Intertwining Markov Processes via Matrix Product Operators

TL;DR

This paper develops a novel out-of-equilibrium matrix product operator framework to implement duality transformations in boundary-driven Markov processes, revealing new insights into non-equilibrium physics.

Contribution

It introduces a new class of duality operators for Markov processes that work out-of-equilibrium, extending the concept of duality beyond local symmetries.

Findings

Exact construction of duality operators for symmetric simple exclusion process

Out-of-equilibrium boundaries dual to equilibrium boundaries satisfying Liggett's condition

Gibbs-Boltzmann measure captures out-of-equilibrium physics via duality operator

Abstract

Duality transformations reveal unexpected equivalences between seemingly distinct models. We introduce an out-of-equilibrium generalisation of matrix product operators to implement duality transformations in one-dimensional boundary-driven Markov processes on lattices. In contrast to local dualities associated with generalised symmetries, here the duality operator intertwines two Markov processes via generalised exchange relations and realises the out-of-equilibrium duality globally. We construct these operators exactly for the symmetric simple exclusion process with distinct out-of-equilibrium boundaries. In this case, out-of-equilibrium boundaries are dual to equilibrium boundaries satisfying Liggett's condition, implying that the Gibbs-Boltzmann measure captures out-of-equilibrium physics when leveraging the duality operator. We illustrate this principle through physical applications.