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arXiv:1812.10373 · Physical review. B./Physical review. B

Dissipative spin chain as a non-Hermitian Kitaev ladder

Naoyuki Shibata, Hosho Katsura

Published May 10, 2019• 118 citations
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Abstract

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.

DOI
10.1103/physrevb.99.174303

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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 \ensuremathγ\ensuremath{\gamma}\ensuremathγ is small, while it decreases monotonically for large \ensuremathγ\ensuremath{\gamma}\ensuremathγ, 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.
We generalize the recent work of Shibata and Katsura [Phys. Rev. B 99, 174303 (2019)], who considered a S=12S=\frac{1}{2}S=21​ chain with alternating XXXXXX and YYYYYY 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∗{\mathbb{Z}}_{2}^{\phantom{*}}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.
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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.
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Dissipative Yao-Lee Spin-Orbital Model: Exact Solvability and PT\mathcal{PT}PT Symmetry Breaking
Exactly solvable dissipative models provide an analytical tool for studying the relaxation dynamics in open quantum systems. In this work, we study an exactly solvable model based on an anisotropic variant of the Yao-Lee spin-orbital model, with dissipation acting in the spin sector. We map Liouvillian dynamics to fermions hopping in a doubled Hilbert space under a non-Hermitian Hamiltonian and demonstrate the model's exact solvability. We analyze the model's strong and weak symmetries, which protect an exponentially large manifold of non-equilibrium steady states, establishing the system as a physically feasible dissipative spin liquid. Furthermore, we analyze the transient dynamics in a translationally invariant sector and discover that the single-particle Liouvillian spectrum hosts an exceptional ring in momentum space. We map out a characteristic PT\mathcal{PT}PT symmetry breaking transition driven by the dissipation strength, which governs the crossover from oscillatory to decaying relaxation of physical observables. Our work provides a physically motivated, solvable setting for exploring the coexistence of dissipative spin liquid physics and Liouvillian spectral singularities.
The dynamics described by the non-Hermitian Hamiltonian typically capture the short-term behavior of open quantum systems before quantum jumps occur. In contrast, the long-term dynamics, characterized by the Lindblad master equation, drive the system towards a nonequilibrium steady state (NESS), which is an eigenstate with zero energy of the Liouvillian superoperator, denoted as L\mathcal{L}L. Conventionally, these two types of evolutions exhibit distinct dynamical behaviors. However, in this study we challenge this common belief and demonstrate that the effective non-Hermitian Hamiltonian can accurately represent the long-term dynamics of a critical two-level open quantum system. The criticality of the system arises from the exceptional point of the effective non-Hermitian Hamiltonian. Additionally, the NESS is identical to the coalescent state of the effective non-Hermitian Hamiltonian. We apply this finding to a series of critical open quantum systems and show that a local dissipation channel can induce collective alignment of all spins in the same direction. This direction can be well controlled by modulating the quantum jump operator. The corresponding NESS is a product state and maintains long-time coherence, facilitating quantum control in open many-body systems. This discovery paves the way for a better understanding of the long-term dynamics of critical open quantum systems.