Off-diagonal observable elements from random matrix theory: distributions, fluctuations, and eigenstate thermalization

Abstract

We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). 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 of generic observables 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.

Quantum chaotic fluctuation-dissipation theorem: Effective Brownian motion in closed quantum systems

“…This is a very common assumption in the treatment of random matrices [37], and is well justified numerically for this model in [31].…”

“…Despite much recent progress on the understanding of thermalization, there has been less work describing the decay process [26][27][28] or the timescales of equilibration [29]. We address both of these using a Random Matrix Theory (RMT) model [16,30], which the current authors have recently shown reproduces the ETH ansatz [31]. We describe the decay to equilibrium of generic nonintegrable quantum systems, and obtain an expression for the time-averaged fluctuations of local observables in terms of their rate of decay to equilibrium; thus observing an emergent classical Fluctuation-Dissipation Theorem (FDT), analogous to those derived from a Langevin * Electronic address: C.Nation@sussex.ac.uk † Electronic address: D.Porras@iff.csic.es equation for Brownian motion.…”

Ergodicity probes: using time-fluctuations to measure the Hilbert space dimension

“…Our theory, developed in Ref. [38] by extending Deutch's RMT model [39][40][41], can be used to obtain arbitrary correlation functions c µ (α)c ν (α) · · · V , where · · · V denotes the ensemble average over an ensemble of random matrix perturbations, V , for a N × N Hamiltonian of the form (2), with (H 0 ) αβ = αω 0 δ αβ , with ω 0 = 1 N and V a random matrix selected from the GOE, with V 2 αβ V = (1+δ αβ )g 2 N . We showed that these may be expressed as sums of products of four-point correlation functions, given by (for µ = ν),…”

“…where ρ µν = ψ µ |ρ|ψ ν , and (σ z ) µν = ψ µ |σ z |ψ ν . We assume that V is well approximated by a random matrix and build on a statistical theory for the manybody wave-functions [38],…”

“…In Ref. [38] we showed that this model can be solved and it allows us to calculate matrix elements in the interacting basis. The initial state can be written in terms of eigenstates of the joint qubit-device system,…”