Alternative technique for complex spectra analysis

Abstract

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...

Eigenfunction statistics of complex systems: A common mathematical formulation

“…Recently it was shown [49,51] that the distribution ρ for both cases (eqs. (1,2)) appear as the non-equilibrium stages of a Brownian type diffusion process in the matrix-space, evolving with respect to a single parameter which is a function of the distribution parameters of the ensemble: ...…”

“…(14,18,25) too. However, as discussed in [49,51], the evolution of the n th order eigenvalue correlations (n > 1) as a function of Y , is abrupt in large N -limit; a smooth crossover can only be seen in terms of a rescaled parameter Λ e where [24,42]). For later reference, it is worth reviewing the role of Λ e , that is, Λ in locating the critical point of level statistics.…”

“…For numerical analysis, we choose three different ensembles (for both cases β = 1, 2); the choice is dictated by the reason (i) the ensembles are prototype models of many physical systems related to different areas [3,8,9,35], (ii) a comparative study of the eigenvalue fluctuations of these systems has already been carried out, with their Λ parameters and other results given in [49]: [3,50]. The ensemble density in this case can be described by eq.…”

“…As mentioned in section II, the ensemble density in this case has the same form as for Rosenzweig-Porter (RP) ensemble [33]; it can also describe an ensemble of Anderson Hamiltonians with very long range, isotropic, random hopping. Further, as discussed in [49], the special case µ = cN 2 corresponds to the critical BEs; their mean level density is given as F (E) = (π) −1/2 e −E 2 and…”

“…One way to show the connection is by describing their various measures by a common mathematical formulation if possible. A recent study [49,50,51], has shown the existence of a similar formulation for the case of level-statistics where system information enters through a single parameter, basically a function of all system parameters. The well-known connection between the statistics of eigenfunctions and eigenvalues in non-ergodic regime [3] motivates us to seek a similar formulation for the eigenfunctions too.…”