of the system. The Hamil-tonian is promoted to a matrix operator form by performing dyadic tensor products of pairs of interactingempire vectors followed by a summation over all permissible configurations. A spectral analysis of theHamiltonian matrix is performed and a theoretical method is presented to find the exact solution of theattractor configuration that is given by the Fibonacci chain as predicted by the simulations. Finally, a pre-cise theoretical explanation is provided which shows that the Fibonacci chain is the most probable groundstate. The proposed Hamiltonian is a mathematical model of the one dimensional Fibonacci quasicrystal.

1 Introduction

The simplest geometrical method of generating a onedimensional quasicrystal, such as the Fibonacci chain, isby the cut and project procedure from a strip embeddedin the two dimensional Z2 lattice and with a slope pro-

portional to the Galois conjugate − 1φ , where φ = 1+

√5

2 is

the golden mean [1,2]. The method can be replicated togenerate higher dimensional quasi-lattices. However, thispurely geometrical construction does not lend a physicalpicture of the underlying energetics of the quasicrystalconfiguration. Specifically, it is of immense importance toexperimentalists and material scientists to understand themicroscopic atomic arrangements in a quasicrystal [3,4].Besides, a physical picture of the emergence of transla-tion asymmetry may be of general interest to scientistsstriving to unravel the laws of nature through the studyof symmetries and conservation principles [5–8].

? Supplementary material in the form of three mpeg filesand one mp4 file available from the Journal web page athttps://doi.org/10.1140/epjb/e2020-100544-y.

?? Contribution to the Topical Issue “Advances in Quasi-Periodic and Non-Commensurate Systems”, edited by TobiasStauber and Sigmund Kohler.

a e-mail: amriksen@thapar.edu

1.1 Brief introduction to quasicrystal models

Quasicrystal models can be broadly categorized as under:

– microscopic tiling models based on matching rules[9–17],

– continuum models [8,18–20], and– computational models based on molecular dynamics

(MD) simulations of Newton’s equations of motions[21,22] or Monte-Carlo (MC) methods [23,24].

Matching rules based tiling models are abstract mathe-matical approaches that are useful to investigate the prop-erties of aperiodic order but by themselves they are notdirectly physically realizable models. Continuum modelslike the density wave approach involve analyzing densityof the condensing liquid phase in Fourier phase space. Thebasic hypothesis in such approaches relies on the con-servation of the free energy of the system upon a rigidtranslation of the cut space. A minimization of the Lan-dau free energy in the Fourier space representation selectsspecific series expansion coefficients that set particularwave vectors and consequently characterize the resultingunderlying stable structure which is aperiodic but ordered.The continuum models do not reveal the nature of inter-atomic interactions responsible for quasicrystal formation.MD simulations involve solving Newton’s classical equa-tions of motion with specifically prescribed pair potential

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interactions (e.g., the Lennard-Jones-Gauss (LJG) poten-tial [23,24]) that have local scope. The aforementionedsimulations have successfully demonstrated quasicrystalgrowth with long range order using local interactions.The experimental plausibility of local interactions driv-ing quasicrystal growth has also been demonstrated forthe decagonal case [25]. Further, in MC simulations usingthe LJG potential [24], in dynamic phase field simu-lations in two dimensions [26], and in entropy-driventiling models [27,28], defect free quasicrystal growth byaccretion is observed. However, in atomic quasicrystals,it is believed that quantum mechanical effects may beimportant [29–31].

1.2 The Fibonacci quasicrystal

We have chosen the Fibonacci model as a subject of inves-tigation because it is the most elementary quasicrystalmodel in one dimension. Even though the Fibonacci chain,defined mathematically in a subsequent section of thispaper, is generally regarded as a theoretical constructto study and conceptualize aperiodic order, it has directrelevance in many physical applications as well.

The Fibonacci system is a widely studied model.Fibonacci quasicrystals have been artificially fabricated[32] and their characteristic physical and topological prop-erties have been investigated [33,34] in the laboratory.Several authors have studied electronic structures andenergy spectrum of Fibonacci quasicrystals by consid-ering tight binding Hamiltonian models with hoppingconstants prescribed by a finite Fibonacci word (Fibonacciapproximant) [35–39].

In this paper, we propose a tiling model in the form ofan appropriate Hamiltonian for stabilizing a one dimen-sional random chain into the Fibonacci quasicrystal. Theproposed mathematical model is based on a Hamilto-nian involving non-local empire-empire interactions. Theempires have a generalized geometrical notion and can beconstructed for any given chain configuration of long andshort tiles in one dimension (not just for the Fibonacciconfiguration). The proposed lattice Hamiltonian has aground state which is the Fibonacci quasicrystal.

1.3 Scope of this work

The stability of quasicrystal structures is still an openproblem. This research paper presents a Hamiltonianarchitecture using vertex configurations (VCs) and theirrespective empires to understand the nature of atomicre-arrangements compatible with a stable quasicrystalconfiguration in one dimension. The theoretical modelpresented here assumes non-local interactions betweenempire-pairs. Non-locality is manifested in the non-localscope of empires of the VCs. The proposed Hamiltonianis promoted to a matrix operator form whose constituentshave a geometrical interpretation. The simulations of theproposed Hamiltonian model demonstrate the manner inwhich a random assortment of tiles interacts and rear-ranges to form a one dimensional Fibonacci quasicrystal.A detailed spectral analysis of the Hamiltonian operatorreveals that the Fibonacci state is the most likely ground

state of the system. This aperiodic Fibonacci ground statecan be generated entirely by the interaction of a collectionof sub-states (empires) of different chain configurationsundergoing energetic relaxation.

1.4 Organization of this paper

Section 2 introduces the definition of Fibonacci words ina recursive manner, and presents three different methodsof constructing empires for a given VC in a Fibonaccichain. While the geometric method is primarily useful forimplementing the Monte Carlo simulations discussed inthe latter section, the algebraic expression of the empiresof a Fibonacci chain will be key to understand the mannerin which the translation symmetry is broken as discussedlater. Section 3 discusses the implementation of the MonteCarlo simulation of the proposed Hamiltonian model andpresents results of these simulations. In Section 4 thematrix operator form of the Hamiltonian and its spec-tral analysis is presented and these analytical results arecompared with the results of the simulations. A directphysical explanation is provided as to why the Fibonaccichain is the most likely ground state of the system as isevident from the simulations. In Sections 5 and 6, a sum-mary of the main contributions of this paper is presentedand future plans to extend the work to two dimensionalquasicrystals are outlined.

