Simulations of fluctuations of quantum statistical systems of electrons

TL;DR

This paper applies random matrix theory to quantum statistical systems, analyzing eigenenergy fluctuations, chaos, and integrability, with a focus on nonhermitean ensembles and complex eigenvalues.

Contribution

It introduces a framework for analyzing dissipative quantum systems using nonhermitean random matrices and extends the space of dynamics with discrete labeling indices.

Findings

Eigenenergy second differences serve as discrete Hessians.

Measures of quantum chaos and integrability are defined and calculated.

Probability distributions are derived from maximum entropy principles.

Abstract

The random matrix ensembles (RMT) of quantum statistical Hamiltonian operators, e.g.Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied to following quantum statistical systems: nuclear systems, molecular systems, and two-dimensional electron systems (Wigner-Dyson's electrostatic analogy). The Ginibre ensemble of nonhermitean random Hamiltonian matrices $K$ is considered. Each quantum system described by $K$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. The second difference of complex eigenenergies is viewed as discrete analog of Hessian with respect to labelling index. The results are considered in view of Wigner and Dyson's electrostatic analogy. An extension of space of dynamics of random magnitudes is performed by introduction of discrete space of labeling indices. The comparison with the Gaussian ensembles of random hermitean Hamiltonian matrices $H$ is performed. Measures of quantum chaos and quantum integrability with respect to eigenergies of quantum systems are defined and they are calculated. Quantum statistical information functional is defined as negentropy (opposite of von Neumann's entropy or minus entropy). The probability distribution functionals for the random matrix ensembles (RMT) are derived from the maximum entropy principle.