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

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...

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Bibliographic Details
Published in:arXiv.org 2018-09
Main Authors: Nation, Charlie, Porras, Diego
Format: Article
Language:English
Subjects:
Eigenvectors
Matrix theory
Subsystems
Thermalization (energy absorption)
Variation
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1803.01650