At the very outset, it must be carefully noted thatunless otherwise specified, by the phrase Fibonacci chain,we refer to the finite Fibonacci approximant in this article.This terminology is interchangeably used with equiva-lent phrases such as Fibonacci configuration and Fibonaccistate.

2 The Fibonacci lattice

A Fibonacci sequence is constructed from the Fibonaccinumbers by using the following recurrence relation,

Fn+1 = Fn + Fn−1, (1)

where F0 = 0, F1 = 1. An interesting property of this

sequence is the golden ratio scaling, limn→∞Fn+1

Fn= φ

where the rapid convergence to φ (the golden ratio) canbe verified in a simple manner. A Fibonacci word isconstructed using the following recurrence relation,

Sn = Sn−1Sn−2, n ≥ 2, (2)

with S0 = 0, S1 = 01. Thus, a Fibonacci word of length nis a finite sequence of 0 and 1 constructed as above [40,41].The relationship between a finite Fibonacci word and theFibonacci sequence stems from the fact that the length ofSn is Fn+2, the (n+ 2)th Fibonacci number. A Fibonaccilattice or Fibonacci quasicrystal of size n is a lattice withgrid spacing encoded by the Fibonacci word Sn where0 → φ and 1 → 1. A standard representation of such aquasi lattice is given by replacing 0 by L and 1 by S wherethe symbols L and S are regarded as tiles. E.g., a section ofa Fibonacci quasicrystal in the L, S representation looks

Eur. Phys. J. B (2020) 93: 67 Page 3 of 14

V : set of VCs (vertex configurations) = L,L, L, S, S,LX : set of coordinates denoted by subscript numerals, i.e. (x0, x1, x2, ...) ≡ (0, 1, 2...)

αj : the VC α located at coordinate j (3)

Eαj ,l =

+1, if tile located at l is S

−1, if tile located at l is L

0, otherwise (i.e. unforced tile)

τi : binary representation of the tiling space with entries 1 for S and −1 for L

Ω : domain of the lattice containing the set αii=2:N where N is the length of the lattice

like ...LSLLSLSLLSLLS... . Thus, a Fibonacci lattice isa quintessential example of a one dimensional quasicrystal.

2.1 Vertex configurations and empires

The model demonstrated in this manuscript is designedbased on VCs and empires. These are standard canonicaldescriptors of quasicrystals [42]. For the 1D Fibonacci qua-sicrystal, there are three VCs, viz., L,L, L, S, S,L.Note that the tile S cannot appear in succession in aFibonacci lattice and hence S, S is not a legal VC.

2.1.1 Notations and definitions

2.1.1.1 Empire

Corresponding to each vertex type (vertex configuration)at each coordinate, there is a set of forced tiles thatconstitute the respective empire [43–48]. The precise for-mulation of an empire for a given VC will become clearthrough the detailed discussions of this section.

Generally speaking, there are three ways of constructingempires of a VC in a quasicrystal. For the one dimen-sional case, a simple substitution rule may be used toenlist the empires for any vertex configuration [2]. Alterna-tively, a geometric method using an irrational projectionfrom a two dimensional lattice may be used to gener-ate the empires of a given VC [49,50]. Finally, a set ofalgebraic formulae for the empires of the one dimensionalFibonacci quasicrystal is developed and presented here.All these methods are discussed in an elaborate manner inthe context of the one dimensional Fibonacci quasicrystal.

In what follows, the mathematical notation and defini-tions of the related terms are given below.

See equation (3) above.

For the sake of notational brevity, a tile referenced by thecoordinate l means that the tile is located between thelattice coordinates l and l + 1 for a right sided entry andbetween the coordinates −l and −l − 1 for a left sidedentry. For example, consider the chain expressed by (5).The tile referenced by l = 1 is S located between l = 1and l+ 1 = 2.1 A VC referenced by coordinate j refers tothe one composed of the tiles on either side of j. Finally,

1 Left and right sided entries in an empire vector are with ref-erence to the coordinate located at the center of the VC, this isanalogous to a radial convention with the center of the VC as theorigin.

the VC-empire parameterization encompasses the localand non-local field of influence of a given VC throughits domain of influence by the forced tiles. Concisely, thisempire field of VC α situated at j is denoted by the empirevector

Eαj=(...EαLj,−2E

αLj,−1E

αLj,0E

αRj,0 E

αRj,1 E

αRj,2 ...

),

where superscripts L and R refer to left and right sidedentries respectively. In the following three sections, we dis-cuss three different methods of finding the empires of VCs.The substitution method and the geometric method ofconstructing empires have been explored by others in thepast but the algebraic prescriptions of the empires are newresults presented here.

2.1.2 Substitution algorithm for constructing empires

This algorithm borrows extensively from detailed dis-cussions on substitution rules for generating aperiodicFibonacci words in chapter 4 of the text by Baake andGrimm [2]. This method is applicable for the one dimen-sional Fibonacci chain but may be extended as a principleto a network of Fibonacci chains. The identification of theempire tiles in a real Fibonacci chain, denoted by upper-case S and L tiles, is undertaken by referencing from avirtual Fibonacci chain, denoted by lowercase s and ltiles. The virtual chain (hitherto referred to as fibonacci)begins with the tile l and is constructed by using the sub-stitution rule l → ls, s→ l iteratively. Thus the first fewiterations give lsllslsl because l → ls → lsl → lslls →lsllslsl. The VCs along with their respective empires canbe expressed algorithmically as follows.

L,Ls+ fibonacci

s→ SL

l→ (LS|SL)L

L, SL+ s+ fibonacci

s→ (LS|SL)L

l→ (LSLS|SLSL|SLLS)L

S,Ls+ fibonacci

s→ (LS|SL)L

l→ (LSLS|SLSL|SLLS)L

(4)

Here, each line of the algorithm must be read as follows.E.g., the first line means: “the empire of a given L,L VCin a real Fibonacci chain is constructed by the substitu-tion L,L → LL, and s→ SL, l→ (LS|SL)L performedon the virtual fibonacci chain s + fibonacci where the

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...S↑L↑SLL↑S↑L↑LSL↑SLL↑S↑ .L↑ .L.S−3L↑−2

S↑−1

L↑ 0

L↑ 1S↑ 2L↑ 3S.L.L↑ .S↑L↑LSL↑SLL↑S↑L↑LSL↑S↑... (5)

...L↑()()L

↑S↑L↑()()L

↑S↑L↑()()L

↑()()L

↑S↑L↑()()L

↑S↑

L↑︸︷︷︸

E1L0,0

L↑︸︷︷︸

E1R0,0

S↑L↑()()L

↑S↑L↑()()L

↑()()L

↑S↑L↑()()L

↑S↑L↑()()L

↑... . (6)

E10 = E1

0,ll∈X,X∈Ω = (... E1L0,−5 E

1L0,−4 E

1L0,−3 E

1L0,−2 E

10,−1 E

1L0,0 E

1R0,0 E

1R0,1 E

1R0,2 E

1R0,3 E

1R0,4 E

1R0,5 ...)

