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Equilibration towards generalized Gibbs ensembles in non-interacting theories

by Marek Gluza, Jens Eisert, Terry Farrelly

Submission summary

As Contributors: Marek Gluza
Arxiv Link: (pdf)
Date accepted: 2019-08-29
Date submitted: 2019-07-17
Submitted by: Gluza, Marek
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Quantum Physics
Approach: Theoretical


Even after almost a century, the foundations of quantum statistical mechanics are still not completely understood. In this work, we provide a precise account on these foundations for a class of systems of paradigmatic importance that appear frequently as mean-field models in condensed matter physics, namely non-interacting lattice models of fermions (with straightforward extension to bosons). We demonstrate that already the translation invariance of the Hamiltonian governing the dynamics and a finite correlation length of the possibly non-Gaussian initial state provide sufficient structure to make mathematically precise statements about the equilibration of the system towards a generalized Gibbs ensemble, even for highly non-translation invariant initial states far from ground states of non-interacting models. Whenever these are given, the system will equilibrate rapidly according to a power-law in time as long as there are no long-wavelength dislocations in the initial second moments that would render the system resilient to relaxation. Our proof technique is rooted in the machinery of Kusmin-Landau bounds. Subsequently, we numerically illustrate our analytical findings by discussing quench scenarios with an initial state corresponding to an Anderson insulator observing power-law equilibration. We discuss the implications of the results for the understanding of current quantum simulators, both in how one can understand the behaviour of equilibration in time, as well as concerning perspectives for realizing distinct instances of generalized Gibbs ensembles in optical lattice-based architectures.

Ontology / Topics

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Anderson insulators Gibbs ensembles Mean-field theory (MFT)

Published as SciPost Phys. 7, 038 (2019)

Reports on this Submission

Anonymous Report 2 on 2019-8-26 Invited Report

  • Cite as: Anonymous, Report on arXiv:1809.08268v2, delivered 2019-08-26, doi: 10.21468/SciPost.Report.1133


1. Exact and new results for local thermalization (in the sense of a generalized Gibbs ensemble), especially regarding the equilibration timescales
2. Very well written paper


1. The results are only valid for non-interacting models, therefore there is no thermalization process that could lead to thermal mode occupation.


In this paper the authors show in a mathematically exact manner how local equilibration occurs in translation-invariant non-interacting models. It is especially noteworthy that a timescale for local equilibration can be extracted (power law behavior), which is often missing in other discussions of equilibration in the literature (as the authors correctly point out).

The main limitation of this paper is that non-interacting models can only thermalize locally, but not in momentum space. The authors point this out as well in Eq. (16) where the momentum occupation numbers are conserved quantities. In solid state physics the momentum occupation numbers are easily accessible observables, for example via ARPES: The main question for thermalization of translation invariant systems is how such mode occupations become thermal. This question can obviously not be addressed in the setting of this paper. So the results in this paper are mathematically nice and important, but do not contribute to the main physical question.

Still this is a very well written paper with exact results, which should certainly be published.

  • validity: top
  • significance: good
  • originality: good
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Author Marek Gluza on 2019-10-07
(in reply to Report 2 on 2019-08-26)
suggestion for further work

We thank the Referee very much for his or her positive comments.

We agree that systems which can be studied with ARPES are outside the scope of our work as we are treating only the limiting cases where e.g. phonons are effectively uncoupled from electrons and there is negligible many-body scattering. Our theory could however be interesting for experiments in optical lattices where one could systematically tune the interactions.

Anonymous Report 1 on 2019-8-17 Invited Report

  • Cite as: Anonymous, Report on arXiv:1809.08268v2, delivered 2019-08-16, doi: 10.21468/SciPost.Report.1118


1. Mathematical rigor.
2. Generality.
3. Breadth of interest.


1. Details of the proofs can be tedious to go through, but this is unavoidable in a rigorous treatment.


This paper treats the approach to a state of thermal equilibrium, as described by a generalized Gibbs ensemble, for a large class of initial states of a system of noninteracting lattice fermions. It unifies, under a general formalism, a number of other results in the literature, and provides an overview for the general mechanism of thermalization. Therefore this work should be of interest to physicists across a number of disciplines.

In my judgment, this is a very high quality paper that deserves a wide audience.

Requested changes


  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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