Level statistics for quantum Hall systems

Level statistics for two classes of disordered systems at criticality are analyzed in terms of different realizations of the Chalker–Coddington network model. These include: 1) Re-examination of the standard U(1) model describing dynamics of electrons on the lowest Landau level in the quantum Hal...

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Datum:	2005
Hauptverfasser:	Kagalovsky, V., Horovitz, B., Avishai, Y.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2005
Schriftenreihe:	Физика низких температур
Schlagworte:	Электpонные свойства металлов и сплавов
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spelling	nasplib_isofts_kiev_ua-123456789-1217582025-06-03T16:29:05Z Level statistics for quantum Hall systems Kagalovsky, V. Horovitz, B. Avishai, Y. Электpонные свойства металлов и сплавов Level statistics for two classes of disordered systems at criticality are analyzed in terms of different realizations of the Chalker–Coddington network model. These include: 1) Re-examination of the standard U(1) model describing dynamics of electrons on the lowest Landau level in the quantum Hall effect, where it is shown that after proper local unfolding the nearest-neighbor spacing distribution (NNSD) at the critical energy follows the Wigner surmise for Gaussian unitary ensembles (GUE). 2) Quasi-particles in disordered superconductors with broken time reversal and spin rotation invariance (in the language of random matrix theory this system is a representative of symmetry class D in the classification scheme of Altland and Zirnbauer). Here again the NNSD obeys the Wigner surmise for GUE, reflecting therefore only «basic» discrete symmetries of the system (time reversal violation) and ignoring particle–hole symmetries and other finer details (criticality). In the localized regime level repulsion is suppressed. The work was supported in part by Sacta-Rashi foundation (VK). VK appreciates stimulating discussions with Hans Weidenmüller, Thomas Seligman, Yoram Althassid, Richard Berkovits, and Alexander Mirlin. 2005 Article Level statistics for quantum Hall systems / V. Kagalovsky, B. Horovitz, Y. Avishai // Физика низких температур. — 2005. — Т. 31, № 3-4. — С. 377-381. — Бібліогр.: 12 назв. — англ. 0132-6414 PACS: 73.20.Fz, 72.15.Rn https://nasplib.isofts.kiev.ua/handle/123456789/121758 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов
spellingShingle	Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов Kagalovsky, V. Horovitz, B. Avishai, Y. Level statistics for quantum Hall systems Физика низких температур
description	Level statistics for two classes of disordered systems at criticality are analyzed in terms of different realizations of the Chalker–Coddington network model. These include: 1) Re-examination of the standard U(1) model describing dynamics of electrons on the lowest Landau level in the quantum Hall effect, where it is shown that after proper local unfolding the nearest-neighbor spacing distribution (NNSD) at the critical energy follows the Wigner surmise for Gaussian unitary ensembles (GUE). 2) Quasi-particles in disordered superconductors with broken time reversal and spin rotation invariance (in the language of random matrix theory this system is a representative of symmetry class D in the classification scheme of Altland and Zirnbauer). Here again the NNSD obeys the Wigner surmise for GUE, reflecting therefore only «basic» discrete symmetries of the system (time reversal violation) and ignoring particle–hole symmetries and other finer details (criticality). In the localized regime level repulsion is suppressed.
format	Article
author	Kagalovsky, V. Horovitz, B. Avishai, Y.
author_facet	Kagalovsky, V. Horovitz, B. Avishai, Y.
author_sort	Kagalovsky, V.
