GOE-GUE-Poisson transitions in the nearest neighbor spacing distribution ofmagnetoexcitons

Frank Schweiner, Jorg Main, and Gunter WunnerInstitut fur Theoretische Physik 1, Universitat Stuttgart, 70550 Stuttgart, Germany

(Dated: March 13, 2017)

Recent investigations on the Hamiltonian of excitons by F. Schweiner et al. [Phys. Rev. Lett. 118,046401 (2017)] revealed that the combined presence of a cubic band structure and external fieldsbreaks all antiunitary symmetries. The nearest neighbor spacing distribution of magnetoexcitons canexhibit Poissonian statistics, the statistics of a Gaussian orthogonal ensemble (GOE) or a Gaussianunitary ensemble (GUE) depending on the system parameters. Hence, magnetoexcitons are an idealsystem to investigate the transitions between these statistics. Here we investigate the transitionsbetween GOE and GUE statistics and between Poissonian and GUE statistics by changing the angleof the magnetic field with respect to the crystal lattice and by changing the scaled energy knownfrom the hydrogen atom in external fields. Comparing our results with analytical formulae for thesetransitions derived with random matrix theory, we obtain a very good agreement and thus confirmthe Wigner surmise for the exciton system.

PACS numbers: 05.30.Ch, 05.45.Mt, 71.35.-y, 61.50.-f

I. INTRODUCTION

Ever since the Bohigas-Giannoni-Schmit conjecture [1],which stated that these quantum systems can be de-scribed by random matrix theory [2, 3], it has been shownthat irregular classical behavior manifests itself in statis-tical quantities of the corresponding quantum system [4].In random matrix theory the Hamiltonian of a system isreplaced by a random matrix with appropriate symme-tries to study the statistical properties of its eigenvaluespectrum [5]; so only universal quantities of a system areconsidered and detailed dynamical properties are irrele-vant. Even though Hamiltonians of dynamical systemsare not random in most cases, it is already understoodthat spectral fluctuations for nonrandom and randomHamiltonians are equivalent [6–8].

All systems with a Hamiltonian leading to global chaosin the classical dynamics can be assigned to one of threeuniversality classes: the orthogonal, the unitary or thesymplectic universality class [7]. To which of these uni-versality classes a given system belongs is determinedby the remaining symmetries in the system. Most of thephysical systems still have time-reversal or at least one re-maining antiunitary symmetry and thus show the statis-tics of a Gaussion orthogonal ensemble (GOE). Some ex-amples of these systems are nuclei in external magneticfields [9–12], microwave billards [13–15], molecular spec-tra [16], impurities [17], and quantum wells [18]. Atomsin constant external fields, in particular, are among themost important physical systems belonging to the orthog-onal universality class [19–21]. They are ideal systems toinvestigate the emergence of quantum chaos both in high-precision experimental measurements and precise quan-tal calculations, possible because of the availability ofthe analytically known Hamiltonian (see Refs. [4, 22] andfurther references therein). Hence, they are a perfectlysuitable physical system to study the transition from thePoissonian level statistics, which describes the classically

integrable case in the absence of the fields [7, 23], to GOEstatistics [21], where the breaking of symmetries due tothe external fields leads to a correlation of levels andhence to a strong suppression of crossings [7].

As regards the other universality classes, examples aremuch rarer since systems without any antiunitary symme-try [Gaussian unitary ensemble (GUE)] or systems withtime-reversal invariance possessing Kramer’s degeneracybut no geometric symmetry at all [Gaussian symplec-tic ensemble (GSE)] have to be found [7]. Until nowGUE statistics was observable in rather exotic systemssuch as microwave cavities with ferrite strips [24], atomsin a static electric field and a resonant microwave fieldof elliptical polarization [25], a kicked rotor or a kickedtop [6, 26, 27], the metal-insulator transition in the An-derson model of disordered systems [28], which can becompared to the Brownian motion model [29], or forbillards in microwave resonators [30], and in graphenequantum dots [31]. Since random matrix theory has al-ready been extended to describe also transitions betweenthe different statistics with analytical functions [5], it ishighly desirable to study these transitions theoreticallyand experimentally. However, due to the small number ofphysical systems showing GUE statistics, there are onlyfew examples, where transitions from Poissonian to GUEstatistics or from GOE to GUE statistics in dependenceof a parameter of the system could be studied [6, 26–28, 32]. Often only mathematical models with specifi-cally designed Hamiltonians are introduced to investigatethese transitions [5].

In this paper we will investigate these transitions inmagnetoexcitons. Excitons are the fundamental opti-cal excitations in the visible or ultraviolet spectrum ofa semiconductor and consist of an electron in the con-duction band and a positively charged hole in the va-lence band. As the interaction between both quasi par-ticles can be described by a screened Coulomb interac-tion, excitons are often regarded as the hydrogen analog

2

of the solid state. Only three years ago T. Kazimier-czuk et al [33] observed in a remarkable high-resolutionabsorption experiment an almost perfect hydrogen-likeabsorption series for the yellow exciton in cuprous oxide(Cu2O) up to a principal quantum number of n = 25.This experiment has opened the field of research of giantRydberg excitons, and has stimulated a large number ofexperimental and theoretical investigations [33–46].

Very recently, we have shown that the Hamiltonianof magnetoexcitons in cubic semiconductors breaks allantiunitary symmetries [47]. This is the first evidence fora spatially homogeneous system breaking all antiunitarysymmetries.

Since in many cases excitons are treated theoreticallyvia a hydrogen-like Hamiltonian, the appearance of GUEstatistics seems surprising as the hydrogen atom in exter-nal fields still shows one antiunitary symmetry. However,it is well known that the hydrogen-like model of excitonsis often too simple to account for the huge number ofeffects in the solid (see, e.g., Refs. [35, 38, 40, 48–52]and further references therein). M. Aßmann et al. [34]attributed the appearance of GUE statistics in a recentexperiment with magnetoexcitons in Cu2O to the inter-action of excitons with phonons.

