Computer Physics Communications 121–122 (1999) 499–501www.elsevier.nl/locate/cpc

Energy levels of quasiperiodic Hamiltonians, spectral unfolding,and random matrix theory

Michael Schreibera,1, Uwe Grimma, Rudolf A. Römera, Jianxin Zhonga,ba Institut für Physik, Technische Universität, D-09107 Chemnitz, Germany

b Department of Physics, Xiangtan University, Xiangtan 411105, People’s Republic of China

Abstract

We consider a tight-binding Hamiltonian defined on the quasiperiodic Ammann–Beenker tiling. Although the density of states(DOS) is rather spiky, the integrated DOS (IDOS) is quite smooth and can be used to perform spectral unfolding. The effectof unfolding on the integrated level-spacing distribution is investigated for various parts of the spectrum which show differentbehaviour of the DOS. For energy intervals with approximately constant DOS, we find good agreement with the distributionof the Gaussian orthogonal random matrix ensemble (GOE) even without unfolding. For energy ranges with fluctuating DOS,we observe deviations from the GOE result. After unfolding, we always recover the GOE distribution. 1999 Elsevier ScienceB.V. All rights reserved.

In two recent papers [1,2], we investigated theenergy-level statistics of two-dimensional quasiperi-odic Hamiltonians, concentrating on the case of theeight-fold Ammann–Beenker tiling [3] shown in Fig. 1.The Hamiltonian contains solely constant hopping el-ements along the edges of the tiles in Fig. 1. Numeri-cal results suggest that typical eigenstates of the modelare multifractal [4]. In [1,2], we numerically calcu-lated the level-spacing distributionP(s) and the∆3andΣ2 statistics [5], and found perfect agreement withthe results for the GOE. One ingredient of the calcula-tion was the so-called unfolding procedure, explainedbelow, which corrects for the fluctuations in the DOSof the model Hamiltonian. It is well known that, fora spectrum with non-constant DOS, it is necessary tounfold the spectrum in order to extract universal levelstatistics which can then be compared to the results ofrandom matrix theory [5].

1 E-mail: schreiber@physik.tu-chemnitz.de.

In the present paper, we trace the question howcrucial the unfolding procedure affects the universalresults, restricting ourselves to the integrated level-spacing distribution (ILSD)

I (s)=∞∫s

P (s′) ds′.

In previous studies of the model [6–8], spectral un-folding had not been used, the main reason being thatthe DOS is rather spiky as shown in Fig. 2. In [9], itwas claimed that the level-spacing distribution (LSD)depends crucially on the unfolding, yielding resultsranging from log-normal to GOE behaviour. In fact,if one considers the IDOS which is also shown inFig. 2, one finds that it is rather smooth, apart froma couple of small gaps and a jump at energyE = 0caused by the existence of 13077 degenerate statesin the band center [1]. These states can be neglectedfor the level-spacing distribution since they only con-

0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved.PII: S0010-4655(99)00391-4

500 M. Schreiber et al. / Computer Physics Communications 121–122 (1999) 499–501

Fig. 1. Octagonal cluster of the Ammann–Beenker tiling with 833vertices and an exactD8 symmetry around the central vertex.

Fig. 2. DOS and IDOS for aD8-symmetric patch with 157369vertices.

tribute toP(0). Therefore, we use the IDOS to unfoldthe energy spectra.

The purpose of the spectral unfolding is to transfer aspectrum with a non-constant DOS to the another witha unit DOS or a linear IDOS. In order to achieve this,one defines the unfolded levels asei =Nav(Ei), whereEi is the ith energy level,Nav(Ei) the smoothenedIDOS [5]. Level spacings are thus given bysi = ei+1−ei . An effective way to calculate theNav(Ei) is to fitthe IDOS to cubic splines by choosing a sequence ofenergy levels(E1,Em+1,E2m+1, . . .) from the energyspectrum(E1,E2, . . . ,En). A meaningful statisticsshould be independent of the unfolding parameterm.Evidently, the value ofm, which does not change the

statistics, depends on the total number of levelsn andthe structure of the DOS. The typical value for theAnderson Hamiltonian is around 100 [10,11].

The eight-fold Ammann–Beenker tiling shown inFig. 1 has the fullD8 symmetry of the regular octagon,hence the Hamiltonian matrix splits into ten blocks ac-cording to the irreducible representations of the dihe-dral groupD8, resulting in seven different independentsubspectra as there are three pairs of identical spec-tra. In order to obtain the underlying universal LSD,one should consider the irreducible subspectra sepa-rately [1,5]. In the following, we demonstrate our re-sults for one of the largest subspectra of the Ammann–BeenkerD8-symmetric tiling of 157369 vertices. Wenote that the derived conclusions hold for all the sub-spectra. Fig. 3 shows level spacing distributions forenergy levels in various parts of the spectrum withoutand with unfolding. It is easy to see from Fig. 3(a) thatenergy levels with an approximately constant DOS(for instance, 3.26 E 6 3.3) exhibit GOE behavioureven without unfolding. However, for energy rangeswith fluctuating DOS,I (s) apparently deviates fromthe IGOE(s) if we do not perform the unfolding. Forinstance,I (s) for the non-unfolded whole spectrum isclose to the log-normal distribution although the log-normal distribution is not the generic level statisticsof the quasiperiodic Hamiltonian. Analyzing the non-

Fig. 3. ILSD I (s) obtained (a) without unfolding and (b) withunfolding for various parts of the spectrum of one sector of theD8-symmetric patch: whole spectrum (dashed line), 0.56E 6 1.5(dotted line), and 3.2 6 E 6 3.3 (solid line). Circles and boxesdenoteIGOE(s) and the log-normal distribution, respectively.I (s)in (b) has been shifted by 0.2 for clarity.

M. Schreiber et al. / Computer Physics Communications 121–122 (1999) 499–501 501

Fig. 4. ILSD I (s) obtained for one sector of theD8-symmetricpatch with different parametersm in the unfolding procedure.The result is compared toIGOE(s) (circles) and the intermediate(“semi-Poisson”) statisticsISP(s) (diamonds).

unfolded spectrum in the range with the largest fluctu-ations, 0.56 E 6 1.5, one can clearly see thatI (s) isneither log-normal nor GOE. From Fig. 3(b), we cansee that, after the unfolding,I (s) for different energyintervals show a very good agreement with the GOEresult.

In our calculation, we find that there exists asequence ofm ranging from 4 to 30, which givesthe same statistics described byIGOE(s), within ournumerical precision. In Fig. 4, we illustrateI (s)obtained withm = 5,10, 20, and 100. One can seethat there is only a small deviation from theIGOE(s)

even for largem up to 100. We emphasize that evenfor such a largem, I (s) is still far from the so-called “semi-Poisson” intermediate statisticsISP(s)=(2s + 1)e−2s [12], which is supposed to be validto describe the level statistics at the metal-insulatortransition with multifractal eigenstates in the three-dimensional Anderson model of localization.

In summary, we have shown that although the DOSof the quasiperiodic tight-binding Hamiltonian definedon the Ammann–Beenker tiling exhibits very spikystructures, the IDOS is in fact rather smooth and can beused to perform spectral unfolding. We demonstratedthat the level spacing distribution is independent ofthe unfolding parameter and is well described by theGOE distribution. We also studied the level statisticsfor energy spectra without unfolding and found that,for energy intervals with approximately constant DOS,I (s) agrees with the GOE distribution; for energyranges with fluctuating DOS,I (s) varies with differentenergy intervals and the log-normal distribution foundin a previous calculation [7] is not generic.

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