title	Level statistics for quantum Hall systems
title_short	Level statistics for quantum Hall systems
title_full	Level statistics for quantum Hall systems
title_fullStr	Level statistics for quantum Hall systems
title_full_unstemmed	Level statistics for quantum Hall systems
title_sort	level statistics for quantum hall systems
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2005
topic_facet	Электpонные свойства металлов и сплавов
url	https://nasplib.isofts.kiev.ua/handle/123456789/121758
citation_txt	Level statistics for quantum Hall systems / V. Kagalovsky, B. Horovitz, Y. Avishai // Физика низких температур. — 2005. — Т. 31, № 3-4. — С. 377-381. — Бібліогр.: 12 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT kagalovskyv levelstatisticsforquantumhallsystems AT horovitzb levelstatisticsforquantumhallsystems AT avishaiy levelstatisticsforquantumhallsystems
first_indexed	2025-11-25T01:21:06Z
last_indexed	2025-11-25T01:21:06Z
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fulltext	Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4, p. 377–381 Level statistics for quantum Hall systems V. Kagalovsky1, B. Horovitz2, and Y. Avishai2 1Negev Academic College of Engineering, Beer-Sheva 84100, Israel E-mail: victork@nace.ac.il 2Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Received July 12, 2004 Level statistics for two classes of disordered systems at criticality are analyzed in terms of dif- ferent realizations of the Chalker–Coddington network model. These include: 1) Re-examination of the standard U( )1 model describing dynamics of electrons on the lowest Landau level in the quantum Hall effect, where it is shown that after proper local unfolding the nearest-neighbor spac- ing distribution (NNSD) at the critical energy follows the Wigner surmise for Gaussian unitary en- sembles (GUE). 2) Quasi-particles in disordered superconductors with broken time reversal and spin rotation invariance (in the language of random matrix theory this system is a representative of symmetry class D in the classification scheme of Altland and Zirnbauer). Here again the NNSD obeys the Wigner surmise for GUE, reflecting therefore only «basic» discrete symmetries of the system (time reversal violation) and ignoring particle–hole symmetries and other finer details (criticality). In the localized regime level repulsion is suppressed. PACS: 73.20.Fz, 72.15.Rn 1. Introduction The statistics of energy levels in a disordered sys- tem is an important tool in determining its transport properties as well as its critical behavior. A central quantity in this study is the nearest-neighbor spacing distribution (NNSD) denoted by p s( ). Here the ran- dom variable s is the (fluctuating) level spacing under the proviso that the local average of the density of states is energy independent (otherwise, a proper un- folding procedure is required). The distribution p s( ) involves all N point correlation functions of the perti- nent Green function and hence, it is generally not available in a closed form. In dealing with disordered systems, it is useful to distinguish between systems undergoing an Anderson type metal–insulator transition, and those character- ized by a quantum Hall (QH)-like transition where, in the thermodynamic limit, critical state energies are isolated points occuring between continuous intervals of localized state energies. As for level statistics per- taining to disordered systems of the Anderson metal–insulater transition kind, there are a couple of important properties which are well established: 1) Under certain conditions it is expected to be repre- sented (on the metallic side) by random matrix spec- tra [1]. To be more precise, it is well described by one of the corresponding Gaussian ensembles, Gaussian orthogonal (GOE), Gaussian unitary (GUE) and Gaussian simplectic (GSE), depending on the symme- try class to which the physical system belongs. The main condition is that the corresponding energy inter- vals are smaller than the Thouless energy. 2) It has been shown [2] that in the limit of an infinite system there are only three types of distributions p s( ). They are the Poisson law for the insulating regime, the Wigner surmise for the metallic domain, and a third one for the critical region. On the other hand, for sys- tems in the second group (such as the quantum Hall effect) there is no similar analysis. The main difficulty is related to the fact that in the absence of a metallic regime, it is not possible to approach the critical point from the metallic regime using the powerful tool of ex- pansion in the small parameter 1/g (here g is the dimensionless conductance). Common sense suggests that the distribution follows the Poisson law in the in- sulating part of the spectra while again, the distribu- tion in the critical region is different, and related to the relevant universality class. For the quantum Hall transition this is supported by numerous numerical calculations [3]. V. Kagalovsky, B. Horovitz, and Y. Avishai, 2005 Recently, it has been noticed that the second group contains, beside the systems belonging to the quantum Hall effect universality