However, we have shown that it is indispensable to ac-count for the complete valence band structure to describethe spectra of excitons in magnetic fields in a theoreti-cally correct way [53]. Without the complete band struc-ture the striking experimental finding of a dependence ofthe magnetoexciton spectra on the direction of the exter-nal magnetic field cannot be explained. It is indeed thesimultaneous presence of the cubic band structure andexternal fields which breaks all antiunitary symmetriesand leads to GUE statistics [47].

In this paper we investigate the symmetry breaking forexcitons in semiconductors with a cubic band structurein dependence on system parameters such as the strengthand the angle of the magnetic field or the scaled en-ergy [4, 54]. Since the eigenvalue spectrum of the magne-toexciton Hamiltonian shows Poissonian, GOE or GUEstatistics depending on these parameters, it is an idealsystem to investigate the transitions between GOE andGUE or Poisson and GUE statistics. To the best of ourknowledge, there are only two more systems where bothtransitions have been studied, i.e., the kicked top [27]and the Anderson model [28]. However, while the kickedtop is a time-dependent system, which has to be treatedwithin Floquet theory [6, 27], the Anderson model israther a model system for a d-dimensional disorderedlattice, where parameters such as the disorder and thehopping rate need to be adjusted [28]. Magnetoexcitonsare a more realistic physical system allowing for a system-atic investigation of transitions between different statis-tics. In particular, the parameters describing these tran-sitions can be easily adjusted in experiments. Comparingour results with analytical functions from random ma-trix theory describing the transitions between the statis-tics [5, 6], we confirm the so-called Wigner surmise [55],

which states that the NNS of large random matrices canbe approximated by the NNS of 2 × 2 matrices of thesame universality class [5].

The paper is organized as follows: In Sec. II we presentthe Hamiltonian of excitons in cubic semiconductors inan external magnetic field and introduce a complete ba-sis to solve the corresponding Schrodinger equation. Themethods of solving the Schrodinger equation for fixed val-ues of the external field strenghts or for a constant scaledenergy are discussed in Secs. II A and II B, respectively.Having shown analytically that the presence of the cubicband structure and external fields breaks all antiunitarysymmetries in Sec. III, we investigate the eigenvalue spec-trum and the level spacing statistics numerically At first,we demonstrate the appearance of GOE or GUE statis-tics for specific directions of an external magnetic fieldin Sec. IV. The transitions between different level spac-ing statistics are then investigated in Secs. V A and V B.Finally, we give a short summary and outlook in Sec. VI.

II. HAMILTONIAN AND COMPLETE BASIS

In this section we briefly discuss the Hamiltonian ofexcitons in direct semiconductors with a cubic valenceband structure and show how to solve the correspond-ing Schrodinger equation in a complete basis. For moredetails see Refs. [40, 53] and further references therein.

When neglecting external fields at first, the Hamilto-nian of excitons in direct semiconductors is given by [56]

H = V (re − rh) +He (pe) +Hh (ph) . (1)

The Coulomb interaction between the electron (e) andthe hole (h) is screened by the dielectric constant ε:

V (re − rh) = − e2

4πε0ε

1

|re − rh|. (2)

Since the conduction band is often parabolic, the ki-netic energy of the electron is similar to that of a freeparticle

He (pe) =p2e

2me. (3)

However, the effective mass me of the electron in thesemiconductor has to be used instead of the free electronmass m0. As regards the valence bands, the situationis more complicated. In general, the uppermost valenceband is threefold degenerate at the center of the Brillouinzone or the Γ point and the kinetic energy of a hole withinthese valence bands is given by [39, 40]

Hh (ph) =(1/2~2m0

) {~2 (γ1 + 4γ2)p2

h

− 6γ2(p2h1I

21 + c.p.

)− 12γ3 ({ph1, ph2} {I1, I2}+ c.p.) (4)

3

with p = (p1, p2, p3), {a, b} = 12 (ab+ ba) and c.p. de-

noting cyclic permutation. The three Luttinger param-eters γi describe the behavior and the anisotropic effec-tive mass of the hole. The matrices Ij denote the threespin matrices of the quasispin I = 1 which describes thethreefold degenerate valence band [57]. The componentsof these matrices Ii read [40, 57]

Ii, jk = −i~εijk (5)

with the Levi-Civita symbol εijk.Note that the expression for Hh (ph) can be separated

in two parts having spherical and cubic symmetry, re-spectively [58]. The coefficients µ′ and δ′ of these partscan be expressed in terms of the three Luttinger pa-rameters: µ′ = (6γ3 + 4γ2) /5γ′1 and δ′ = (γ3 − γ2) /γ′1with γ′1 = γ1 + m0/me [40, 51, 58]. The spin-orbit cou-pling Hso, which generally enters the kinetic energy of thehole (4), is neglected here since it is spherically symmet-ric and therefore does not affect the symmetry propertiesof the exciton Hamiltonian.

When applying external fields, the correspondingHamiltonian is obtained via the minimal substitution.After introducing relative and center of mass coordi-nates [59, 60] and setting the position and momentumof the center of mass to zero, the complete Hamiltonianof the relative motion reads [50, 59–64]

H = V (r) + eΦ (r)

+ He (p + eA (r)) +Hh (−p + eA (r)) (6)

with the relative coordinate r = re − rh and the relativemomentum p = (pe − ph) /2 of electron and hole. Weuse the vector potential A = (B × r) /2 of a constantmagnetic field B and the electrostatic potential Φ (r) =−F · r of a constant electric field F .

