# Non-compact quantum spin chains as integrable stochastic particle processes

## TL;DR

This paper links integrable quantum spin chains to stochastic particle processes, revealing new boundary conditions, dual models, and connections to supersymmetric gauge theories, advancing understanding of non-equilibrium systems and their quantum analogs.

## Contribution

It demonstrates the integrability of certain stochastic particle models via their mapping to $ ext{sl}(2)$ spin chains, including boundary terms and dual models, with connections to supersymmetric gauge theories.

## Key findings

- Identification of boundary reservoir terms maintaining integrability.
- Construction of a dual probabilistic model.
- Connection of fluctuating hydrodynamics to superstring evolution.

## Abstract

In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in [Sasamoto-Wadati], [Barraquand-Corwin] and [Povolotsky] in the context of KPZ universality class. We show that they may be mapped onto an integrable $\mathfrak{sl}(2)$ Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a "dual model" of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of $\mathcal{N}=4$ super Yang-Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the $\mathfrak{sl}(2|1)$ superstring that has been derived directly from $\mathcal{N}=4$ SYM.