class, other disordered systems whose phase diagram is much richer. They are related to the physics of disordered superconductors [4] and comprise of four novel universality classes, deter- mined according to the symmetry properties of the corresponding Bogoliubov–de Gennes Hamiltonian under spin rotation and time reversal. Some of these new phase diagrams have already been exposed, mainly in class C (where time reversal symmetry is broken while spin rotation invarinace is preserved). So far, the level statistics in the critical regions of the four new universality classes has not yet been studied. The goal of the present work is to fill this gap, start- ing by elucidating the level statistics of one of these new classes, namely class D, for which both symme- tries (spin rotation and time reversal) are violated. Some of the disordered systems in the second group can be mapped on a network model. The most studied one is the quantum Hall system which is mapped on the Chalker–Coddington network model (CCNM) [5]. The CCNM is designed to describe transition be- tween plateaus in the QH system using transfer matrix algorithm in an infinite cylinder geometry (in the Landauer sense). It was later suggested that if, in- stead of studying transport properties, the system is closed up as a torus, then the eigenvalue problem can be addressed and the level statistics can be studied [6] (although no Hamiltonian is specified). In our previ- ous works, a somewhat modified CCNM has been con- structed which can describe noninteracting quasi- particles in disordered superconductors [7,8]. It appears that such a description can serve as an appro- priate physical realization of the new random matrix universality classes [4]. It is then natural to attempt an investigation of level statistics of these new symmetry classes by using the CCNM. Our main results are summarized below. 1) As a starting reference point we revisit the familiar QH system by repeating calculations for the original U( )1 model [6]. The results of Ref. 6 are indeed repro- duced ( p s( ) deviates slightly from the Wigner sur- mise for GUE). Moreover, we show that after proper local unfolding, the NNSD of the U( )1 model at the critical energy is exactly identical with the Wigner surmise. In order to stress the necessity of unfolding we argue that the density of states (DOS) averaged over all samples is indeed uniform, whereas it has some structure for each sample. 2) We then present re- sults for class D of disordered superconductors that have neither time-reversal nor spin-rotation inva- riance. Once again NNSD at the critical energy (after unfolding) coincides with the Wigner surmise. We find the DOS to have a periodic structure (period �/2), as one would expect from the form of the uni- tary operator. Beside the critical region, we also pres- ent results for the localized regime and show that level repulsion disappears. Thus, although the U( )1 and class D models considered here describe different sys- tems and have different phase diagrams, yet the NNSD in the critical region is the same, depending only on the broken time-reversal symmetry. The fact that class D obeys particle–hole symmetry is not reflected in its NNSD. Our findings are in agreement with recent works [9] where it is argued that k-body embedded Gaussian ensembles of random matrices for suffi- ciently high rank k of the random interaction behave generically (i.e., in order to have exact RMT results it is not necessary for the Hamiltonian to be a full ran- dom matrix). 2. The U( )1 network model In the original CCNM, electrons move along unidi- rectional links forming closed loops in analogy with semiclassical motion on equipotential contours. Scat- tering between links is allowed at nodes in order to map tunneling through saddle point potentials. Propa- gation along links yields a random phase �, thus links are presented by diagonal matrices with elements in the form exp( )i� (hence the notation U( )1 model). The transfer matrix for one node relates a pair of in- coming and outgoing amplitudes on the left to a corre- sponding pair on the right; it has the form T � � � � � � cosh sinh sinh cosh . (1) The node parameter is related to the electron energy in the following way � � � � 2 ln ( )sinh , (2) where � is a relative distance between the electron en- ergy and the barrier height. If the network forms a torus, then on every link the electron motion appears once as an outgoing one and once as an incoming one. The collection of relations between incoming and outgoing amplitudes defines the system’s S matrix or rather, a discrete-time unitary evolution operator, U( )� [6]. The eigenphases of U serve as input for level statistics analysis. For a square network of N N nodes,U is an ( ) ( )2 22 2 N N unitary matrix. The action ofU on a vector � of flux amplitudes defined on the start of each link maps the system onto itself, providing therefore an implicit eigenvalue equationU( )� � �� . Since the dependence of the matrix elements of U on � is comlicated, it is 378 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 V. Kagalovsky, B. Horovitz, and