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

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.