As we will show in Sec. III, the symmetry breakingin the system depends on the orientation of the fieldswith respect to the crystal lattice. We will denote theorientation of B and F in spherical coordinates via

B (ϕ, ϑ) = B

cosϕ sinϑ,sinϕ sinϑ

cosϑ

(7)

and similar for F in what follows.Before we solve the Schrodinger equation correspond-

ing to the Hamiltonian (6), we rotate the coordinate sys-tem to make the quantization axis coincide with the di-rection of the magnetic field (see Appendix A) and thenexpress the Hamiltonian (6) in terms of irreducible ten-sors [58, 64, 65]. We can then calculate a matrix repre-sentation of the Schrodinger equation using a completebasis.

Note that the Hamiltonian (6) is a model system formagnetoexcitons since we neglect the spin-orbit couplingbetween the quasi spin I and the hole spin Sh, whichappears, e.g., in Cu2O [40, 53]. Furthermore, we neglectan additional term in Eq. (6), which describes the energy

of the electron and hole spin in the magnetic field but isinvariant under the symmetry operations considered be-low. Therefore, we can disregard these spins in our basis.As regards the angular momentum part of the basis, wehave to consider that the Hamiltonian (6) couples the an-gular momentum L of the exciton and the quasi spin I.Hence, we introduce the total momentum G = L+I withthe z component MG. For the radial part of the excitonwave function we use the Coulomb-Sturmian functions ofRef. [66]

UNL (r) = NNL (2ρ)Le−ρL2L+1

N (2ρ) (8)

with ρ = r/α, a normalization factor NNL, the associatedLaguerre polynomials Lmn (x) and an arbitrary scaling pa-rameter α. Note that we use the radial quantum numberN , which is related to the principal quantum number nvia n = N +L+1. Finally, we make the following ansatzfor the exciton wave function

|Ψ〉 =∑

NLGMG

cNLGMG|Π〉 , (9a)

|Π〉 = |N, L, I, G, MG〉 , (9b)

with complex coefficients c.The Schrodinger equation can now be solved for fixed

values of the external field strengths or for a fixedvalue of the scaled energy known from atoms in exter-nal fields [54]. Both methods will be presented in thefollowing

A. Constant field strengths

Inserting the ansatz (9) in the Schrodinger equa-tion HΨ = EΨ yields a matrix representation of theSchrodinger equation of the form [47]

Dc = EMc, (10)

where the external field strengths are assumed to be con-stant. The vector c contains the coefficients of the ex-pansion (9). Since the functions UNL (r) actually de-pend on the coordinate ρ = r/α, we substitute r → ραin the Hamiltonian (6) and multiply the correspondingSchrodinger equation by α2. All matrix elements whichenter the hermitian matrices D and M can be calculatedsimilarly to the matrix elements given in Refs. [40, 53].The generalized eigenvalue problem (10) is finally solvedusing an appropriate LAPACK routine [67].

Since in numerical calculations the basis cannot be in-finitely large, the values of the quantum numbers arechosen in the following way: For each value of n =N + L+ 1 ≤ nmax we use

L = 0, . . . , n− 1,

G = |L− 1| , . . . , min (L+ 1, Gmax) , (11)

MG = −G, . . . , G.

4

The values Gmax and nmax are chosen appropriately largeso that as many eigenvalues as possible converge. Addi-tionally, we can use the scaling parameter α to enhanceconvergence. In particular, if the eigenvalues of excitonicstates with principal quantum number n are to be becalculated, we can set α = nγ′1εa0 according to Ref. [66],where a0 denotes the Bohr radius.

Note that without an external electric field, parityis a good quantum number and the operators in theSchrodinger equation couple only basis states with evenor with odd values of L. In this case we consideronly basis states with odd values of L as these excitonstates can be observed in parity-forbidden semiconduc-tors [38, 40, 44].

B. Constant scaled energy

Besides solving the Schrodinger equation or the gener-alized eigenvalue problem (10) for fixed values of the ex-ternal field strength, it is also possible to use the conceptof scaled energy [54]. In classical mechanics the Hamil-tonian of a hydrogen atom in external fields possesses ascaling property which allows reducing the three parame-ters energy E, magnetic field B and electric field F to twoparameters [68, 69]. The corresponding transformationreads

r = γ2/3r, p = γ−1/3p,

F = γ−4/3F , E = γ−2/3E, (12)

with γ = B/B0 and B0 = 2.3505×105 T [4]. This scalingis not applicable in quantum mechanics since [ri, pj ] =

i~γ1/3δij 6= i~δij holds. However, it is possible to define

a scaled quantum Hamiltonian by substituting r = γ2/3rin the Schrodinger equation and introducing the scaledenergy E = γ−2/3E.

We will now apply this scaling to the exciton system.Let us write the Hamiltonian of excitons (6) in the form

H = − e2

4πε0ε

1

r− eF · r

+ H0 + (eB)H1 + (eB)2H2, (13)

with the Hi given in Appendix A. Due to the effectivemasses of electron and hole and due to the scaling of theCoulomb energy by the dielectric constant, we introduceexciton Hartree units so that the hydrogen-like part ofthe Hamiltonian is exactly of the same form as that ofthe hydrogen Hamiltonian in normal Hartree units [37](see Appendix B). Variables in exciton Hartree units willbe indicated by a tilde sign.

Performing the substitution r = γ2/3r/α in the cor-responding Schrodinger equation, where we now have touse γ = B/B0 with B0 = 2.3505 × 105 T/

(γ′21 ε

2), and

multiplying the resulting equation with α2γ2/3, we ob-

tain

−αr− γ4/3α3F · r

+ γ2/3H0 + γ1/3α2H1 + α4H2

= γ−2/3α2E. (14)

As for the hydrogen atom, we define the scaled en-ergy E = γ−2/3E and scaled electric field strengthF = γ4/3F . When using the complete basis of Eq. (9),Eq. (14) represents a quadratic eigenvalue problem of theform

Ac = τBc + τ2Cc (15)

with hermitian matrices A, B, and C and an eigenvalueτ = γ1/3. The eigenvalue problem can be changed toa standard generalized eigenvalue problem by defining avector d = τc:(

A B0 1

)(cd

)= τ

(0 C1 0

)(cd

). (16)

This eigenvalue problem is solved for constant scaled en-ergies E using an appropriate LAPACK routine [67].

