A study of dissipative models based on Dirac matrices

TL;DR

This paper extends a model of dissipative quantum spin chains to a 2D lattice using Dirac matrices, analyzing non-equilibrium steady states and decay modes with a genetic algorithm, revealing phase transition behavior.

Contribution

It introduces a generalized 2D Dirac matrix-based model that is solvable and studies its non-equilibrium properties and decay modes using numerical methods.

Findings

Identification of exponentially many non-equilibrium steady states.

Observation of a transition in the first decay modes.

Consistency with perturbative analysis for different dissipation strengths.

Abstract

We generalize the recent work of Shibata and Katsura, who considered a S=1/2 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 in terms of a single Majorana species hopping on a two-leg ladder in the presence of a nondynamical Z_2 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 non-equilibrium steady states in this model. We identify how the spin degrees of freedom can be accounted for in the 2d model in terms of the gauge-invariant quantities and then proceed to study the Liouvillian spectrum. We use a genetic algorithm 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 found by Shibata and Katsura. The results we obtain are consistent with a perturbative analysis for small and large values of the dissipation strength.