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pub34 - VOLUME 79 NUMBER 10 PHYSICAL REVIEW LETTERS 8...

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V OLUME 79, N UMBER 10 P H Y S I C A L R E V I E W L E T T E R S 8 S EPTEMBER 1997 New Class of Random Matrix Ensembles with Multifractal Eigenvectors V. E. Kravtsov 1,2 and K. A. Muttalib 3 1 International Center for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy 2 Landau Institute for Theoretical Physics, Kosygina str. 2, 117940 Moscow, Russia 3 Department of Physics, University of Florida, Gainesville, Florida 32611 ( Received 12 March 1997 ) Three recently suggested random matrix ensembles (RME) are linked together to represent a class of RME with multifractal eigenfunction statistics. The generic form of the two-level correlation function for the case of weak and extremely strong multifractality is suggested. [S0031-9007(97)03707-1] PACS numbers: 72.15.Rn, 05.40.+j, 05.45.+b Random matrix ensembles turn out to be a natural and convenient language to formulate generic statistical prop- erties of energy levels and transmission matrix elements in complex quantum systems. Gaussian random matrix en- sembles, first introduced by Wigner and Dyson [1,2] for describing the spectrum of complex nuclei, became very popular in solid state physics as one of the main theo- retical tools to study mesoscopic fluctuations [3] in small disordered electronic systems. The success of the ran- dom matrix theory (RMT) [2] in mesoscopic physics is due to its extension to the problem of electronic transport based on the Landauer-Büttiker formula [4] and the statis- tical theory of transmission eigenvalues [5]. Another field where the RMT is exploited very intensively is the prob- lem of the semiclassical approximation in quantum sys- tems whose classical counterpart is chaotic [6]. It turns out [6] that the energy level statistics in true chaotic sys- tems is described by the RMT, in contrast to that in the integrable systems where in most cases it is close to the Poisson statistics. Apparently the nature of the energy level statistics is related to the structure of eigenfunctions, and more pre- cisely, to the overlapping between different eigenfunc- tions. This is well illustrated by spectral statistics in a system of noninteracting electrons in a random poten- tial which exhibits the Anderson metal-insulator transition with increasing disorder. At small disorder the electron wave functions are extended and essentially structureless. They overlap very well with each other, resulting in en- ergy level repulsion characteristic of the Wigner-Dyson statistics. On the other hand, in the localized phase elec- trons are typically localized at different points of the sample, and in the thermodynamic limit where the sys- tem size L ! ` they “do not talk to each other.” In this case there is no correlation between eigenvalues, and the energy levels follow the Poisson statistics. The energy level statistics in the critical region near the Anderson transition turns out to be universal and different from both Wigner-Dyson statistics and the Poisson statis- tics [7,8]. A remarkable feature of the critical level statis- tics is that the level number variance S 2 s N d ks d N d 2 l x N
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