We finally note that due to the substitution r =γ2/3r/α and due to the use of exciton Hartree units, adifferent value of the free convergence parameter α thanin Sec. II has to be used to obtain convergence for theexciton states with principal quantum number n. Thisvalue is given by α ≈ nγ2/3.

III. DISCUSSION OF ANTIUNITARYSYMMETRIES

In a previous paper [47] we have shown analyticallythat the last remaining antiunitary symmetry knownfrom the hydrogen atom in external fields is broken forthe exciton Hamiltonian (6) for most orientations of theexternal fields. For the reader’s convenience we recapit-ulate the most important steps as some of the results areimportant for the following discussions.

The matrices Ii of the quasi-spin I = 1 given by Eq. (5)are not the standard spin matrices Si of spin one [70].However, these matrices obey the commutation rules [57]

[Ii, Ij ] = i~3∑k=1

εijkIk, (17)

for which reason a unitary transformation can be foundso that U †IiU = Si holds. Since in Ref. [70] the behaviorof the standard spin matrices under symmetry operationssuch as time reversal and reflections are given, we will usethe matrices Si instead of the Ii in the following.

In the special case of vanishing Luttinger parametersγ2 = γ3 = 0, the exciton Hamiltonian (6) is of the same

5

form as the Hamiltonian of a hydrogen atom in externalfields. It is well known that for this Hamiltonian there isstill one antiunitary symmetry left, i.e., that it is invari-ant under the combined symmetry of time inversion Kfollowed by a reflection Sn at the specific plane spannedby both fields [7]. This plane is given by the normalvector

n = (B × F ) / |B × F | (18)

or n ⊥ B = B/B if F = 0 holds. Therefore, thehydrogen-like system shows GOE statistics in the chaoticregime.

As the hydrogen atom is spherically symmetric in thefield-free case, it makes no difference whether the mag-netic field is oriented in z direction or not. However, ina semiconductor with δ′ 6= 0 the Hamiltonian has cu-bic symmetry and the orientation of the external fieldswith respect to the crystal axis of the lattice becomesimportant. Any rotation of the coordinate system withthe aim of making the z axis coincide with the directionof the magnetic field will also rotate the cubic crystallattice. The only remaining antiunitary symmetry men-tioned above is now broken for the exciton Hamiltonian ifthe plane spanned by both fields is not identical to one ofthe symmetry planes of the cubic lattice. Even withoutan external electric field the symmetry is broken if themagnetic field is not oriented in one of these symmetryplanes. Only if the plane spanned by both fields is iden-tical to one of the symmetry planes of the cubic lattice,the antiunitary symmetry KSn with n given by Eq. (20)is present since only then the reflection Sn transformsthe lattice into itself.

This criterion can also be expressed in a different way:The antiunitary symmetry known from the hydrogenatom is broken if none of the normal vectors ni of the 9symmetry planes of the cubic lattice given by

n1 = (1, 0, 0)T,

n2 = (0, 1, 0)T,

n3 = (0, 0, 1)T,

n4 = (1, 1, 0)T/√

2,

n5 = (0, 1, 1)T/√

2,

n6 = (1, 0, 1)T/√

2,

n7 = (1, −1, 0)T/√

2,

n8 = (0, 1, −1)T/√

2,

n9 = (−1, 0, 1)T/√

2, (19)

is parallel to

n = (B × F ) / |B × F | , (20)

or, in the case of F = 0, if none of these vectors is per-pendicular to

B = B/B. (21)

Since the breaking of all antiunitary symmetries de-pends on the relative orientation of the external fieldsto all normal vectors ni, we can introduce a parameterwhich is a qualitative measure for the deviation from thecases with antiunitary symmetry:

σ =

[9∑i=1

|B × F |2

|ni × (B × F )|2

]− 12

. (22)

For the special case of F = 0 we define

σ =

[9∑i=1

(ni · B

)−2]− 12

. (23)

We have σ = 0 for the cases with antiunitary symme-try; and that symmetry is more and more broken withincreasing values of σ.

Under time inversion K and reflections Sn at a planeperpendicular to a normal vector n the vectors of positionr, momentum p and spin S transform according to [70]

KrK† = r, (24a)

KpK† = −p, (24b)

KSK† = −S, (24c)

and

SnrS†n = r − 2n (n · r) , (25a)

SnpS†n = p− 2n (n · p) , (25b)

SnSS†n = −S + 2n (n · S) . (25c)

For all orientations of the external fields the hydrogen-like part of the Hamiltonian (6) is invariant under KSn

with n given by Eq. (20). However, other parts of theHamiltonian such as Hc =

(p21S

21 + c.p.

)[see Eq. (4)] are

not invariant if σ 6= 0 holds. For example, for the casewith B (0, 0) and F (π/6, π/2), we obtain

SnKHcK†S†n −Hc

= 1/8[2√

3(S22 − S2

1

)p1p2

+ 3(S21p

22 + S2

2p21

)− 3

(S21p

21 + S2

2p22

)+ {S1,S2}

(2√

3(p22 − p21

)+ 12p1p2

)]6= 0 (26)

with n =(−1/2,

√3/2, 0

)T. Note that even though Hc

does not depend on the external fields, the normal vectorn is determined by these fields via Eq. (20). Otherwise,the hydrogen-like part of the Hamiltonian would not beinvariant under KSn.

