Exactly solvable dissipative dynamics and one-form strong-to-weak spontaneous symmetry breaking in interacting two-dimensional spin systems

TL;DR

This paper introduces an exactly solvable model of dissipative two-dimensional spin systems that exhibits mixed-state topological order and spontaneous symmetry breaking, providing insights into nonequilibrium quantum phases and relaxation dynamics.

Contribution

It constructs a solvable Lindbladian model linking spin dynamics to non-Hermitian Majorana fermions, revealing universal steady states and relaxation behaviors independent of the underlying graph.

Findings

Steady states exhibit mixed-state topological order.

Relaxation rates are finite but can vanish in certain symmetry sectors.

Exact solutions constrain dissipation forms and elucidate relaxation mechanisms.

Abstract

We study the dissipative dynamics of a class of interacting ``gamma-matrix'' spin models coupled to a Markovian environment. For spins on an arbitrary graph, we construct a Lindbladian that maps to a non-Hermitian model of free Majorana fermions hopping on the graph with a background classical $\mathbb{Z}_2$ gauge field. We show, analytically and numerically, that the steady states and relaxation dynamics are qualitatively independent of the choice of the underlying graph, in stark contrast to the Hamiltonian case. We also show that the exponentially many steady states provide a concrete example of mixed-state topological order, in the sense of strong-to-weak spontaneous symmetry breaking of a one-form symmetry. While encoding only classical information, the steady states still exhibit long-range quantum correlations. Afterward, we examine the relaxation processes toward the steady state by numerically computing decay rates, which we generically find to be finite, even in the dissipationless limit. However, we identify symmetry sectors where fermion-parity conservation is enhanced to fermion-number conservation, where we can analytically bound the decay rates and prove that they vanish in the limits of both infinitely weak and infinitely strong dissipation. Finally, we show that while the choice of coherent dynamics is very flexible, exact solvability strongly constrains the allowed form of dissipation. Our work establishes an analytically tractable framework to explore nonequilibrium quantum phases of matter and the relaxation mechanisms toward them.