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Dynamics of Fluctuations in Quantum Simple Exclusion Processes

by Denis Bernard, Fabian H. L. Essler, Ludwig Hruza, Marko Medenjak

Submission summary

As Contributors: Fabian Essler · Ludwig Hruza
Arxiv Link: (pdf)
Date accepted: 2021-11-22
Date submitted: 2021-07-19 16:52
Submitted by: Hruza, Ludwig
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We consider the dynamics of fluctuations in the quantum asymmetric simple exclusion process (Q-ASEP) with periodic boundary conditions. The Q-ASEP describes a chain of spinless fermions with random hoppings that are induced by a Markovian environment. We show that fluctuations of the fermionic degrees of freedom obey evolution equations of Lindblad type, and derive the corresponding Lindbladians. We identify the underlying algebraic structure by mapping them to non-Hermitian spin chains and demonstrate that the operator space fragments into exponentially many (in system size) sectors that are invariant under time evolution. At the level of quadratic fluctuations we consider the Lindbladian on the sectors that determine the late time dynamics for the particular case of the quantum symmetric simple exclusion process (Q-SSEP). We show that the corresponding blocks in some cases correspond to known Yang-Baxter integrable models and investigate the level-spacing statistics in others. We carry out a detailed analysis of the steady states and slow modes that govern the late time behaviour and show that the dynamics of fluctuations of observables is described in terms of closed sets of coupled linear differential-difference equations. The behaviour of the solutions to these equations is essentially diffusive but with relevant deviations, that at sufficiently late times and large distances can be described in terms of a continuum scaling limit which we construct. We numerically check the validity of this scaling limit over a significant range of time and space scales. These results are then applied to the study of operator spreading at large scales, focusing on out-of-time ordered correlators and operator entanglement.

Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-11-8 (Invited Report)


This paper is dedicated to a thorough study of the quadratic fluctuations of the quantum symmetric simple exclusion process (Q-SSEP). After the introduction, in section 2 and 3 the authors set up the formalism allowing to study arbitrary order fluctuations of the more general quantum asymmetric simple exclusion process (Q-ASEP), by writing the Lindblad equation of the model as an higher rank spin chain, and work out the splitting of the dynamics into different
U (1) sectors.
In the rest of the paper the focus is restricted to the symmetric case and to quadratic fluctuations. In section 4 it is discussed the integrability or non integrability of the resulting spin chains in the different symmetry sectors. In section 5 the steady state and low lying modes. In section 6 the scaling limit of the model is worked out. In the final section it is considered the
large scale dynamics of operator spreading, with a particular focus on the out-of-time ordered correlators and on the hydrodynamics of of the operator entanglement spreading.

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

Anonymous Report 1 on 2021-10-3 (Invited Report)


This excellent paper reports substantial advances on understanding the dynamical properties of the Quantum Simple Exclusion Processes, which can be described in terms of a Lindblad evolution equation. A partial integrability - which allows for obtaining exact results - is derived and many significant properties of the late-time behaviour are studied. The paper is very long due to detailed derivations and pedagogical appendices. This is very helpful for readers who are not fully familiar with the approach taken. Readers interested only in the results can skip the appendices without loosing the thread. Indeed, the paper is well-structured and well-written and can be published without revision.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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