= (... 0 0 − 1 1 − 1 0 0 − 1 1 −1︸︷︷︸E1L

0,0

−1︸︷︷︸E1R

0,0

1 − 1 0 0 − 1 1 − 1 0 0 ...) (7)

forced tiles are expressed without parenthesis, the opera-tion | means conditional OR, and stands for the abovementioned substitution.”Above, only a right sided empireidentification procedure has been shown but extension tothe left of the VC is straightforward by symmetry consid-eration. This is made clear in the example presented inthe following section.

2.1.2.1 Illustration of the algorithm

Consider a given section of a 1D Fibonacci chain (labelledas the real chain):

See equation (5) above.

where the bold font tiles L0L constitute the VC of inter-

est located at the coordinate x0 ≡ 0 in this example.The underset numerals denote coordinate locations (recallthat the tiles are indexed by the coordinates to theirimmediate left). The significance of the underset upar-rows ↑ is explained after expression (7). The 1D Fibonaccichain in (5) is represented in tiling space. The equiva-lent binary representation of the chain is a sequence τiwith entries ±1 depending on the corresponding entriesin the tiling space, i.e., −1 for L and 1 for S. Thus, wehave τ−3 = 1, τ−2 = −1, τ−1 = 1, τ0 = −1, τ1 = −1, τ2 =1, τ3 = −1, . . . etc.

The first step involves the construction of the virtualFibonacci chain that serves as a guide to identify theforced tiles in the real chain and thereby the correspondingempire. The algorithm prescribes the form of the virtualchain to be s+ fibonacci where fibonacci begins withthe tile l and is constructed by using the substitutionl → ls, s → l iteratively as mentioned earlier. Thus wehave the following virtual chain,

s+ lsllslsl... = slsllslsl... .

The substitution rule for L,L is applied to the abovevirtual chain, i.e. s→ SL, l → ()()L where each () corre-sponds to one unforced tile. Further, the unforced tiles arelabelled by 0s for convenience. This entails the followingempire form.

See equations (6) and (7) above.

Fig. 1. (Not to scale) Proof of concept of the cut and projectmethod to find the empire of a given vertex configuration,L, S, bounded by the lattice coordinates (x3, y3), (x4, y4) and(x5, y5). The empire window depicted by the horizontal dashedlines is bounded by the maximum and minimum y-coordinatesof the lattice points defining the VC, shown here by solid dou-ble lines (oblique) bounded by the horizontal dashed lines.Tiles that have both their bounding coordinates within thisempire window constitute the empire of the given VC. Thus, forthe string shown here, the section of the empire correspondingto the L, S VC is: ...L()()LSL()....

The tiles bearing the underset uparrows ↑ are the forcedtiles of the VC LL. The corresponding tiles in the realchain (5) are consequently indexed by the uparrows ↑ andconstitute the forced tiles of the empire of the VC LL.The coordinate locations for the chain in (6) are impliedas in (5) and are not shown again. In terms of the vectorialnotation, the empire for L,L located at j = 0 is givenby equation (7).

2.1.3 Geometric method of constructing empires

The cut and project method of constructing the empire ofa given vertex configuration is illustrated in this section.This method can be applied for any chain (not neces-sarily a Fibonacci chain). It may be recalled that a 1DFibonacci chain can be constructed by projecting a two-dimensional cylindrical section of the Z2 lattice onto a1D real line by using an irrational angle [1,2]. Hence, itis reasonable to search for the forced tiles that consti-tute the empire of a given vertex type within this bandof the original Z2 lattice. For convenience of calculation,we consider a rotated frame of reference of the originalZ2 lattice (a.k.a. mother lattice) such that the horizontalx-axis of the new 2D plane coincides with the projectedquasicrystal space of the Fibonacci chain (refer Fig. 1).In this frame, the horizontal and vertical arms of theZ2 lattice appear at an irrational angle with respect to

Eur. Phys. J. B (2020) 93: 67 Page 5 of 14

the x-axis as is further illustrated in the graphic depictedin Figure 1. The lattice arm joining points (x0, y0) and(x1, y1) makes an angle θ = tan−1 1

φ with the horizontal

axis. Here φ = 1+√

52 . Moreover, the lattice arms are each

of unit length and orthogonal to each other. The arm join-ing the points (x2, y2) and (x1, y1) also makes the sameangle θ with the vertical.

Given the state of a two-alphabet chain (Fibonacci orotherwise) in 1D, the first step involves reconstructing thecorresponding section of the Z2 lattice, i.e., the (x, y) coor-dinates in the original 2D space. Following Figure 1 andsimple trigonometry, we obtain the following relations,

yi = yi−1 − τi sin(Θiπ/2 + θ), Θi =1 + τi

2,

xi = xi−1 − κiτi, κi =

1, if τi = −1

− 1φ , if τi = 1.

(8)

The starting point of the chain is located at the origin,i.e., (x0, y0) = (0, 0).

Next, the calculation of the empire is undertaken byidentifying the bounding (x, y) coordinates of the VC inquestion for which we intend to find the empire. Theempire window is constructed in the 2D plane by find-ing the minimum and the maximum of the y-coordinatesassociated with the VC. This is illustrated graphically inFigure 1. The tiles with both bounding (x, y) coordinateswithin the empire window constitute the empire of thegiven VC. It is important to emphasize that this methodof finding the empires is not restricted to a Fibonacci chainalone but is more general and works for any 2-alphabetchain.

2.1.4 Algebraic formulae of empires

This section presents analytic expressions of the empiresfor the Fibonacci quasicrystal and thereby prescribes theexact form by which the translation symmetry is absent. Itmust not be surprising that the expressions are related tothe Fibonacci word. The empires obtained by using theseformulae are validated against the corresponding empiresconstructed by two other methods explained above. In thefollowing paragraphs, we illustrate the algebraic forms ofthe empires for each VC of the chain.

