Intertwining Markov Processes via Matrix Product Operators

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.

💡 Research Summary

This paper introduces a novel framework for understanding duality transformations in out-of-equilibrium Markov processes, leveraging the power of tensor networks. The core idea is to generalize the concept of Matrix Product Operators (MPOs) to construct explicit “duality operators” that intertwine two distinct Markovian generators.

The authors begin by contextualizing their work within the rich history of dualities in both equilibrium statistical mechanics and quantum many-body systems. They highlight the parallel between the efficient description of quantum ground states using Matrix Product States (MPS) and the exact solution of certain non-equilibrium steady states using the Matrix Product Ansatz (MPA), as pioneered by Derrida et al. for the asymmetric simple exclusion process (ASEP). This motivates the use of tensor network methods for stochastic processes.

The key theoretical advancement is the definition and construction of a duality operator D, formulated as an MPO, which satisfies an intertwining relation between the Markov generators L and L’ of two different processes: D L = L’ D. This is a global duality transformation, distinct from local ones tied to symmetries. The MPO form provides a compact and operational representation of this potentially complex transformation acting on the entire lattice.

To demonstrate the power and concrete utility of this framework, the authors apply it to the paradigmatic Symmetric Simple Exclusion Process (SSEP) with open boundaries driven out of equilibrium by different injection/extraction rates at the left and right ends. They successfully construct the exact MPO duality operator for this specific case. A profound consequence of this construction is revealed: the original non-equilibrium SSEP is mapped via this duality to an SSEP with boundary conditions that satisfy Liggett’s condition for equilibrium. This equilibrium system admits a simple Gibbs-Boltzmann steady state.

The practical implication is significant. When calculating expectation values in the hard-to-solve non-equilibrium steady state, one can instead use the duality operator to transform the problem into a calculation involving the much simpler equilibrium Gibbs state of the dual model. This provides a powerful computational shortcut and a deep conceptual link, showing how equilibrium statistical mechanics can, through the lens of a suitably chosen duality, capture the physics of a driven non-equilibrium system.

In summary, the work bridges tensor network techniques from quantum information theory with non-equilibrium statistical physics. It provides a systematic MPO-based approach to duality, uncovers a specific hidden equivalence between a non-equilibrium and an equilibrium SSEP, and opens new avenues for solving and understanding driven open systems by connecting them to their simpler equilibrium counterparts.