Solution to the Quantum Symmetric Simple Exclusion Process: The Continuous Case

Denis Bernard,Tony Jin

doi:10.1007/s00220-021-04087-x

Abstract

The quantum symmetric simple exclusion process (Q-SSEP) is a model for quantum stochastic dynamics of fermions hopping along the edges of a graph with Brownian noisy amplitudes and driven out-of-equilibrium by injection-extraction processes at a few vertices. We present a solution for the invariant probability measure of the one dimensional Q-SSEP in the infinite size limit by constructing the steady correlation functions of the system density matrix and quantum expectation values. These correlation functions code for a rich structure of fluctuating quantum correlations and coherences. Although our construction does not rely on the standard techniques from the theory of integrable systems, it is based on a remarkable interplay between the permutation groups and polynomials. We incidentally point out a possible combinatorial interpretation of the Q-SSEP correlation functions via a surprising connexion with geometric combinatorics and the associahedron polytopes.

Highlights

Non-equilibrium phenomena, classical or quantum, are ubiquitous in Nature, but their understanding is more difficult, and yet less profound, than the equilibrium ones
The quantum Symmetric Simple Exclusion Process (SSEP) is a model for stochastic quantum many-body dynamics of fermions hopping on the edges of a graph but with Brownian amplitudes and injection/extraction processes at a few vertices modelling interaction with external reservoirs
We shall be interested in the one dimensional case (1D) defined on a line interval with injection/extraction processes at the two ends of the interval

Summary

Introduction

Non-equilibrium phenomena, classical or quantum, are ubiquitous in Nature, but their understanding is more difficult, and yet less profound, than the equilibrium ones. Other pieces of information on a possible form of such theory for diffusive systems has recently been gained by studying model systems based say on random quantum circuits for which a membrane picture [17,18,19,20,21] for entanglement production in many-body systems is starting to emerge Another route has been taken in a series of works [22,23,24] consisting in analysing model systems which provide quantum extensions of the classical exclusion processes SSEP or ASEP.

Summary

Preliminaries

Solution to Q-SSEP in the Continuum

The Regular Loop

Simple Deformations of the Regular Loop

General Deformations of the Regular Loop