2.1.4.1 Symmetric VC, L,LWe provide a direct analytical expression for every elementof the empire vector Eα

j . For example, the (l − j − 1)th

entry of the empire vector E1Rj (for VC L,L) is given

by

E1Rj,l ≡ E1R

n = fn+1

(fn + fn−1

)− fn,

n = l − j > 0,

E1R0,0 = E1R

0 = −1 (by construction), (9)

where fn is the nth bit of an infinite Fibonacci word2

fn = 2 + un − un+1, un = bnφc,

φ is the golden ratio 1+√

52 . The superscript R denotes

empire entries to the right of the VC located at j includ-ing the right tile of the concerned VC. Also note thatf0 = 1, i.e. the 0th bit of a Fibonacci word is taken as1 by prepending 1 to the Fibonacci word defined by therecursion (2), f1 = 0, f2 = 1, etc.. bc is the standard flooroperator that outputs the integer part of the argument.For consistency, entries from equation (9) are comparedwith that of entries from equation (7). E1R

0,0 = −1, E1R0,1 =

1(0 + 1)− 0 = 1, E1R0,2 = 0(1 + 0)− 1 = −1, E1R

0,3 = 0(0 +

1)− 0 = 0, E1R0,4 = 1(0 + 0)− 0 = 0, E1R

0,5 = 0(1 + 0)− 1 =−1.

Owing to the symmetry of the L,L VC, using f−n =fn in conjunction with the formula given by equation (9),gives the entries of the empire vector to the left of LL. Fora symmetric VC like L,L, the following is trivially true:E1L−n = E1R

n . Here L corresponds to left and the conventionfor referencing the tiles is the following:

– tiles to the right of the VC are referenced by thecoordinate xl to their immediate left, and

– tiles to the left of the VC are referenced by thecoordinate xl to their immediate right,

i.e. the convention follows a radial indexing scheme withthe center of the VC located at xj (x0 in the aboveexample) as the origin. Additionally, E1R

0,0 = E1L0,0, each

referencing the right and left tiles that respectively con-stitute the VC. Finally, E1

j = E1Lj + E1R

j constitutes thefull empire vector. Here

E1Lj=0 = (... E1L

0,−5 E1L0,−4 E

1L0,−3 E

1L0,−2 E

10,−1 E

1L0,0 0 0 0 0 0 0...)

and likewise for E1Rj=0. For brevity, the superscripts L and

R are often omitted and the sign of the subscript n = l− jis sufficient to identify the left and right tiles.

2.1.4.2 Asymmetric VC, L, S, S,LSimilarly, exact formulations of the asymmetric VCs aregiven below.

E3Rn =

−1; n = 0

3 + un − un+2; n ≥ 1,

E2Rn =

1; n = 0

−1; n = 1

E3Rn−1; n > 1.

(10)

By symmetry, the left sided entries are

E3L−n = E2R

n , n ≥ 0

E2L−n = E3R

n , n ≥ 0. (11)

2 The first few entries of an infinite Fibonacci word are1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1... after prepending by 1 that isnow taken as the 0th bit of an infinite Fibonacci word.

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Clearly, the non-trivial empire tiles obey the followingrelations E3

−n = E2n, E

2−n = E3

n, n > 0. Moreover, Eαj =

EαLj + EαR

j constitutes the full empire vector for the VCs

α = 2, 3. The definitions of un, EαLj and EαR

j are asdescribed earlier. Note that for an asymmetric VC, the0th mode empire tiles EαL

0,0 6= EαR0,0 are defined as above.

This completes the parametrization of the 1D Fibonaccichain in terms of the VCs and the corresponding empires.

2.2 Random flips

The Monte Carlo simulations of Section 3 below employrandom flips to span the different chain configurations. Arandom flip can be categorized as follows:

– symmetric flip: For the 1D case, this refers to theflips of the type L, S ↔ S,L. A flip of this kindpreserves the local length at the location of the flip.

– asymmetric flip: This refers to the flips of thetype L,L ↔ S,L or S,L ↔ L,L, etc. Aflip of this kind does not preserve the length of thequasicrystal section locally.

3 Hamiltonian

The canonical Hamiltonian HΩ for the 1D Fibonaccisystem is constructed as follows:

HΩ = − 1

N

∑k,i∈X

Bk,iEαk,i︸ ︷︷ ︸interaction free terms

− 1

N

∑j,i∈X

Jj,i〈Eαj |Eαi〉︸ ︷︷ ︸interaction terms

≡ − 1

NBk,iEαk,i −

1

NJj,i〈Eαj |Eαi〉︸ ︷︷ ︸

Einstein summation notation

, (12)

where Bk,i and Jj,i are the free parameters of the modelwith units of energy and Eαj

≡ Eαj is a dimensionless

vector of 0s and ±1s denoting the empire of the VC αlocated at j. Here, 〈·|·〉 denotes an inner product oper-ation: 〈A|B〉 :=

∑iAiBi, where A,B are vectors with

entries Ai and Bi respectively. The locations of the forcedtiles are referenced by the elements ±1 in the vector Eαj

depending on whether the forced tile in question is S or L.The Hamiltonian in equation (12) has two terms, viz., theinteraction free term which expresses the energy of a givenVC (in conjunction with its empire field) in an externalfield or in vacuum, and the interaction term that encom-passes the mutual energy of interaction of two distinctVCs through the interaction of their respective empires.It is important to note that by construction, the empireinteractions are non-local because the values of m and ncan be far apart.

3.1 Metropolis-Hastings simulation of theHamiltonian

The algorithm first chooses selection probabilities ps(µ, ν)which represent the probability that state ν is selected by

the algorithm out of all states given that the previousstate is µ. It then uses acceptance probabilities pa(µ, ν)that ensure the detailed balance condition.

3.1.1 Simulation steps

– A lattice site is randomly picked from the Fibonaccigrid using selection probability ps(µ, ν) and the con-tribution to the total energy involving the VC at thissite is calculated.

– The VC is flipped and the new contribution tothe energy is calculated. The flip may be LL →LS,LS → SL, SL→ LL, etc.

– If the new energy is less, then the flipped value isretained.

– If the new energy is more, then the flipped value isretained with probability e−β(Hν−Hµ). Here β = 1

kBT

where kB is the Boltzmann’s constant and T is thetemperature.

– The process is repeated until a global minimum ofthe total energy is attained.

3.2 Simulation parameters

Parameter Value

Bk,i ≡ B 0 or 1Jj,i ≡ J 1T 0.6N 21†

†N = 21 corresponds to the simulation shown in Figures 2and 3. The source files used for developing and running thesimulations are freely available in a GitHub repository.3

The choice of unit coupling constants Jj,i is natural andensures equal weight to the interactions between empiresof pairs of VCs irrespective of their location in the chain.