Since the expression in Eq. (26) is not equal to zero, wehave shown for B (0, 0) and F (π/6, π/2) that the gen-eralized time-reversal symmetry of the hydrogen atom isbroken for excitons due to the cubic symmetry of thesemiconductor. The same calculation can also be per-formed for other orientations of the external fields. Aswe have stated above, the antiunitary symmetry remainsunbroken only for specific orientations of the fields.

6

P(s

)

0.0

0.5

1.0

(a) ϕ = 0

F(s

)

0.0

0.5

1.0

(b) ϕ = 0P

(s)

s

0.0

0.5

1.0

0.5 1.5 2.5

(c) ϕ = π/6

F(s

)

s

num. results

FP(s)

FGOE(s)

FGUE(s)0.0

0.5

1.0

0.5 1.5 2.5

(d) ϕ = π/6

FIG. 1: Level spacing probability distribution functions P (s) (left) and cumulative distribution functions F (s) (right) forδ′ = −0.15, B = 3 T, ϑ = π/6, and two different values of ϕ. Besides the numerical data (red boxes or red dots), we alsoshow the corresponding functions of a Poissonian ensemble (black dashed line), GOE (blue dash-dotted line), and GUE (greensolid line). Only if the magnetic field is oriented in one of the symmetry planes of the lattice, one antiunitary symmetry ispresent and GOE statistics can be observed (a,b). In all other cases, all antiunitary symmetries are broken and GUE statisticsappears (c,d).

IV. APPEARANCE OF GOE AND GUESTATISTICS

We will now demonstrate the breaking of all an-tiunitary symmetries by analyzing the nearest-neighborspacings of the energy eigenvalues corresponding to theHamiltonian (6) [21] for a model system with the ar-bitrarily chosen set of parameters Eg = 0, ε = 7.5,me = m0, γ′1 = 2, µ′ = 0, and δ′ = −0.15. If we setF = 0, we expect to obtain GUE statistics in the limitof high energies as long as the magnetic field is not ori-ented in one of the symmetry planes of the lattice.

Before analyzing the nearest-neighbor spacings, wehave to unfold the spectra to obtain a constant meanspacing [1, 7, 21, 71]. The unfolding procedure sepa-rates the average behavior of the non-universal spectraldensity from universal spectral fluctuations and yieldsa spectrum in which the mean level spacing is equal tounity [5]. Furthermore, we have to leave out a certainnumber of low-lying sparse levels to remove individualbut nontypical fluctuations [21].

Since the magnetic field breaks all symmetries in the

system and limits the convergence of the solutions of thegeneralized eigenvalue problem with high energies [40],the number of level spacings analyzed here is compar-atively small and comprises about 250 to 500 excitonstates. In this case, the cumulative distribution func-tion [72]

F (s) =

∫ s

0

P (x) dx (27)

is often more meaningful than histograms of the levelspacing probability distribution function P (s).

We will compare our results with the distribution func-tions known from random matrix theory [1, 34]: the Pois-sonian distribution

PP(s) = e−s (28)

for non-interacting energy levels, the Wigner distribution

PGOE(s) =π

2se−πs

2/4, (29)

and the distribution

PGUE(s) =32

π2s2e−4s

2/π (30)

7

P(s

)PP(s)

PGOE(s)

0.0

0.5

1.0

P → GOE

F(s

)

FP(s)

FGOE(s)0.0

0.5

1.0

F(s

)-F

P(s

)

s

FP(s)

FGOE(s)-0.2

0.0

0.2

0.0 0.5 1.0 1.5 2.0

P(s

)

PP(s)

PGUE(s)

0.0

0.5

1.0

P → GUE

F(s

)

FP(s)

FGUE(s)0.0

0.5

1.0

F(s

)-F

P(s

)

s

FP(s)

FGUE(s)-0.2

0.0

0.2

0.0 0.5 1.0 1.5 2.0

P(s

)

PGOE(s)

PGUE(s)

0.0

0.5

1.0

GOE → GUE

F(s

)

FGOE(s)

FGUE(s)0.0

0.5

1.0

F(s

)-F

GO

E(s

)

s

FGOE(s)

FGUE(s)-0.1

0.0

0.1

0.0 0.5 1.0 1.5 2.0

FIG. 2: (Color online) Level spacing probability distribution functions P (s) (first row) and cumulative distribution functionsF (s) (second row) for the transitions P → GOE, P → GUE, and GOE → GUE. The blue dotted lines show the transitionfunctions of Eqs. (31), (32), and (33) for λ = 0.1, 0.2, 0.4, 0.6, 0.8. To visualize the differences between the cumulativedistribution functions more clearly, especially for the GOE → GUE transition, we also show in the third row the differencebetween the distributions for a fixed value of λ and the initial distribution, respectively.

for systems without any antiunitary symmetry. It canbe seen that the most striking difference between thethree distributions is the behavior for small values of s.While for the Poissonian distribution the probability oflevel crossings in nonzero and thus PP(0) 6= 0 holds, inchaotic spectra the symmetry reduction leads to a cor-relation of levels and hence to a strong suppression ofcrossings. Note that the most characteristic feature ofGOE or GUE statistics is the linear or quadratic levelrepulsion for small s, respectively.

In Fig. 1 we show the results for level spacing probabil-ity distribution function and the cumulative distributionfunction for B (0, π/6) and B (π/6, π/6) obtained witha constant magnetic field strength of B = 3 T and ex-citon states within a certain energy range. While forB (0, π/6) the magnetic field is oriented in one of thesymmetry planes of the lattice and thus only GOE statis-tics can be observed, we see clear evidence for GUEstatistics as regards the case with B (π/6, π/6). Notethat we have chosen the values δ′ = −0.15 and B = 3 Tto be fixed. It is well known from atomic physics thatchaotic effects become more apparent in higher magneticfields or by using states of higher energies for the analysis.

