Statistics of Energy Levels and Eigenfunctions in Disordered Systems
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A.D. Mirlin / Physics Reports 326 (2000) 259}382 259 STATISTICS OF ENERGY LEVELS AND EIGENFUNCTIONS IN DISORDERED SYSTEMS Alexander D. MIRLIN Institut fu( r Theorie der kondensierten Materie, Universita( t Karlsruhe, 76128 Karlsruhe, Germany AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO Physics Reports 326 (2000) 259}382 Statistics of energy levels and eigenfunctions in disordered systems Alexander D. Mirlin1 Institut fu( r Theorie der kondensierten Materie, Postfach 6980, Universita( t Karlsruhe, 76128 Karlsruhe, Germany Received July 1999; editor: C.W.J. Beenakker Contents 1. Introduction 262 5.2. Strong correlations of eigenfunctions near 2. Energy level statistics: random matrix theory the Anderson transition 325 and beyond 266 5.3. Power-law random banded matrix 2.1. Supersymmetric p-model formalism 266 ensemble: Anderson transition in 1D 328 2.2. Deviations from universality 269 6. Conductance #uctuations in quasi-one- 3. Statistics of eigenfunctions 273 dimensional wires 344 3.1. Eigenfunction statistics in terms of the 6.1. Modeling a disordered wire and mapping supersymmetric p-model 273 onto 1D p-model 345 3.2. Quasi-one-dimensional geometry 277 6.2. Conductance #uctuations 348 3.3. Arbitrary dimensionality: metallic 7. Statistics of wave intensity in optics 353 regime 283 8. Statistics of energy levels and eigenfunctions in 4. Asymptotic behavior of distribution functions a ballistic system with surface scattering 360 and anomalously localized states 294 8.1. Level statistics, low frequencies 362 4.1. Long-time relaxation 294 8.2. Level statistics, high frequencies 363 4.2. Distribution of eigenfunction 8.3. The level number variance 364 amplitudes 303 8.4. Eigenfunction statistics 365 4.3. Distribution of local density of states 309 9. Electron}electron interaction in disordered 4.4. Distribution of inverse participation mesoscopic systems 366 ratio 312 9.1. Coulomb blockade: #uctuations in the 4.5. 3D systems 317 addition spectra of quantum dots 367 4.6. Discussion 319 10. Summary and outlook 373 5. Statistics of energy levels and eigenfunctions at Acknowledgements 374 the Anderson transition 320 Appendix A. Abbreviations 374 5.1. Level statistics. Level number variance 320 References 375 1 Tel.: #49-721-6083368; fax: #49-721-698150. Also at Petersburg Nuclear Physics Institute, 188350 Gatchina, St. Petersburg, Russia. E-mail address: [email protected] (A.D. Mirlin) 0370-1573/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 573(99)00091-5 A.D. Mirlin / Physics Reports 326 (2000) 259}382 261 Abstract The article reviews recent developments in the theory of #uctuations and correlations of energy levels and eigenfunction amplitudes in di!usive mesoscopic samples. Various spatial geometries are considered, with emphasis on low-dimensional (quasi-1D and 2D) systems. Calculations are based on the supermatrix p-model approach. The method reproduces, in so-called zero-mode approximation, the universal random matrix theory (RMT) results for the energy-level and eigenfunction #uctuations. Going beyond this approxi- mation allows us to study system-speci"c deviations from universality, which are determined by the di!usive classical dynamics in the system. These deviations are especially strong in the far `tailsa of the distribution function of the eigenfunction amplitudes (as well as of some related quantities, such as local density of states, relaxation time, etc.). These asymptotic `tailsa are governed by anomalously localized states which are formed in rare realizations of the random potential. The deviations of the level and eigenfunction statistics from their RMT form strengthen with increasing disorder and become especially pronounced at the Anderson metal}insulator transition. In this regime, the wave functions are multifractal, while the level statistics acquires a scale-independent form with distinct critical features. Fluctuations of the conductance and of the local intensity of a classical wave radiated by a point-like source in the quasi-1D geometry are also studied within the p-model approach. For a ballistic system with rough surface an appropriately modi"ed (`ballistica) p-model is used. Finally, the interplay of the #uctuations and the electron}electron interaction in small samples is discussed, with application to the Coulomb blockade spectra. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 05.45.Mt; 71.23.An; 71.30.