3.3 Results of simulations

Several simulation runs with chains of different lengths(N = 5, 8, 13, 21, 34, 55, 89, 144, and 233) were performedstarting with different initial states of the chain. In eachcase, the attractor, that minimized the total Hamiltonian,was found to be the Fibonacci chain. In Figures 2 and 3, anexample of one such simulation with N = 21 is presented.A selected number of movies of the simulations can befound in the first author’s youtube channel.4

3.4 Distribution of the attractor and invariance of theHamiltonian to initial conditions

It is important to note that by establishing the Fibonaccichain (a.k.a. the Fibonacci state) as the attractor of theHamiltonian (12), as shown by the simulations, we areessentially interested in the relative distribution of the Land S tiles. In the next section, we invoke the ansatz that

3 https://github.com/amriksen/Fibonacci Ham code web4 https://www.youtube.com/watch?v=MirQPchbo7Q

https://www.youtube.com/watch?v=1n-je95jOlkhttps://www.youtube.com/watch?v= KsToOQs5hc

Eur. Phys. J. B (2020) 93: 67 Page 7 of 14

Fig. 2. The set of plots corresponds to the initial state of a chain of length N = 21 (bottom left panel) at the start of thesimulation where the tiles L and S are color coded as green and red respectively. The VC pairs being flipped are shown by theblack colored stem pairs displayed in the top panels. The number of defects in the current state of the chain corresponds to thenumber of forbidden elementary configurations (LLL and SS) in a Fibonacci chain. The coordinates of the 2D lattice (motherlattice, ref. Fig. 1), from which the current state of the chain may be constructed, are shown in the bottom right panel. Thetotal Hamiltonian of the current configuration is displayed in the middle right panel.

the distribution of the sign of the entries of the most rele-vant eigenvector of the Hamiltonian matrix prescribes theattractor configuration. The applicability of this ansatzis found to be consistent by treating the Hamiltonian asa quantum mechanical system and verifying one of themain axioms of probability as discussed in the subsequentsection. It must be noted that the choice of labels for theternary system used in the definition of the empires inSection 2.1 is entirely arbitrary and a matter of com-putational convenience. It must be emphasized that theresults of the simulations are invariant to the initial stateof the chain as is shown by the above cited movies of thesimulations and Figures 2 and 3.

4 Empire dyads, spectral analysis, and theFibonacci state

In order to formulate an algebraic representation ofthe Fibonacci system and the associated empire dyads,it is essential to promote the Hamiltonian defined byequation (12) to a matrix operator form. Moreover, since

the external field Bαk,i, plays no influence on the finalattractor configuration, the interaction free term will beomitted from consideration in the theoretical analysis ofthe Hamiltonian presented in this section. This is a mat-ter of convenience without any loss of generality. Thus, inthe absence of the external field B ≡ 0, we will present aninterpretation akin to quantum mechanics in the spirit ofthe discussions in the literature [51–53].

4.1 Construction of empire dyads and theHamiltonian matrix operator

Consider any chain (not necessarily Fibonacci) of fixedlength N (i.e. N lattice sites) where N is moderately large.The matrix operator, corresponding to a pair of interact-ing empire vectors5 of VCs α and β located at positions

5 Note that the geometric method of constructing empires enablesus to find the empire of a VC located at a certain coordinate of anychain configuration. Hence, the notion of empires is not restrictedto a Fibonacci quasicrystal but applies to any configuration of thechain.

Page 8 of 14 Eur. Phys. J. B (2020) 93: 67

Fig. 3. The set of plots here corresponds to the final relaxed state of the chain shown in Figure 2. The arrangement of thegreen (L) and red (S) balls clearly shows that the attractor configuration is the Fibonacci chain. The evolution of the totalHamiltonian of the system is also displayed clearly demonstrating that the attractor configuration has the minimum energy.The defect counter shows the absence of any forbidden configuration in the final state and hence is consistent with the fact thatthe attractor is the Fibonacci chain.

m and n respectively, is a dyad and is defined as

Eαmβn := Eαm ⊗Eβn ≡ EαmEβn

T . (13)

Unlike in the earlier sections, here the coordinate loca-tions of the individual tiles are referenced from the left(beginning of the chain), e.g. the subscript above identi-fies the location of the α VC as the mth coordinate and soon. The empire matrix Eαmβn is a dyadic product of twovectors resulting in a tensor of rank two and dimensionsN ×N .

The Hamiltonian operator can be then written as thedistributed sum of the symmetrized Empire matricesEαmβn over all possible VCs and location pairs as follows:

H := − 1

N

∑m,n∈X

Eαmβn + (Eαmβn)T

2, (14)

where the sum is over all possible pairs of interactingempires of the given chain. H is an N × N symmet-ric matrix whose trace gives the Hamiltonian HΩ defined

by equation (12), i.e. HΩ := Tr(H ). Since the inter-action free term of the Hamiltonian prescribed earlierby equation (12) plays no discernible influence on theevolution of the tiled chain into the Fibonacci configu-ration, it is omitted from consideration here and only thebilinear interaction terms are retained. Further, the inter-action coefficients Jm,n that appear in the equation (12)are set to unity. Among all possible chain configurations,the Fibonacci chain corresponds to the configuration thatminimizes the trace HΩ = Tr(H ) as demonstrated by theMonte Carlo simulations of Section 3.3.

4.2 Spectrum of the Hamiltonian H and theoreticalanalysis

In this section we will establish a few key points:

– the matrix operator form of the Hamiltonian H willenable us to theoretically validate the results of thesimulations presented in the previous section,

– the construction of H given by expression (14) iscorrect and consistent with the construction of theHamiltonian HΩ given by expression (12),

Eur. Phys. J. B (2020) 93: 67 Page 9 of 14

Fig. 4. Cumulative eigenspectrum of H for different chain configurations clearly shows that the total energy given by HΩ =Tr(H ) is minimized by the Fibonacci configuration. Length of the chain N = 8 (left) and N = 9 (right). In each case, all 2N

different configurations are considered.

– the matrix H encodes the energetics of differ-ent chain configurations succinctly in the form ofempires, and

– spectral analysis of H gives us the analytic solutionof the stable chain configuration which turns out tobe the Fibonacci configuration.

Further, it must be emphasized that since the geomet-ric method allows us to construct empires of a given VCfor any chain configuration of long and short tiles, theconstruction of H is generally applicable to any chainconfiguration. Consequently, the analysis here demon-strates that the Fibonacci state emerges naturally fromthe model Hamiltonian as the stable solution.

4.2.1 Eigenspectrum of H for all chain configurations

The Hamiltonian H can be diagonalized as ΨDΨ−1

where the energy eigenvalues are given by the diagonalmatrix

D =

λ(1) 0 · · · 00 λ(2) · · · 0·· ·· · · · ··0 ·· · · · λ(N)

≡E1 0 · · · 0

0 E2 · · · 0·· ·· · · · ··0 ·· · · · EN

,

and Ψ is a matrix whose columns are the eigenvectorsψn corresponding to the eigenvalues λ(n) ≡ En. By con-vention, we have |λ(1)| ≡ |E1| ≥ |λ(2)| ≡ |E2| ≥ |λ(3)| ≡|E3| ≥ ... . Since H is a real symmetric matrix, it hasreal eigenvalues with λ(1) = λmin < 0 corresponding tothe ground state energy level E1 ≡ Emin.

