[Paper Review] Towards local equilibration in closed interacting quantum many-body systems

Henrik Wilming, M. Goihl+2 more|arXiv (Cornell University)|Apr 20, 2017

Advanced Thermodynamics and Statistical Mechanics17 citations

TL;DR

This paper provides a physical intuition for why local observables in closed, interacting quantum many-body systems equilibrate rapidly, independent of system size, by leveraging dephasing and harmonic analysis. It argues that equilibration arises from the delocalization of quantum information and the structure of local Hamiltonians, supported by numerical examples and connections to established concepts like the eigenstate thermalization hypothesis and typicality.

ABSTRACT

One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system size, but only the precision with which this occurs does. In this discussion paper, we aim at giving very simple and plausible arguments for why this happens. While our discussion does not yield rigorous results about equilibration time scales, we believe that it helps to clarify the essential underlying mechanisms, the intuition and important figure of merits behind equilibration. We then connect our arguments to common assumptions and numerical results in the field of equilibration and thermalization of closed quantum systems, such as the eigenstate thermalization hypothesis as well as rigorous results on interacting quantum many-body systems. Finally, we complement our discussions with numerical results - both in the case of examples and counter-examples of equilibrating systems.

Motivation & Objective

• To clarify the physical mechanism behind fast local equilibration in closed quantum many-body systems.
• To explain why equilibration times are expected to be weakly dependent on system size, contrary to crude theoretical bounds.
• To provide intuitive, physically motivated arguments grounded in harmonic analysis and system structure.
• To connect these arguments to established concepts such as the eigenstate thermalization hypothesis and typicality in quantum systems.
• To demonstrate the behavior through numerical examples, including both equilibrating and non-equilibrating cases.

Proposed method

• Uses a simple argument from harmonic analysis to explain how dephasing leads to local equilibration in interacting systems.
• Introduces regularization of time-averaged observables to meaningfully assess equilibration in finite systems.
• Analyzes the role of the effective dimension of the initial state in determining equilibration behavior.
• Compares the proposed mechanism to rigorous results and conjectures, including the eigenstate thermalization hypothesis.
• Employs numerical simulations to illustrate equilibration in systems with and without translational invariance, and in many-body localized systems.
• Contrasts the behavior of non-interacting, localized, and generic interacting systems to highlight the conditions for equilibration.

Experimental results

Research questions

• RQ1 Why do local observables in generic interacting quantum many-body systems equilibrate rapidly, independent of system size?
• RQ2 What physical mechanisms underlie the apparent independence of equilibration time from system size?
• RQ3 How do the structure of local Hamiltonians and the delocalization of quantum information contribute to equilibration?
• RQ4 In what ways do typicality and random matrix theory assumptions relate to the observed equilibration behavior?
• RQ5 What are the conditions under which equilibration fails, and how can such counter-examples be identified numerically?

Key findings

• Local equilibration occurs due to dephasing of energy eigenstate contributions, driven by the structure of local Hamiltonians and the delocalization of quantum information.
• The equilibration time is not strongly dependent on system size, as supported by numerical examples of both equilibrating and non-equilibrating systems.
• Systems with translational invariance can fail to equilibrate, while systems breaking this symmetry can still equilibrate, indicating that symmetry alone is not a determining factor.
• Many-body localized systems, despite strong localization, still equilibrate due to dephasing, though with a logarithmic light-cone instead of a linear one.
• Non-interacting, disordered systems such as Anderson insulators do not equilibrate locally, as information does not propagate beyond the localization length.
• Regularization of time-averaged observables is essential for meaningful numerical assessment of equilibration in finite systems.