# Lattice SUSY for the DiSSEP at $\lambda^2=1$ (and $\lambda^2 = -3 $)

## TL;DR

This paper explores the existence of lattice supersymmetry in the DiSSEP Markov matrix, showing it can be realized for specific parameters by relating it to supersymmetric quantum spin chains, revealing a novel symmetry in exclusion processes.

## Contribution

It demonstrates that the DiSSEP Markov matrix exhibits lattice supersymmetry for certain parameters by establishing a conjugation relation to supersymmetric quantum spin chains.

## Key findings

- DiSSEP Markov matrix admits supersymmetry at specific parameters.
- Supersymmetry relates to length-changing supercharges similar to transfer matrix symmetries.
- The supersymmetry is connected to conjugation with quantum spin chain Hamiltonians.

## Abstract

We investigate whether the dynamical lattice supersymmetry discussed for various Hamiltonians, including one-dimensional quantum spin chains, by Fendley et.al. and Hagendorf et.al. might also exist for the Markov matrices of any one-dimensional exclusion processes, since these can be related by conjugation to quantum spin chain Hamiltonians. We find that the DiSSEP (Dissipative Symmetric Simple Exclusion Process), introduced by Crampe et.al. provides one such example for suitably chosen parameters. The DiSSEP Markov matrix admits the supersymmetry in these cases because it is conjugate to spin chain Hamiltonians which also possess the supersymmetry. We note that the length-changing supersymmetry relation for the DiSSEP Markov matrix and the supercharge is reminiscent of a "transfer matrix" symmetry that has been observed in other exclusion processes and discuss the similarity.