To illustrate the distinct energy levels of different con-figurations of a chain of fixed length N , we calculatethe matrix Hamiltonian H as prescribed by the expres-sion (14) for every chain configuration. This is followedby computing the eigenspectrum of H for each of 2N

possible configurations. The cumulative eigenspectrumcorresponding to each such configuration is plotted in

Figure 4 for N = 8 and N = 9. This theoretical analysisis consistent with the results of the simulation presentedin the previous section. In what follows here, we showthat the ground state prescribes the stable Fibonacciconfiguration.

4.2.2 Eigenstates of H prescribe stable chainconfiguration

The results and analysis of Sections 3 and 4.2.1 demon-strate that the stable chain configuration corresponds tothe state that minimizes HΩ = Tr(H ). Having obtainedthis energetically favored configuration, we next demon-strate how the eigenstates corresponding to this optimalconfiguration prescribe the Fibonacci solution state. Ofcourse, we could simply read off the configuration with theleast energy and show that it is the Fibonacci state. How-ever, we wish to clarify two subtle points in the followinganalysis:

– the collection of empires corresponding to the mini-mum energy state would be sufficient to recover thefull stable solution because H is built entirely andsolely from empires, and

– the Fibonacci configuration not only minimizes thetotal energy as shown above but also minimizes theground state among all possible configurations.

The analysis presented below reveals that the chain hasthe highest probability of being in the ground state.This fact along with the second point above imply thatthe Fibonacci configuration is the most probable groundstate of the system. So the presentation here incorporatesrandomness as a key ingredient of the model.

It may be easily checked that the eigenvector corre-sponding to λmin prescribes the distribution of L andS tiles of a Fibonacci chain (up to our chosen signconvention). This corroborates the observations made inSection 3.3 earlier that the attractor of the chain is theFibonacci state. In order to obtain the Fibonacci chain

Page 10 of 14 Eur. Phys. J. B (2020) 93: 67

Table 1. rank(H ) vs N .

Length of chain, N rank(H )

5 28 413 821 1234 2155 3389 55144 88233 144

exactly in terms of the signed unitary digits, −1 and1 (and not only in their relative distribution of signs),a superposition of the relevant eigenvectors of H mustbe considered. Thus, if one were to posit the Fibonaccistate as a quantum state, the Fibonacci state ψF may bewritten as a linear superposition of r+ 1 eigenvectors cor-responding to the r + 1 dominant eigenvalues (includingthe zero eigenvalue) of the matrix H whose rank is r < N ,resulting in the following ansatz,

ψF =∑

n≤r+1

cnψn, (15)

where the ψns are the orthonormal eigenvectors of Hand ψF := ± 1√

Nsignψ1 (see Eqs. (17) and (18)). The

Fibonacci state ψF is degenerate as manifested by theappearance of ± in the ansatz because the choice of thelabels L → −1 and S → +1 is arbitrary. Both ψF and−ψF are solutions of H |ψ〉 = E|ψ〉 [54]. This underscoresthe fact that it is the relative ordering of the signs of theentries of ψ1 that essentially dictates the attractor config-uration. Further, the matrix H is rank deficient becausethe resulting truncated Fibonacci chain is not an exactquasicrystal but an approximant with periodicity equal toN . Only in the infinite length chain, a full rank matrixHamiltonian can be obtained. Another interesting obser-vation from Table 1 is that the rank(H ) for a chain oflength N (corresponding to, say, Fibonacci word Sm) isequal to the length of the previous Fibonacci word (Sm−1)except in some cases where the result is off by one due tonumerical imprecision in computing the diagonalization ofH owing to infinitesimally small numbers involved result-ing in roundoff errors. This is likely due to the manner inwhich the higher Fibonacci words may be generated recur-sively like Sm = Sm−1Sm−2, ∀m ≥ 2 as mentioned earlierin equation (2).

It may be insightful to provide an analogy with quan-tum mechanical systems. The energy of the Fibonaccistate can be obtained as

〈ψF |H |ψF 〉 = EψF =∑n

|cn|2En, (16)

where En is the nth eigenvalue of H and denotes the nthenergy eigenstate and the coefficient |cn|2 is the probabil-ity of being in state ψn. The latter demands the restriction∑n≥1 |cn|2 = 1. It may be interesting to note that if

|cn|2 = 1N , then the value of EψF would be the ensem-

ble average of the energy eigenvalues prescribed by HΩ

whence all the energy eigenstates are equally probable,i.e. NEψF =

∑nEn = Tr(H ) = HΩ. This equiparti-

tion of energy eigenmodes is a signature of the principleof eigenstate thermalization [55–57]. Further, while theenergy eigenstates may be equally probable in occur-rence, their magnitudes are distinctly different because|E1| > |E2| > |E3| > and so on, unless there is a degener-acy of eigenstates. In the present case, as is shown below,|E1| |Ei| ∀i 6= 1 which guarantees the prominence ofthe most relevant eigenstate ψ1. In the next section, wewill discuss if the condition of equiprobable eigenstates,given above, holds or not in the large N thermodynamiclimit; and if not, what inference one may draw. In sum-mary, two questions remain central to the inquiry in thispaper.

– How to find the attractor of the Hamiltonian model?– Why does the attractor configuration coincide with

the Fibonacci chain as demonstrated by the simula-tions of the previous section?

The answer to the first question is essential to justify theobservations of the simulations presented in Section 3.3.The answer to the second question is important in orderto understand the physics of quasicrystal growth and theorigin of aperiodicity in physical systems.

