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arXiv:2308.05245 · Physical Review A

Dissipative Dirac matrix spin model in two dimensions

Jyotsna Gidugu, Daniel P. Arovas

Published Feb 9, 2024• 4 citations
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Abstract

We generalize the recent work of Shibata and Katsura [Phys. Rev. B 99, 174303 (2019)], who considered a $S=\frac{1}{2}$ 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 [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 ${\mathbb{Z}}_{2}^{\phantom{*}}$ 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.

DOI
10.1103/physreva.109.022212

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We derive exact results for the Lindblad equation for a quantum spin chain (one-dimensional quantum compass model) with dephasing noise. The system possesses doubly degenerate nonequilibrium steady states due to the presence of a conserved charge commuting with the Hamiltonian and Lindblad operators. We show that the system can be mapped to a non-Hermitian Kitaev model on a two-leg ladder, which is solvable by representing the spins in terms of Majorana fermions. This allows us to study the Liouvillian gap, the inverse of relaxation time, in detail. We find that the Liouvillian gap increases monotonically when the dissipation strength $\ensuremath{\gamma}$ is small, while it decreases monotonically for large $\ensuremath{\gamma}$, implying a kind of phase transition in the first decay mode. The Liouvillian gap and the transition point are obtained in closed form in the case where the spin chain is critical. We also obtain the explicit expression for the autocorrelator of the edge spin. The result implies the suppression of decoherence when the spin chain is in the topological regime.
Dissipative Yao-Lee Spin-Orbital Model: Exact Solvability and PT\mathcal{PT}PT Symmetry Breaking
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A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
We address dissipative dynamics of the one-dimensional nearest-neighbour XX spin-1/21/21/2 chain governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the absence of dissipation, the model is integrable. We identify a broad class of dissipative terms that generically destroy integrability but leave the operator space of the model fragmented into an extensive number of dynamically disjoint subspaces of varying dimensions. In sufficiently small subspaces, the GKSL equation in the Heisenberg representation can be easily solved, sometimes in a closed analytical form. We provide an example of such an exact solution for a specific choice of dissipative terms. It is found that observables experience the Wannier-Stark localization in the corresponding operator subspace. As a result, the expectation values of the observables are linear combinations of essentially a few discrete decay modes, the long time dynamics being governed by the slowest mode. We examine the complex Liouvillian eigenvalue corresponding to this latter mode as a function of the dissipation strength. We find an exceptional point at a critical dissipation strength that separates oscillating and nonoscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace. Finally, we point out that our exact solutions of the GKSL equation entail exact solutions of the Schr\"odinger equation describing the quench dynamics in closed spin ladders dual to the dissipative spin chains.
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 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.
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