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arXiv:1608.06636 · Entropy

Realistic Many-Body Quantum Systems vs. Full Random Matrices: Static and Dynamical Properties

Eduardo Jonathan Torres-Herrera, Jonathan Karp, Marco Távora, Léa Ferreira dos Santos

Published Oct 8, 2016• 50 citations
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Abstract

We study the static and dynamical properties of isolated many-body quantum systems and compare them with the results for full random matrices. In doing so, we link concepts from quantum information theory with those from quantum chaos. In particular, we relate the von Neumann entanglement entropy with the Shannon information entropy and discuss their relevance for the analysis of the degree of complexity of the eigenstates, the behavior of the system at different time scales and the conditions for thermalization. A main advantage of full random matrices is that they enable the derivation of analytical expressions that agree extremely well with the numerics and provide bounds for realistic many-body quantum systems.

DOI
10.3390/e18100359

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This work supports the existence of extended nonergodic states in the intermediate region between the chaotic (thermal) and the many‐body localized phases. These states are identified through an extensive analysis of static and dynamical properties of a finite one‐dimensional system with onsite random disorder. The long‐time dynamics is particularly sensitive to changes in the spectrum and in the structures of the eigenstates. The study of the evolution of the survival probability, Shannon information entropy, and von Neumann entanglement entropy enables the distinction between the chaotic and the intermediate region. image
We define correlational (von Neumann) entropy for an individual quantum state of a system whose time-independent Hamiltonian contains random parameters and is treated as a member of a statistical ensemble. This entropy is representation independent, and can be calculated as a trace functional of the density matrix which describes the system in its interaction with the noise source. We analyze perturbation theory in order to show the evolution from the pure state to the mixed one. Exactly solvable examples illustrate the use of correlational entropy as a measure of the degree of complexity in comparison with other available suggestions such as basis-dependent information entropy. It is shown in particular that a harmonic oscillator in a uniform field of random strength comes to a quasithermal equilibrium; we discuss the relation between effective temperature and canonical equilibrium temperature. The notion of correlational entropy is applied to a realistic numerical calculation in the framework of the nuclear shell model. In this system, which reveals generic signatures of quantum chaos, correlational entropy and information entropy calculated in the mean field basis display similar qualitative behavior.