4.2.3 How to find the attractor of the Hamiltonian?

Simulations of Section 3 prescribe a solution strategyfor finding the attractor configuration. Is this attractorsolution consistent with our quantum mechanical interpre-tation of the system? The central idea to find the explicitform of the solution of the final relaxed state of the Hamil-tonian (12) is postulated as a superposition of eigenstates(15). Recall the ansatz (15): the attractor configurationis prescribed by signψ1 where ψ1 is the eigenvector cor-responding to the most dominant energy eigenvalue E1

and sign is the well known signum function. Since theeigenvector corresponding to the most dominant eigen-value prescribes the distribution of the tiles in the finalstate up to the chosen sign convention, the exact form ofthe final Fibonacci state of the chain can be obtained bywriting a linear combination of the first (r + 1) eigenvec-tors with the associated coefficients as mentioned in theexpression (15). This will prescribe the correct exact formof the attractor configuration provided

〈cr+1, cr+1〉 =r+1∑i=1

|ci|2 = 1,

where cr+1 =(c1, c2, c3, . . . , cr, cr+1

)T. Importantly, this

attractor turns out to be the Fibonacci chain of length

N . Figure 5 presents the values ofr+1∑i=1

|ci|2 for chains of

different lengths by considering only the first (r+ 1) coef-

ficients. In all cases, the value ofr+1∑i=1

|ci|2 is very close to

Eur. Phys. J. B (2020) 93: 67 Page 11 of 14

unity thereby validating the suitability of equation (15)to find the exact solution of the final relaxed state ofthe Hamiltonian. The solution obtained is the Fibonaccichain.

The solution strategy mentioned above is explained herethrough an example. Consider a chain of length N = 13.Under similarity transformation, H = ΨDΨ−1 and therank of H is r = 8. The attractor of this system isobtained by considering the eigenvector

ψ1 =

0.1962−0.1962

0.29170.2403−0.1941

0.5313−0.1455

0.14550.3860−0.2452

0.43660.0954−0.0954

(17)

and using the following ansatz and solving for the coeffi-cients cn,

−1√13

signψ1 =−1√13

1−1

11−1

1−1

11−1

11−1

=∑

n≤8+1

cnψn. (18)

This gives us

cr+1 =

c1c2c3c4c5c6c7c8c9

=

−0.8873−0.0335

0.0107−0.1524−0.2994

0.21100.1909−0.1328

0.0000

whence it may be checked that indeed

〈cr+1, cr+1〉 =∑i≤8+1

|ci|2 = 1.

Finally, the attractor configuration ψF is expressed as asuperposition of eigenstates (15) by using the coefficientscomputed above. This solution corresponds to the sixth

Fig. 5. This plot shows that the probability of the Fibonaccistate being the ground state is significantly higher than beingin any other excited state. It may be interesting to point outthat the probability of being in the ground state asymptoticallyapproaches the value of the golden ratio φ. Further, it is shownthat the solution prescribed by equation (15) is consistent withthe fundamental axiom of probability,

∑i≤r+1 |ci|

2 = 1.

Fibonacci word S5 which can be found using the algorith-mic recursion Sn = Sn−1Sn−2, n ≥ 2, S0 = 0, S1 = 01.This paper presents a physical model of the emergence ofFibonacci quasicrystal that can also be constructed algo-rithmically as explained in several papers cited earlier.The solution of H is an NP-hard optimization problemthat essentially involves finding the most probable groundstate energy [54,58,59].

4.2.4 Why is the attractor configuration given by theFibonacci chain?

While the ansatz allows us to find the attractor solu-tion consistent with the postulates of quantum mechanics,a natural question that follows is: why is the attrac-tor configuration given by the Fibonacci chain? Afterall, shouldn’t one expect the relaxation of the Hamilto-nian reminisce thermalization resulting in equipartition ofenergy among the different eigenstates? In fact, to the con-trary, eigenstate thermalization would imply the absenceof an attractor solution as multiple final states would beequally probable. So if one is looking for an attractoras a solution, and consequently a model of quasicrystalgrowth, thermalization is undesirable. So it works to ouradvantage that the Hamiltonian H does not thermalizeto equiprobable eigenstates.

In fact, let us rewrite the spectral decomposition givenby equation (16) as

EψF =∑n

|cn|2En =∑(m)

|c(m)|2E(m), (19)

where by convention |c(1)|2 ≥ |c(2)|2 ≥ |c(3)|2 ≥ . . . andE(m) ∈ ρ(H ) (spectrum of H ) is such that its coef-ficient is c(m). In other words, the second equality is

Page 12 of 14 Eur. Phys. J. B (2020) 93: 67

Fig. 6. Top: energy eigen values E1 and E2 of the HamiltonianH are plotted along with the trace of the Hamiltonian whichgives the total energy of the system. Clearly, |E1| |E2|.Bottom: this plot shows that the relative error in the approx-imation of Tr(H ) by E1 decreases as N increases and ispractically miniscule. The plots show that for moderately largeN , Tr(H ) can be well approximated by E1.

simply a re-arrangement of the summands in∑n|cn|2En

that obeys the above inequalities and sorts the energyeigenstates in decreasing order of the probability of find-ing the Fibonacci chain in that state. It turns out thatc(1) ≡ c1 and consequently E(1) ≡ E1. More importantly,

the Hamiltonian H is such that |c(1)|2 |c(2)|2 as shownclearly in Figure 5. In Section 3, we have presented resultsof numerical experiments that established that of all thepossible configurations, it is the Fibonacci configurationthat minimizes the total Hamiltonian HΩ = Tr(H ). Notethat practically almost all of the total energy of the sys-tem is concentrated in the ground state as illustrated byFigure 6. In essence, since E1 → HΩ as N becomes largeenough, it follows that the Fibonacci configuration notonly minimizes HΩ among all possible chain configura-tions but also minimizes the ground state energy amongall possible chain configurations. This establishes that theFibonacci chain is the most probable ground state of theHamiltonian H and by a big margin (see Fig. 5).

It must be emphasized here that the ability of theempires, through simple non-local interactions as pre-scribed by the r.h.s. of equations (12) and (14), to formthe final Fibonacci state of the chain renders the empiresas the generators of the Fibonacci chain and fundamentalelements in a model of aperiodic order. It can be shownthat the simple nearest neighbor Ising model does not gen-erate the Fibonacci chain.6 Essentially, it is the non-localscope of the empires that is the quintessential element ofthe model.

6 https://www.youtube.com/watch?v=N85 aDD lUI

4.3 Symmetries and comparison with the Ising model

The most fundamental symmetry in crystalline struc-tures is translation symmetry. This symmetry is absentin quasicrystalline structures. In the current case underinvestigation here, the precise algebraic form in whicha translation asymmetry pertaining to the Fibonacciquasicrystal may be conceptualized is illustrated by equa-tions (9), (10) and (11). These empires collectively charac-terize all forbidden configurations of the Fibonacci chain.The Hamiltonian H encodes the exact physical mecha-nism which manifests this translation asymmetry throughthe interaction of the empires.

Of course, this beckons a natural comparison with otherlattice models especially with the most widely-knownnearest neighbor Ising Hamiltonian. At the outset, it mustbe clarified that the Monte Carlo simulation of the nearestneighbor Ising Hamiltonian, that relies entirely on localspin–spin interactions, does not generate the Fibonaccichain or any one dimensional quasicrystal. This clearlyunderscores the necessary role of non-local interactions ina model of quasicrystal growth.

