Microscopic foundation of the Eigenstate Thermalization Hypothesis

by Nils O. Abeling
Doctoral thesis
Date of Examination:2022-03-29
Date of issue:2023-03-20
Advisor:Prof. Dr. Stefan Kehrein
Referee:Prof. Dr. Stefan Kehrein
Referee:Prof. Dr. Fabian Heidrich-Meisner
Persistent Address: http://dx.doi.org/10.53846/goediss-9799

 

 

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Abstract

English

The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deutsch, who showed how a small interaction, which is modeled by a random matrix, leads to ETH (J.M. Deutsch, Phys. Rev. A 43, 1991). This thesis aims to find justification for the assumptions made by Deutsch from a microscopic point of view. The idea is to analyze whether a generic quantum system can be brought to a form that satisfies Deutsch's ansatz. The method is to use flow equations to define unitary transformations to map the initial Hamiltonian to an effective one, which can be compared to a random matrix. The test system is a one-dimensional hardcore boson model, which is set up using exact diagonalization and numerous symmetries. We analyze the statistical properties of the numerically obtained Hamiltonian matrices and test for normality and compare them to a random transformation. In a second step we verify that the continuous unitary transformations do not destroy the structure of typical few-body observables by studying the scaling of the inverse participation ratio. The obtained results support our hypothesis that it is possible to extend Deutsch's argument to generic microscopic many-body Hamiltonians.
Keywords: Eigenstate Thermalization Hypothesis; Flow equations; Isolated quantum systems; Non-equilibrium dynamics; Quantum chaos; Quantum many-body systems
 
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