Perturbation theory for the Rosenzweig-Porter matrix model

Altland, Alexander; Janssen, Martin; Shapiro, Boris

doi:10.1103/physreve.56.1471

Cited by 31 publications

(58 citation statements)

References 17 publications

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“…For µ > cN , Λ → 0 and for µ < cN , Λ → ∞ for N → ∞ and thus the GGE behaves like a Poisson ensemble in the first case and like a GOE in the second; this result is in agreement with the one obtained, in [25], by using NLSM technique. (Note in ref.…”

Section: B 1-d Quasi 1-d Periodic 1-d Disordered and Chaotic Systemssupporting

confidence: 81%

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Alternative technique for complex spectra analysis

Shukla

2000

Phys. Rev. E

117

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The choice of a suitable random matrix model of a complex system is very sensitive to the nature of its complexity. The statistical spectral analysis of various complex systems requires, therefore, a thorough probing of a wide range of random matrix ensembles which is not an easy task. It is highly desirable, if possible, to identify a common mathematical structure among all the ensembles and analyze it to gain information about the ensemble-properties. Our successful search in this direction leads to Calogero Hamiltonian, a one-dimensional quantum Hamiltonian with inverse-square interaction, as the common base. This is because both, the eigenvalues of the ensembles, and, a general state of Calogero Hamiltonian, evolve in an analogous way for arbitrary initial conditions. The varying nature of the complexity is reflected in different form of the evolution parameter in each case. A complete investigation of Calogero Hamiltonian can then help us in the spectral analysis of complex systems. Recent statistical studies in various branches of theoretical physics, ranging from Calogero model of 1-d fermionic system [1], random matrix (RM) model of disordered systems, matrix models of random surfaces to non-linear sigma model of quantum chaotic systems have revealed the presence of a common mathematical structure [2][3][4]. The connecting-web of these various models with each other is well-described in [3]. However, so far, the connection of RM model with other models was established only for standard Gaussian ensembles (SGE), that is, Gaussian ensembles invariant under unitary transformation. This was achieved by showing that distribution of the eigenvalues of the ensemble is governed by a Fokker-Planck (F-P) equation [6,7] similar to that of Dyson's "Brownian" motion model [5]. Through the reduction of F-P equation to the Schrodinger equation, the latter model is already known to be connected to Calogero Hamiltonian and thereby to various other models [6][7][8]. In this paper, we explore RM models with non-invariant distributions, and, following the same route as in the case of SGE, connect them to Calogero Hamiltonian. This gives us a new technique to analyze the spectral behavior of the quantum operators of complex systems.The connection between Complex systems and Calogero Hamiltonian seems to be wide-ranging. The eigenvalue dynamics of Hermitian operators, for example, Hamiltonians of complex quantum systems e.g. chaotic systems, disordered systems seems to have an intimate connection with the particle-dynamics of Calogero-Moser (CM) Hamiltonian. The latter describes the dynamics of particles interacting via pairwise inverse square interaction and confined to move along a real line [1]here µ i is the position of the i th particle and V (µ i ) is the confining potential. Similarly the level-dynamics of the unitary operators e.g. time-evolution operator is connected to Calogero-Sutherland (CS) Hamiltonian [9]:where particles are confined to move in a circle thus mimicking the similar confinement of eigenvalues due t...

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Section: B 1-d Quasi 1-d Periodic 1-d Disordered and Chaotic Systemssupporting

confidence: 81%

“…(Note in ref. [25], D is taken as D ∝ 1/N , which gives Λ ≈ N 2 2µ and therefore GOE and Poisson ensemble result for µ < cN 2 and µ > cN 2 respectively).…”

Section: B 1-d Quasi 1-d Periodic 1-d Disordered and Chaotic Systemsmentioning

confidence: 99%

Alternative technique for complex spectra analysis

Shukla

2000

Phys. Rev. E

117

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“…Comparing our formula (2.22) with the expression (15) in Ref. 21 it seems that the results of Ref. 21 are qualitatively, but not quantitatively, correct.…”

Section: Discussionmentioning

confidence: 52%

“…III E and Appendix C) which are completely neglected in Ref. 21. To understand this in detail, we have also to take into account that Altland et al considered a model where the diagonal matrix elements of the perturbation αH 1 are zero.…”

Section: Discussionmentioning

confidence: 99%

“…This problem, and some surprises which we encountered while studying it, have been our original motivation in this project. The random matrix model (1.1) has been considered by many authors [14][15][16][17][18]20,21 . Recently, one of us succeeded 18 in deriving exact integral representations for the spectral k-point functions of the Hamiltonian (1.1).…”

Section: Introductionmentioning

confidence: 99%

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Between Poisson and GUE Statistics: Role of the Breit–Wigner Width

1998

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We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, the coset method and the graded eigenvalue method, we derive the two-level correlation function X2(r) and the number variance Σ 2 (r). The graded eigenvalue approach leads to an expression for X2(r) which is valid for all values of the parameter λ governing the strength of the GUE admixture on the unfolded scale. A new twofold integration representation is found which can be easily evaluated numerically. For λ ≫ 1 the Breit-Wigner width Γ1 measured in units of the mean level spacing D is much larger than unity. In this limit, closed analytical expression for X2(r) and Σ 2 (r) can be derived by (i) evaluating the double integral perturbatively or (ii) an ab initio perturbative calculation employing the coset method. The instructive comparison between both approaches reveals that random fluctuations of Γ1 manifest themselves in modifications of the spectral statistics. The energy scale which determines the deviation of the statistical properties from GUE behavior is given by √ Γ1. This is rigorously shown and discussed in great detail. The Breit-Wigner Γ1 width itself governs the approach to the Poisson limit for r → ∞. Our analytical findings are confirmed by numerical simulations of an ensemble of 500 × 500 matrices, which demonstrate the universal validity of our results after proper unfolding.

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Eigenvalue correlations for generalized Gaussian ensembles

Shukla

2000

Physica A: Statistical Mechanics and its Applications

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Perturbation theory for the Rosenzweig-Porter matrix model

Cited by 31 publications

References 17 publications

Alternative technique for complex spectra analysis

Alternative technique for complex spectra analysis

Between Poisson and GUE Statistics: Role of the Breit–Wigner Width

Eigenvalue correlations for generalized Gaussian ensembles

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