Hence, by increasing B or investigating the statistics ofexciton states with higher energies, GUE statistics couldprobably be observed also for smaller values of |δ′|. Atthis point we have to note that an evaluation of numer-ical spectra for δ′ > 0 shows the same appearance ofGUE statistics. This is expected since the analyticallyshown breaking of all antiunitary symmetries in Sec. IIIis independent of the sign of the material parameters.

V. TRANSITIONS BETWEEN SPACINGDISTRIBUTIONS

To the best of our knowledge, there are only two phys-ical systems where both the transition from Poissonianto GUE statistics and the transition from GOE to GUEstatistics in dependence of a parameter of the systemcould be studied [27, 28]. As we have already stated inSecs. III and IV, our system shows Poisson, GOE or GUEstatistics in dependence on the energy, the magnetic fieldstrength and the angles ϑ and ϕ, i.e., in dependence ofexperimentally adjustable parameters. Thus, our sys-tem is perfectly suited to investigate transitions between

8

-0.1

0.0

0.1ϕ = 0π/48

λ = 0.000

ϕ = 3π/48

λ = 0.552

ϕ = 6π/48

λ = 0.595

F(s

)-F

GO

E(s

)

-0.1

0.0

0.1ϕ = 1π/48

λ = 0.236

ϕ = 4π/48

λ = 0.599

ϕ = 7π/48

λ = 0.537

-0.1

0.0

0.1

0.0 0.5 1.0 1.5 2.0

ϕ = 2π/48

λ = 0.430

s

0.0 0.5 1.0 1.5 2.0

ϕ = 5π/48

λ = 0.698

FGOE(s)

FGUE(s)

FGOE-GUE(s; λ)

0.0 0.5 1.0 1.5 2.0

ϕ = 8π/48

λ = 0.518

FIG. 3: Transition from GOE to GUE statistics for fixed values of the magnetic field strength B and increasing values ofthe angle ϕ in B (ϕ, ϑ = π/6). The results are presented in the same way as in the bottom most panel of Fig. 2 to showthe differences between FGOE (s) and FGUE (s) more clearly. The data points (red) were fitted with the analytical functionFGOE→GUE (s; λ). The optimum values of the fit parameter λ are given in each panel, but also shown in Fig. 4. One canobserve a good agreement between the numerical data and the analytical function describing the transition between the twostatistics in dependence on λ. Only for ϕ = 0 the data shows a slight slight admixture of Poissonian statistics to the expectedGOE statistics. For further information see text.

the different statistics or different symmetry classes whenchanging one or more of these parameters.

In Ref. [5] analytical expressions for the spacing distri-bution functions in the transition region between the dif-ferent statistics have been derived using random matrixtheory for 2×2 matrices. The transition from Poissonianto GOE statistics is described by

PP→GOE (s; λ) = Cse−D2s2

×∫ ∞0

dx e−x2/4λ2−xI0 (z) (31a)

with z = xDs/λ and

D (λ) =

√π

2λU

(−1

2, 0, λ2

), (31b)

C (λ) = 2D (λ)2, (31c)

a parameter λ, the Tricomi confluent hypergeometricfunction U (a, b z) [73] and the modified Bessel functionI0 (z) [73]. For the special cases of λ→ 0 or λ→∞ Pois-

sonian or GOE statistics is obtained, respectively. How-ever, already for λ >∼ 0.7 the transition to GOE statisticsis almost completed [5]. Note that besides the transi-tion formula (31) derived within random matrix theoryalso other interpolating distributions for the transitionP→ GOE have been proposed in the literature [74–78].

The transition from Poissonian to GUE statistics isdescribed by

PP→GUE (s; λ) = Cs2e−D2s2

×∫ ∞0

dx e−x2/4λ2−x sinh (z)

z(32a)

with z = xDs/λ and

D (λ) =1√π

+1

2λeλ

2

erfc (λ)− λ

2Ei(λ2)

+2λ2√π

2F2

(1

2, 1;

3

2,

3

2; λ2

), (32b)

C (λ) =4D (λ)

3

√π

, (32c)

9

λ(ϕ

)

0

0.2

0.4

0.6

0.8(a)σ

(ϕ)

ϕ

0

0.04

0.08

0.12

0π/48 2π/48 4π/48 6π/48 8π/48

(b)

FIG. 4: (a) Optimum values of the fit parameter λ in depen-dence on the angle ϕ for the situation presented in Fig. 3. Theblue dashed line only serves as a guide to the eye. (b) Thefunction σ (ϕ) of Eq. (23) for ϑ = π/6. We obtain a qualita-tively good agreement between both curves, i.e., as expected,both values λ (ϕ) and σ (ϕ) increase from zero to a certainvalue and then decrease for ϕ >∼ π/8.

the complementary error function erfc [73], the expo-nential integral Ei [73] and a generalized hypergeometricfunction 2F2 [79].

Finally, the transition from GOE to GUE statistics isgiven by

PGOE→GUE (s; λ) = Cse−D2s2erf

(Ds

λ

)(33a)

with

D (λ) =

√1 + λ2√π

(λ

1 + λ2+ arccot (λ)

), (33b)

C (λ) = 2√

1 + λ2D (λ)2. (33c)

As in Ref. [5], we calculate the distribution functionsfor λ = 0.01 × 1000(k−1)/999 with k = 1, . . . , 1000 andthen numerically integrate the results to obtain the corre-sponding cumulative distribution functions F (s; λ). Allthese functions are shown for different values of λ inFig. 2.

As the transition from Poissonian to GOE statisticshas been investigated in detail for the hydrogen atom inexternal fields [21], we will treat the two other transitionsin the following.

