Dynamics of fluctuations in quantum simple exclusion processes

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. (C) Copyright D. Bernard et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation.

Automated summary

This study investigates the dynamics of fluctuations in the Q-ASEP model, finding that fluctuations of fermionic degrees of freedom follow evolution equations of Lindblad type and deriving the corresponding Lindbladians. The study shows that Lindbladian correspond to known Yang-Baxter integrable models in the case of Q-SSEP and investigates level-spacing statistics. The dynamics of observable fluctuations are described by closed sets of coupled linear differential-difference equations, and a continuum scaling limit is constructed. The results are applied to the study of operator spreading and operator entanglement.