Dissipative Dirac matrix spin model in two dimensions

We generalize the recent work of Shibata and Katsura Phys. Rev. B 99, 174303 (2019), who considered a S=12 chain with alternating XX and YY couplings in the presence of dephasing, the dynamics of which are described by the GKLS master equation. Their model is equivalent to a non-Hermitian system described by the Kitaev formulation Kitaev, Ann. Phys. 321, 2 (2006) in terms of a single Majorana species hopping on a two-leg ladder in the presence of a nondynamical Z₂^* gauge field. Our generalization involves Dirac gamma matrix ``spin'' operators on the square lattice and maps onto a non-Hermitian square lattice bilayer which is also Kitaev solvable. We describe the exponentially many nonequilibrium steady states in this model. We identify how the spin degrees of freedom can be accounted for in the two-dimensional model in terms of the gauge-invariant quantities and then proceed to study the Liouvillian spectrum. We use simulated annealing to estimate the Liouvillian gap and the first decay modes for large system sizes. We observe a transition in the first decay modes, similar to that in the work of Shibata and Katsura. The results we obtain are compared to the results we obtained from a perturbative analysis for small and large values of the dissipation strength.