The definition of the empires (as illustrated by theexamples (5) and (6), and the top and bottom figureson the left panel of Fig. 3) imply that the inner productJm,n〈Eαm |Eαn〉 in equation (12) is essentially equivalent

to∑Ni=1 J

im,ns

mi s

ni =

∑Ni=1 J

im,ns

2i because smi = sni = si

where si labels the ith tile of the chain under consider-ation. Hence, one may attempt to devise an equivalentzero-field lattice model of the form

H(1)Ω = − 1

N

∑m,n∈X

N∑i=1

J im,nsmi s

ni

= − 1

N

∑m,n∈X

N∑i=1

J im,nsisi

= − 1

N

∑m,n∈X

N∑i=1

J im,ns2i (20)

that generates the Fibonacci chain. Here si = ±1 andJm,n ≡ J im,n is a vector for every m and n with non-zero

unit entries at locations where the effect of the spins (s2i )

must be accounted for. Note that in the above equationwe are no longer using the Einstein’s summation conven-tion. A careful observation of the interaction terms in theHamiltonian (12) and the definition of the empire vec-tors reveals the formal similarities with the Hamiltonian(20). However, such an attempt to engineer the couplingconstants does not encode the information of the pro-jection from a higher dimensional space. Moreover, theredoes not appear a simple way to construct the couplingconstants J im,n and hence using model (20) to performthe analysis and the simulations, instead of the equiva-lent models (12) and (14), would be very challenging. Onthe other hand, the empires provide us with a tool tomodel non-local interactions very naturally and share ageometrical significance as the generators of the Fibonaccichain because they are derived by projection from a higher

Eur. Phys. J. B (2020) 93: 67 Page 13 of 14

dimension which has translation symmetry. In any casethe model (12) presented here is clearly not an Ising modelfor reasons mentioned above.

Like the nearest neighbor Ising model, the HamiltonianHΩ is invariant under spin inversion as mentioned earlier.This is so because the inner product between the empirevectors is invariant under spin inversion. Eg., considerEαm = (s1s2s3 · · · sN ) and Eβn = (s′1s

′2s′3 · · · s′N ) where

si, s′i = −1, 1 or 0 depending on xi = L, S or if it is an

unforced tile. Clearly, the spin inversion si → −si ands′i → −s′i, which is essentially flipping the labels of L andS, preserves 〈Eαm |Eβn〉.

However, unlike the zero external field nearest neigh-bor Ising model in one-dimension or the nearest neighborIsing model in a square lattice in two-dimensions, thezero-external field (Bαk,i ≡ 0) partition function ZΩ =Tr(e−βHΩ) based on equation (12), and thereby the freeenergy density, are not invariant under sign inversion ofthe coupling coefficients Jj,i. This is because the non-zero entries of the interacting empire vectors change signconcurrently. This is further illustrated by the equivalentmodel (20). Therefore, the thermodynamic properties ofthe model presented here are not the same under thereversal of sign of the coupling constants Jj,i.

5 Summary of main results

The main contributions are summarized below.

– Algebraic forms of the empires of the VCs ofa Fibonacci chain are provided. The closed formexpressions are verified in agreement with earlierknown methods of computing empires of VCs usinggeometric and algorithmic methods. These algebraicforms enable us to know the exact manner by whichthe translation symmetry is absent in a Fibonaccichain.

– A Hamiltonian is constructed using the empiresmentioned above and Monte Carlo simulations areperformed. The simulations show that the Fibonaccichain is an attractor of the model.

– The Hamiltonian is promoted to a matrix operatorform and a spectral analysis is performed to find therelevant eigenstates. An ansatz is provided to findthe attractor configuration analytically. It is verifiedto be true and consistent for a chain of any finitelength. A quantum mechanical interpretation revealsthat the Fibonacci chain is the most probable groundstate of the system and hence provides a theoreti-cal explanation of the attractor configuration. Thetheoretical analysis shows that the Fibonacci config-uration not only minimizes the total energy of thesystem but also minimizes the ground state of thesystem.

We have devised a matrix Hamiltonian model of theFibonacci quasicrystal. This approach may lay thefoundation for formulating similar matrix models of morecomplex higher dimensional quasicrystals.

6 Future work

Firstly, there seems a formative correspondence betweenthe empire Hamiltonian presented in this paper and thewell known spin-1 Ising model with the spin-1 angu-lar momentum matrices Lx, Ly, Lz, each of dimension3 × 3, and with diagonal entries of Lz as 1, 0,−1. [60,61]. This similarity stems from the use of ternary ele-ments 1,−1, 0. One may propose the following zero-fieldHamiltonian

H (∗) = −∑

a=x,y,z

∑m,n∈X

Jm,nLamL a

n (21)

as a non-local generalization of the spin-1 nearest neigh-bor Hamiltonian. Here, a, b = 1, 2, 3 correspond to x, y, zand Jm,n are the non-local coupling constants betweenlattice sites m and n. By defining L a

m := I⊗m−1 ⊗ La ⊗I⊗N−m, a = x, y, z, as 3N × 3N matrices and appropri-ately engineering Jm,n, one may investigate the correspon-

dence between H (∗) and H given by equation (14). HereI is a 3× 3 identity matrix.

Secondly, the one-dimensional model described heremay be extended to a two dimensional model of quasicrys-tal by relaxing a network of chains (fibers). The networkof chains may be relaxed using a Monte Carlo simulationunder specific topological constraints to obtain a matrixmodel of a two dimensional quasicrystal. A Fibonaccitopological network based on a quantum string-net Hamil-tonian has been studied recently [62].

The first author wishes to acknowledge some useful discussionswith Raymond Aschheim regarding empires of vertex configu-rations. The second author wishes to thank Meredith Bowersfor her support. Prashant Singh Rana and Rajendra KumarSharma of Thapar Institute are acknowledged gratefully forproviding us access to their advanced computing laboratoriesthat enabled us to perform some of the simulations and anal-ysis presented in the revised version of the manuscript. Theauthors would also like to acknowledge the generous hospital-ity of Narendra Sharma and his team of Thapar Institute formaking the stay of the second author a very pleasant experi-ence. Finally, the authors would like to thank one anonymousreferee for comments and suggestions to improve this paper.

Author contribution statement

Both authors were involved in the development of themodel and in the preparation of the manuscript. Bothauthors have read and approved the final manuscript.

Publisher’s Note The EPJ Publishers remain neutral withregard to jurisdictional claims in published maps and institu-tional affiliations.

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