A study of dissipative models based on Dirac matrices

In this work, we generalize Shibata and Katsura's study (DOI: 10.1103/PhysRevB.99.174303) of a dissipative S=1/2 1d chain to two dimensions. Their model had a Hamiltonian with alternating XX and YY couplings on a 1d chain and Lindblad operators at each site describing the dissipation. They studied the dynamics of this system using the GKLS master equation and by identifying that it could be modeled as a non-hermitian Kitaev system on a two-leg ladder with a single Majorana species in the presence of a static Z2 gauge field.

In our generalization to 2d, we consider a dissipative square lattice based on Gamma matrix spin operators which can be modeled as a non-Hermitian square lattice bilayer. It is again Kitaev-solvable. We identify the non-equilibrium steady states in the model. We identify the gauge invariant quantities and the constraints relating them and show how they can account for all the spin degrees of freedom. We show how a gauge can be chosen and proceed to look at the Liouvillian spectrum. We use a genetic algorithm to estimate the Liouvillian gap and the first decay modes for larger system sizes. We see a change in the dynamics as we change the dissipation strength and observe a transition in the first decay modes similar to that seen in Shibata and Katsura's work. To understand the behavior for small and large values of dissipation strength, we perform a perturbative analysis and compare the results obtained with our computational results.