Y. Avishai practically impossible to find solutions of that equa- tion (even numerically). Instead, it has been sug- gested [6] to find the eigenvalues of the equation U in n n� �� exp [ ( )]� � (3) and to study statistics of �n for a given �. The ratio- nale behind it is twofold. First, there are sufficiently many states even in a narrow window near a particu- lar energy � to provide good statistics. Second, the be- havior of the curves � �n ( ) is rather smooth, and therefore the statistics of �n for a given � is expected to be the same as the statistics of �n for � � 0 (which are the true energy eigenvalues). We argue that the second hypothesis is justified only after a proper un- folding procedure is executed. Indeed, from the RMT point of view the eigenvalue problem of Eq. (3) be- longs to the circular unitary ensemble (CUE). In the standard CUE all the eigenvalues lie on the unit cir- cle and are equally spaced, so there is no need for un- folding of the spectra. However, Eq. (3) represents a physical problem in which not all the elements of U are independent random variables. Therefore, the question of whether the U( )1 model is a bona fide CUE should be examined. Our calculations show that the averaged (over 50 samples) DOS is indeed uni- form, whereas it has some pronounced structure for each particular sample. Since the level statistics should be manifested for each individual sample (as in the study of nuclear spectra), the spectrum of each sample should then be properly unfolded. Here we make use of the fact that the dimensionless unfolded distance between two levels is �s k E E E En n n n k n k � � � � � � 2 1 , (4) where k is a number of neighbors to be optimized by the requirement of having a constant DOS. This pro- cedure encodes the important local fluctuations of level spacing. We have checked that for k � 6 the re- sult is practically independent of k, provided of course that k N�� 2 2. To substantiate this point we plot in Fig. 1 the DOS of a single sample (at the crit- Level statistics for quantum Hall systems Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 379 0 1 2 3 4 5 30 20 10 6 Fig. 1. Histogram for the DOS of a single sample at the critical point � � 0 for U( )1 model without unfolding. 4000 3000 2000 1000 0 1 2 3 4 5 6 Fig. 2. Histogram for the DOS of 50 samples at the criti- cal point � � 0 for U( )1 model. 1.0 0.8 0.6 0.4 0.2 0 P 1 2 3 s Fig. 3. Nearest neighbor spacing distribution P(s) for U( )1 model at � � 0 (critical regime). The curve is the Wigner surmise for the GUE; without unfolding Mettzer–Klesse (�), �3 neighbor local unfolding (�). ical point � � 0) without unfolding. It shows indeed that the averaged DOS is not constant in energy. On the other hand, after averaging over 50 samples, Fig. 2 indicates that the average DOS is constant, as expected for random matrices belonging to CUE. The corresponding NNSD are displayed in Fig. 3. Raw data without unfolding reproduce the results ob- tained in [6], whereas after local unfolding the NNSD nearly coincides with the Wigner surmise for GUE (expected to be true also for CUE at large N). Finally, we assert in Fig. 4 that in the localized re- gime (� � 1) the NNSD follows the Poisson statistics p s e s( ) � � . 3. Disordered superconductors: Class D The properties of quasiparticles in disordered su- perconductors have been the subject of much recent interest. The Hamiltonians of such systems are repre- sentatives of a set of symmetry classes different from the three classes which are familiar both in normal dis- ordered conductors and in the Wigner–Dyson random matrix ensembles. A list of additional random matrix ensembles, determined by these new symmetry classes, has been established [4]. Below we present numerical results on statistics of energy levels for a certain two-dimensional system with a particularly rich phase diagram. In the numenclature of Ref. 4 the corre- sponding symmetry is denoted as class D. It can be re- alized in superconductors with broken time-reversal invariance, and either broken spin-rotation invariance (as in d-wave superconductors with spin–orbit scat- tering) or spinless or spin-polarized fermions (as in certain p-wave states). A particular realization of class D (which will be adopted here) is the Cho–Fisher (CF) model [10] which has a rich phase diagram. Each realization has two parameters: a disorder strength W(0 1� �W ), and a tunneling amplitude �, which controls the value of the thermal Hall conduc- tance at short distances. The phases on the links on the two sides of the same node (which are either � with probability W or 0 with probability 1 � W) are corre- lated: the same random phase appears on both sides. It is equivalent to attributing random sign to the off-di- agonal elements of the transfer matrix. The phase dia- gram in the ( , )� W plane contains a region of metallic states, and two distinct localized domains, which can be identified as regions with different quantized ther- mal Hall conductance. There is a critical state at � � 0 for any W. The detailed structure of the phase dia- gram has been presented elsewhere [8]. In our numerical simulations we have studied 50 dif- ferent network systems, of size ( )( )2 32 2 322 2 on the critical line � � 0 and disorder strength parameter W � 0.1. The raw DOS appears to be a periodic func- tion of � with period �/2 reflecting the cubic symme- try of the CF model (Fig. 5). The NNSD is presented in Fig. 6 and are compared with the Wigner surmise for GUE. The agreement is rather evident, and it could not be achieved without unfolding. We are thus convinced that the critical form of the NNSD at the critical line of the CF model (in fact of class D in gen- eral) coicides with the GUE. Next, we move away from the critical line and put � � 1, keeping the same value W � 01. which, according to our phase diagram, is well within a localized domain. The results are 380 Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 V. Kagalovsky, B. Horovitz, and Y. Avishai 0 1 2 3 4 5 s 0.8 0.6 0.4 0.2 P Fig. 4. Nearest neighbor spacing distribution P(s) for U( )1 model at � � 1 (localized regime). The curve is the Poissonian statistics, data after unfolding (�). 800 600 400 200 0 1 2 3 4 5 6 Fig. 5. Histogram for the DOS of 50 samples for CF model at the critical point � � 0 and W � 01. . shown in Fig. 7. They are fitted by the Berry–Robnick approximation [11]. Usually, the large s behavior is more sensitive to localization than the small s beha- vior. In other words, even deep inside the localized re- gime one still finds level repulsion p( )0 0� . Remark- ably, for the CF model we find p( . ) .0 025 0 53� , which cannot be just attributed to statistical error in view of the fact that we study almost 106 energy levels. We have also calculated the compressibility � of the spec- trum and have found an extremely small value � 0 01. , which is in agreement with the classical result for the GUE, where � �� for large system sizes. In conclusion, although mapping of a physical problem on a network model results in correlated and sparse matrices of unitary evolution operators, the re- sults for NNSD seem to agree with the predictions of RMT which assume non-sparse matrices with uncorrelated matrix elements [9]. In the cases studied here this agreement is achieved after a proper local un- folding of the spectra is executed. The main physical result is the following: Despite the occurrence of ten different random matrix symme- try classes according to time-reversal, spin-rotation, and particle–hole symmetries, with many different physical properties, some basic characteristics remain intact, depending only on time reversibility and spin rotation invariance. There have been numerous at- tempts to check whether the form of p s( ) in QH like systems deviates from that of GUE [12]. Our results indicate that as far the network model realization is concerned, p s( ) is satisfactorily accounted for by the Wigner surmise for the unitary ensemble. The viola- tion of time reversal invariance either by a magnetic field (in QH systems) or spontaneously in unconven- tional superconductors is the dominant factor, which masks finer details such as quantum criticality. The work was supported in part by Sacta-Rashi foundation (VK). VK appreciates stimulating discus- sions with Hans Weidenmüller, Thomas Seligman, Yoram Althassid, Richard Berkovits, and Alexander Mirlin. 1. B.L. Altshuler and B.I. Shklovskii, Zh. Eksp. Teor. Fiz. 9, 220 (1986) [Sov. Phys. JETP 64, 127 (1986)]. 2. B.I. Shklovskii, B. Shapiro, B.R. Sears, P. Lambria- nides, and H.B. Shore, Phys. Rev. B47, 11487 (1993). 3. B. Huckestein, Rev. Mod. Phys. 67, 357 (1995). 4. A. Altland and M.R. Zirnbauer, Phys. Rev. B55, 1142 (1997); M.R. Zirnbauer, J. Math. Phys. 37, 4986 (1996). 5. J.T. Chalker and P.D. Coddington, J. Phys. C21, 2665 (1988). 6. R. Klesse and M. Metzler, Europhys. Lett. 32, 229 (1995); R. Klesse and M. Metzler, Phys. Rev. Lett. 79, 721 (1997). 7. V. Kagalovsky, B. Horovitz, Y. Avishai, and J.T. Chalker, Phys. Rev. Lett. 82, 3516 (1999). 8. J.T. Chalker, N. Read, V. Kagalovsky, B. Horovitz, Y. Avishai, and A.W. W. Ludwig, Phys. Rev. B65, 012506 (2001). 9. L. Benet, T. Rupp, and H.A. Weidenmüller, Phys. Rev. Lett. 87, 010601 (2001); L. Benet, T. Rupp, and H.A. Weidenmüller, Ann. Phys. 292, 67 (2001). 10. S. Cho and M.P.A. Fisher, Phys. Rev. B55, 1025 (1997). 11. M. Robnik, J. Phys. A20, 495 (1987). 12. M. Feingold, Y. Avishai, and R. Berkovits, Phys. Rev. B52, 8400 (1995); V.E. Kravtsov, I.V. Lerner, B.L. Altshuler, and A.G. Aronov, Phys. Rev. Lett. 72, 888 (1994). Level statistics for quantum Hall systems Fizika Nizkikh Temperatur, 2005, v. 31, Nos. 3/4 381 1.0 0.8 0.6 0.4 0.2 0 P 3.02.52.01.51.00.5 s Fig. 6. Nearest-neighbor spacing distribution P(s) for CF model at � � 0 and W � 01. (metallic regime). The curve is the Wigner surmise for the GUE. 0 1 2 3 4 5 s P 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Fig. 7. Nearest-neighbor spacing distribution P(s) for CF model at � � 1 and W � 0.1 (localized regime). Curve is the Berry–Robnick fit for the transition from GUE to Poissonian statistics.

Level statistics for quantum Hall systems

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