A. GOE → GUE

Let us start with the transition from GOE to GUEstatistics. For this case we solve the generalized eigen-

value problem (10) for different orientations of the mag-netic field B (ϕ, ϑ) by setting ϑ = π/6 and graduallyincreasing the angle ϕ from 0 to π/4. To increase thestatistical significance, we analyze and merge the levelspacings for B = 2.8 T, B = 3.0 T, and B = 3.2 T for agiven value of ϕ [21]. The results are finally fitted by thefunction FGOE→GUE (s; λ) and shown in Fig. 3.

For the special case of ϕ = 0 we obtain GOE statisticsas expected since the magnetic field is oriented in the

symmetry plane of the solid with n = (0, 1, 0)T

. Whenincreasing the angle ϕ, the parameter λ changes rapidlyfrom 0 to 0.5 and hence the transition from GOE to GUEstatistics is almost completed for ϕ >∼ 3π/48 (see Fig. 4).

The decrease of the parameter λ for ϕ >∼ π/8 in Fig. 4can be explained by considering the orientation of B withrespect to all symmetry planes of the lattice. Hence, wecalculate the value of the parameter σ of Eq. (23) forϑ = π/6 and increasing values of ϕ. It is obvious thatthe value of σ increases for 0 ≤ ϕ ≤ π/8 and decreasesfor π/8 ≤ ϕ ≤ π/4 since the magnetic field moves awayfrom the plane with n2 and then approaches the planewith n7. Therefore, the fact that B approaches the planewith n7 for ϕ ≥ π/8 explains the decrease of λ in Fig. 3.

B. Poisson → GUE

Let us now treat the transition from Poissonian toGUE statistics. It is known from the hydrogen atomin external fields that for fixed values of the magneticfield strength B the low-energy part of the eigenvaluespectrum will show Poissonian statistics while the high-energy part already shows GOE statistics. For a betterlevel statistics it is appropriate to analyze the spectrawith a constant scaled energy E.

For fixed small values of the scaled energy the cor-responding classical dynamics becomes regular and en-ergy eigenvalues of the quantum mechanical system willshow purely Poissonian statistics. On the other hand,as we have shown above, GUE statistics is observed bestat large energies and for angles ϕ and ϑ, for which themagnetic field is oriented exactly between two symmetryplanes of the lattice. Hence, keeping the values ϕ = π/8,ϑ = π/6, and δ = −0.15 fixed and increasing the scaledenergy, we expect to observe a transition from Poissonianto GUE statistics.

Having unfolded the spectra according to Ref. [21], wefit the numerical results by the function FP→GUE (s; λ)given in Eq. (32). It can be seen from Fig. 5 that weobtain a good agreement between the results for our sys-tem and the analytical function for all scaled energiesE > −0.9. The transition from Poissonian to GUE statis-tics takes place already at very small values of the scaledenergy −1.2 <∼ E <∼ −0.6 (see Fig. 6). This differs fromthe hydrogen atom in external fields where the statisticsis still Poisson-like for E <∼ −0.6 [21] and can be ex-plained by the presence of the cubic band structure here.Therefore, the presence of the cubic band structure in-

10

0.0

0.5

1.0

E^

= -1.009

λ = 0.221

E^

= -0.567

λ = 0.608

E^

= -0.363

λ = 0.651

F(s

)

0.0

0.5

E^

= -0.817

λ = 0.352

E^

= -0.484

λ = 0.642

E^

= -0.319

λ = 0.718

0.0

0.5

0.0 0.5 1.0 1.5 2.0

E^

= -0.675

λ = 0.540

s

0.0 0.5 1.0 1.5 2.0

E^

= -0.417

λ = 0.656

FP(s)

FGUE(s)

FP-GUE(s; λ)

0.0 0.5 1.0 1.5 2.0

E^

= -0.283

λ = 0.693

FIG. 5: Transition from Poissonian to GUE statistics for fixed values of the angles ϕ = π/8, ϑ = π/6 and increasing values

of the scaled energy E. Except for E = −1.009 a good agreement between the numerical data and the analytical functionFP→GUE (s; λ) is obtained. For further information see text.

λ(E^

)

E^

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2

FIG. 6: Optimum values of the fit parameter λ in dependenceon the scaled energy E for the situation presented in Fig. 5.The blue dashed line only serves as a guide to the eye. Thevalue of λ increases from a small value at low scaled energiesto about λ ≈ 0.7, where the function FP→GUE (s; λ) almostdescribes GUE statistics.

creases the chaos in comparison with the hydrogen atom.

For very small values of the scaled energy E <∼ −0.8

a reasonable analysis of the spectra is possible only ifenough eigenvalues corresponding to small values of γor B converge [cf. Eq. (15)] since the number of low-lying sparse levels, which lead to individual but nontyp-ical fluctuations as already stated above, increases withdecreasing values of E. Since the maximum number ofbasis states used in our calculation and thus the numberof eigenvalues for which convergence can be obtained islimited due to the required computer memory, we can-not present reasonable results for E <∼ −1.0. The effectof the low-lying sparse levels can already be observed forE = −1.009 in Fig. 5.

It is generally assumed that the NNS of large ran-dom matrices can be approximated by the NNS of2 × 2 matrices of the same universality class [5]. Sincewe obtained a good agreement when fitting the func-tions FGOE→GUE (s; λ) and FP→GUE (s; λ), which werederived for 2 × 2 matrices, to our numerical results, wecould prove the Wigner surmise [55] for our system.

