Microscopic Foundations of the Eigenstate Thermalization Hypothesis

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A fundamental assumption for the modeling of classical or quantum mechanical many-particle systems is that they return to an equilibrium state after a sufficiently long waiting time. This equilibrium state can then be described as a Gibbs state with only the temperature as a parameter (in the simplest case). All theoretical calculations of properties of many-particle systems like transport properties, thermodynamic properties, etc., are built on this thermalization assumption. The textbook derivation of this assumption proceeds via the coupling to a large environment. In the past decade experiments in ultracold atomic gases, which are extremely well isolated from their environment, became possible so that now one also needs to address the question of thermalization in closed quantum systems. For generic quantum many-particle systems, which are non-integrable (meaning they have only a finite number of conserved quantities), the so called eigenstate thermalization hypothesis (ETH) has been put forward as an explanation for thermalization of such closed non-integrable systems. Most publications regarding this topic proceed numerically, that is they establish ETH for certain model Hamiltonians. However, it should be mentioned that there are unresolved questions, so even the numerical situation is not entirely clear. In this project we aim to pursue a complementary approach, which is at least partially analytic. The starting point is an older publication by J. M. Deutsch [Phys. Rev. A 43, 2046 (1991)], who could show analytically that ETH is fulfilled if one works with a random matrix model. Deutsch's work plays a somehow less prominent role in the current literature since realistic microscopic Hamiltonians are not random matrices. This project aims at closing this gap by mapping a realistic microscopic Hamiltonian to a random matrix Hamiltonian using a sequence of infinitesimal unitary transformations (Wegner flow equations). In this way the analytical line of argument by J. M. Deutsch can be pulled back to the original realistic microscopic Hamiltonian. Additionally, one can learn something about the conditions under which ETH holds, both for the Hamiltonian and the observables, especially also for correlations functions which are nonlocal in space and/or time. Ideally, the analytic approach that we want to pursue could serve as a unifying bracket for the various numerical approaches and thereby be a contribution to a deeper understanding of the thermalization assumption for the description of quantum mechanical many-particle systems.