Physica A 266 (1999) 477–480

Application of random matrix theory toquasiperiodic systems

Michael Schreiber a ;∗, Uwe Grimma , Rudolf A. R�omer a ,Jian-Xin Zhonga;b

aInstitut f�ur Physik, Technische Universit�at, D–09107 Chemnitz, GermanybDepartment of Physics, Xiangtan University, Xiangtan 411105, People’s Republic of China

Abstract

We study statistical properties of energy spectra of a tight-binding model on the two-dimensional quasiperiodic Ammann–Beenker tiling. Taking into account the symmetries of �niteapproximants, we �nd that the underlying universal level-spacing distribution is given by theGaussian orthogonal random matrix ensemble, and thus di�ers from the critical level-spacingdistribution observed at the metal–insulator transition in the three-dimensional Anderson modelof disorder. Our data allow us to see the di�erence to the Wigner surmise. c© 1999 ElsevierScience B.V. All rights reserved.

PACS: 71.23.Ft; 05.45.+b; 71.30.+h; 72.15.Rn

Keywords: Random matrix theory; Wigner surmise; Level-spacing distribution; Quasiperiodicsystems

In a recent paper [1], we investigated energy spectra of quasiperiodic tight-bindingmodels, concentrating on the case of the octagonal Ammann–Beenker tiling [2,10]shown in Fig. 1. The Hamiltonian is restricted to constant hopping matrix elementsalong the edges of the tiles in Fig. 1. Previous studies of the same model had ledto diverging results on the level statistics: For periodic approximants, level repulsionwas observed [3,4], and the level-spacing distribution P(s) was argued to follow alog-normal distribution [4]. On the other hand, for octagonal patches with an ex-act eightfold symmetry and free boundary conditions, level clustering was found [5].On the basis of our numerical results for P(s) and the spectral rigidity �3 [6,11],

∗ Corresponding author. Fax: +49-371-531-3143; e-mail: schreiber@physik.tu-chemnitz.de.

0378-4371/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(98)00634 -7

478 M. Schreiber et al. / Physica A 266 (1999) 477–480

Fig. 1. Octagonal cluster of the Ammann–Beenker tiling with 833 vertices and exact D8-symmetry aroundthe central vertex as indicated by the solid and dashed lines. Shadings indicate successive in ation steps ofthe central octagon.

compiled in Ref. [1], we concluded that the underlying universal level-spacing distri-bution of this system is given by the Gaussian orthogonal random matrix ensemble(GOE) [6,7,11,12]. Concerning the contradictory results of previous investigations, weattribute these to the non-trivial symmetry properties of the octagonal tiling. The peri-odic approximants studied in Refs. [3,4] show, besides an exact re ection symmetry,an “almost symmetry” under rotation by 90◦ which may in uence the level statistics[6,11], whereas the octagonal patches used in Ref. [5] possess the full D8-symmetryof the regular octagon. Hence the level statistics observed in this case is that of asuperposition of seven completely independent subspectra, and therefore rather closeto a Poisson law.To arrive at this conclusion, we considered in Ref. [1] di�erent patches that approx-

imate the in�nite quasiperiodic tiling, both with free and periodic boundary conditions.Exact symmetries were either exploited to block-diagonalize the Hamiltonian, thus split-ting the spectrum into its irreducible parts, or avoided altogether by choosing patcheswithout any symmetries.Here, we concentrate on the D8-symmetric octagonal patch shown in Fig. 1. For

this case, the Hamiltonian matrix splits into 10 blocks according to the irreduciblerepresentations of the dihedral group D8, resulting in seven di�erent independent sub-spectra as there are three pairs of identical spectra. In Fig. 2, we show the integrateddensity of states (IDOS) for a patch, which contains N = 157 369 vertices and cor-responds to three more in ation steps performed on the patch of Fig. 1. Apparently,the IDOS is rather smooth, and the only prominent feature that shows up, apart froma few small gaps, is the huge fraction (13 077 of 157 369, hence about 8.3%) ofexactly degenerate eigenvalues in the band centre. For the level-spacing distribution

M. Schreiber et al. / Physica A 266 (1999) 477–480 479

Fig. 2. IDOS (inset) for the D8-symmetric patch with N = 157 369 vertices, and P(s) averaged over thethree largest sectors. The smooth line denotes PGOE(s).

Fig. 3. Small- and large-s behaviour of P(s) of Fig. 2, compared to PGOE(s) (solid line) and PW(s) (dottedline).

P(s), these do not matter as they would only contribute to P(0), therefore we canneglect them completely. The IDOS shown in Fig. 2 is �tted to a cubic spline whichis then used to “unfold” the spectrum [8,13,14], i.e., to correct for the non-constantdensity of states, what is necessary if we want to compare to results of random matrixtheory.The level-spacing distribution P(s) for the unfolded spectra is shown in Figs. 2 and 3,

measured in units of the mean level spacing. Here, we averaged over the three largestsubspectra, each of which contains 18 043 levels after removing the degenerate states inthe band centre. The resulting histogram is compared to the GOE distribution PGOE(s)(solid lines in Figs. 2 and 3) and, focusing on the small- and large-s behaviour, alsoto the Wigner surmise PW(s) (dotted lines in Fig. 3). Apparently, the level-spacing

480 M. Schreiber et al. / Physica A 266 (1999) 477–480

Fig. 4. �2 statistics for the seven independent subspectra of the D8-symmetric octagonal patch with 157 369vertices. The lines indicate the GOE (solid) and Poisson (dashed) behaviour.

distribution of the quasiperiodic Hamiltonian is well described by random matrix theory,and one can clearly see that PGOE(s) �ts the numerical data even better than PW(s).Fig. 4 shows the corresponding �2 statistics [6,11], compared to the exact GOE

result. The �2 statistics measures the uctuation in the number of energy levels nin an energy range L, i.e., �2 = 〈n2〉 − 〈n〉2 where 〈:〉 denotes the spectral average.Again, the agreement with our numerical results is good, supporting the conclusionthat the underlying universal level statistics is described by the GOE. Because typicaleigenstates in our model are expected to be multifractal, one might have expectedthat one �nds a “critical” level-spacing distribution as observed at the metal–insulatortransition in the three-dimensional Anderson model of disorder [9] – however, this isclearly not the case.

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