VI. SUMMARY AND OUTLOOK

Investigating the Hamiltonian of excitons in cubicsemiconductors we could show analytically and numer-

11

ically that the simultaneous presence of the cubic bandstructure and external fields can break all antiunitarysymmetries in the system. The level spacing statisticsof the quantum mechanical spectrum depends on the en-ergy, the field strengths, the field orientations and on thevalue of the parameter δ′, which determines the strengthof the cubic deformation of the band structure. Thismakes excitons in external fields a prime system to in-vestigate the transitions between different level spacingstatistics. Keeping the parameter δ′ fixed, we analyzedthe transition from GOE to GUE statistics and fromPoissonian to GUE statistics. A comparison with ana-lytical formulae for these transitions derived for 2 × 2matrices within random matrix theory showed very goodagreements. Hence, we could confirm the Wigner sur-mise for our model system. Since we changed only pa-rameters such as the angles of the magnetic field or thescaled energy, which can also be varied in experiments,we think that the transition between the different levelstatistics could also be investigated experimentally. How-ever, changing the two parameters δ′ and the scaled en-ergy E in numerical calculations will allow us to investi-gate arbitrary transitions of the level statistics in the tri-angle between Poissonian (arbitrary δ′, small E), GOE

(δ′ = 0, large E), and GUE statistics (δ′ 6= 0, large E)in the future. As for arbitrary transitions within thistriangle no analytical formulae have been derived withinrandom matrix theory so far, the corresponding func-tions P (s; λ1, λ2) also have to be found. Finally, weare certain that the discovery of GUE statistics for giantRydberg excitons may pave the way to a deeper under-standing of the connection between quantum and classi-cal chaos.

Acknowledgments

We thank D. Frohlich, M. Aßmann and M. Bayer forhelpful discussions.

Appendix A: Hamiltonian

In this section we give the complete Hamiltonian ofEq. (6) and describe the rotation necessary to make thequantization axis coincide with the direction of the mag-netic field. Let us write the Hamiltonian (6) in the form

H = Eg −e2

4πε0ε

1

r

+ H0 + (eB)H1 + (eB)2H2 − eF · r (A1)

with B = |B|. Using Bi = Bi/B with the componentsBi of B, the terms H0, H1, and H2 are given by

H0 =1

2m0(γ′1 + 4γ2)p2 − 3γ2

~2m0

[I21p

21 + c.p.

]− 6γ3

~2m0[{I1, I2} p1p2 + c.p.] , (A2)

H1 =1

2m0

(2m0

me− γ′1 + 4γ2

)B ·L

+3γ2~2m0

[I21

(B2r3p1 − B3r2p1

)+ c.p.

]+

3γ3~2m0

[{I1, I2}

(B2r3p2 − B1r3p1

+B3r1p1 − B3r2p2

)+ c.p.

], (A3)

H2 =1

8m0(γ′1 + 4γ2)

[B2r2 −

(B · r

)2]

− 3γ24~2m0

[I21

(B2r3 − B3r2

)2+ c.p.

]− 3γ3

2~2m0

[{I1, I2}

(B2r3 − B3r2

)×(B3r1 − B1r3

)+ c.p.

]. (A4)

In our calculations, we express the magnetic field inspherical coordinates [see Eq. (7)]. For the different ori-entations of the magnetic field we rotate the coordinatesystem by

R =

cosϕ cosϑ sinϕ cosϑ − sinϑ− sinϕ cosϕ 0

cosϕ sinϑ sinϕ sinϑ cosϑ

, (A5)

i.e., we replace x→ x′ = RTx with x ∈ {r, p, L, I, S}to make the quantization axis coincide with the direc-tion of the magnetic field [64, 65]. Finally we expressthe Hamiltonian in terms of irreducible tensors (see, e.g.,Refs. [40, 53, 58, 65]) and calculate the matrix elementsof the matrices D and M in the generalized eigenvalueproblem (10) or the matrices A, B, and C in the gener-alized eigenvalue problem (15).

Appendix B: Exciton Hartree units

When performing numerical calculations for the hy-drogen atom in external fields, often Hartree units areused [37, 80]. These units are obtained by setting thefundamental physical constants e, m0, ~ as well as theBohr radius a0 to one. As the effective masses of theelectron and hole differ from the free electron mass and

12

TABLE I: Exciton Hartree units converted to SI-units for γ′1 = 2 and ε = 7.5. For a comparison, we also give the values fornormal Hartree units, which are obtained by setting γ′1 = ε = 1.

quantity symbol exc. Hartree unit SI (γ′1 = 2, ε = 7.5) SI (γ′1 = 1, ε = 1)

charge q e 1.6022× 10−19 C 1.6022× 10−19 C

action S ~ 1.0546× 10−34 Js 1.0546× 10−34 Js

mass m m0/γ′1 4.5547× 10−31 kg 9.1094× 10−31 kg

length r γ′1εa0 7.9377× 10−10 m 5.2918× 10−11 m

momentum p ~/γ′1εa0 1.3286× 10−25 kg m/s 1.9929× 10−24 kg m/s

time t γ′1ε2a20m0/~ 2.7213× 10−15 s 2.4189× 10−17 s

energy E ~2/γ′1ε2a20m0 3.8753× 10−20 J 4.3597× 10−18 J

magn. flux density B ~/γ′21 ε2a20e 1.0447× 10+3 T 2.3505× 10+5 T

el. field strength F ~2/γ′21 ε3a30m0e 3.0472× 10+8 V/m 5.1422× 10+11 V/m

since the Coulomb interaction is scaled by the dielectricconstant ε, we introduce exciton Hartree units. Withinthese units the hydrogen-like part of the Hamiltonian (6)is exactly of the same form as the Hamiltonian of thehydrogen atom in Hartree units [37] and the values ofthe scaled energies in Sec. II B can be compared directlywith the values of the scaled energies used in calcula-

tions for the hydrogen atom [21]. The exciton Hartreeunits are obtained by setting e = ~ = 1, m0 = γ′1 andaexc = γ′1εa0 = 1. Since all other physical quantities haveto be converted to exciton Hartree units as well, we givethe according scaling factors in Table I. Variables givenin exciton Hartree units are marked by a tilde sign, e.g.,r → r, throughout the paper.

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