#h; 72.15.Rn; 73.23.!b; 73.23.Ad; 73.23.Hk Keywords: Level correlations; Wave function statistics; Disordered mesoscopic systems; Supermatrix sigma model 262 A.D. Mirlin / Physics Reports 326 (2000) 259}382 1. Introduction Statistical properties of energy levels and eigenfunctions of complex quantum systems have been attracting a lot of interest of physicists since the work of Wigner [1], who formulated a statistical point of view on nuclear spectra. In order to describe excitation spectra of complex nuclei, Wigner proposed to replace a complicated and unknown Hamiltonian by a large N]N random matrix. This was a beginning of the random matrix theory (RMT) further developed by Dyson and Mehta in the early 1960s [2,3]. This theory predicts a universal form of the spectral correlation functions determined solely by some global symmetries of the system (time-reversal invariance and value of the spin). Later it was realized that the random matrix theory is not restricted to strongly interacting many-body systems, but has a much broader range of applicability. In particular, Bohigas et al. [4] put forward a conjecture (strongly supported by accumulated numerical evidence) that the RMT describes adequately statistical properties of spectra of quantum systems whose classical analogs are chaotic. Another class of systems to which the RMT applies and which is of special interest to us here is that of disordered systems. More speci"cally, we mean a quantum particle (an electron) moving in a random potential created by some kind of impurities. It was conjectured by Gor'kov and Eliashberg [5] that statistical properties of the energy levels in such a disordered granule can be described by the random matrix theory. This statement had remained in the status of conjecture until 1982, when it was proved by Efetov [6]. This became possible due to development by Efetov of a very powerful tool of treatment of the disordered systems under consideration } the supersym- metry method (see the review [6] and the recent book [7]). This method allows one to map the problem of the particle in a random potential onto a certain deterministic "eld-theoretical model (supermatrix p-model), which generates the disorder-averaged correlation functions of the original problem. As Efetov showed, under certain conditions one can neglect spatial variation of the p-model supermatrix "eld (so-called zero-mode approximation), which allows one to calculate the correlation functions. The corresponding results for the two-level correlation function reproduced precisely the RMT results of Dyson. The supersymmetry method can be also applied to the problems of the RMT-type. In this connection, we refer the reader to the paper [8], where the technical aspects of the method are discussed in detail. More recently, focus of the research interest was shifted from the proof of the applicability of RMT to the study of system-speci"c deviations from the universal (RMT) behavior. For the problem of level correlations in a disordered system, this question was addressed for the "rst time by Altshuler and Shklovskii [9] in the framework of the di!uson-cooperon diagrammatic per- turbation theory. They showed that the di!usive motion of the particle leads to a high-frequency behavior of the level correlation function completely di!erent from its RMT form. Their pertur- bative treatment was however restricted to frequencies much larger than the level spacing and was not able to reproduce the oscillatory contribution to the level correlation function. Inclusion of non-zero spatial modes (which means going beyond universality) within the p-model treatment of the level correlation function was performed in Ref. [10]. The method developed in [10] was later used for calculation of deviations from the RMT of various statistical characteristics of a dis- ordered system. For the case of level statistics, the calculation of [10] valid for not too large A.D. Mirlin / Physics Reports 326 (2000) 259}382 263 frequencies (below the Thouless energy equal to the inverse time of di!usion through the system) was complemented by Andreev and Altshuler [11] whose saddle-point treatment was, in contrast, applicable for large frequencies. Level statistics in di!usive disordered samples is discussed in detail in Section 2 of the present article. Not only the energy levels statistics but also the statistical properties of wave functions are of considerable interest. In the case of nuclear spectra, they determine #uctuations of widths and heights of the resonances [12]. In the case of disordered (or chaotic) electronic systems, eigenfunc- tion #uctuations govern, in particular, statistics of the tunnel conductance in the Coulomb blockade regime [13]. Note also that the eigenfunction amplitude can be directly measured in microwave cavity experiments [14}16] (though in this case one considers the intensity of a classical wave rather than of a quantum particle, all the results are equally applicable; see also Section 7). Within the random matrix theory, the distribution of eigenvector amplitudes is simply Gaussian, s2 ` a Dt D2 leading to distribution of the intensities i (Porter}Thomas distribution) [12]. A theoretical study of the eigenfunction statistics in a disordered system is again possible with use of the supersymmetry method. The corresponding formalism, which was developed in Refs. [17}20] (see Section 3.1), allows one to express various distribution functions characterizing the eigenfunction statistics through the p-model correlators. As in the case of the level correlation function, the zero-mode approximation to the p-model reproduces the RMT results, in particular the Porter}Thomas distribution of eigenfunction amplitudes.