Off-Diagonal Observable Elements from Random Matrix Theory: Distributions, Fluctuations, and Eigenstate Thermalization

TL;DR

This paper derives the Eigenstate Thermalization Hypothesis (ETH) from a random matrix model, revealing how off-diagonal observable elements behave and fluctuate with system size, supported by analytical and numerical results.

Contribution

It introduces a consistent method to derive ETH from random matrix theory by incorporating orthonormality constraints, improving understanding of off-diagonal elements in quantum chaotic systems.

Findings

Off-diagonal matrix elements follow a specific distribution derived analytically.

Fluctuations scale with system size and are expressed via the Inverse Participation Ratio.

Analytical predictions agree with numerical simulations of quantum spin chains.

Abstract

We derive the Eigenstate Thermalization Hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by J. M. Deutsch [Phys. Rev. A 43, 2046 (1991)]. We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations with system size for which we calculate an analytic form in terms of the Inverse Participation Ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.