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arXiv:1904.12505 · Physical review. B./Physical review. B

Dissipative quantum Ising chain as a non-Hermitian Ashkin-Teller model

Naoyuki Shibata, Hosho Katsura

Published Jun 28, 2019• 63 citations
arXivHTMLPDF

Abstract

We study a quantum Ising chain with tailored bulk dissipation, which can be mapped onto a non-Hermitian Ashkin-Teller model. By exploiting the Kohmoto-den Nijs-Kadanoff transformation, we further map it to a staggered XXZ spin chain with pure-imaginary anisotropy parameters. This allows us to study the eigenstates of the original Liouvillian in great detail. We show that the steady state in each parity sector is a completely mixed state. The uniqueness of each is proved rigorously. We then study the decay modes on the self-dual line corresponding to the uniform XXZ chain and obtain an exact formula for the Liouvillian gap $g$, the inverse relaxation time, in the thermodynamic limit. The gap $g$ as a function of dissipation strength $\mathrm{\ensuremath{\Delta}}$ has a cusp, implying a kind of phase transition for the first decay mode.

DOI
10.1103/physrevb.99.224432

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We study a quantum Ising chain with tailored bulk dissipation, which can be mapped onto a non-Hermitian Ashkin-Teller model. By exploiting the Kohmoto-den Nijs-Kadanoff transformation, we further map it to a staggered XXZ spin chain with pure-imaginary anisotropy parameters. This allows us to study the eigenstates of the original Liouvillian in great detail. We show that the steady state in each parity sector is a completely mixed state. The uniqueness of each is proved rigorously. We then study the decay modes on the self-dual line corresponding to the uniform XXZ chain and obtain an exact formula for the Liouvillian gap ggg, the inverse relaxation time, in the thermodynamic limit. The gap ggg as a function of dissipation strength \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ has a cusp, implying a kind of phase transition for the first decay mode.
In specific open systems with collective dissipation the Liouvillian can be mapped to a non-Hermitian Hamiltonian. We here consider such a system where the Liouvillian is mapped to an XXZ Richardson-Gaudin integrable model and detail its exact Bethe ansatz solution. While no longer Hermitian, the Hamiltonian is pseudo-Hermitian/PT\text{pseudo-Hermitian}/\mathcal{P}\mathcal{T}pseudo-Hermitian/PT symmetric, and as the strength of the coupling to the environment is increased, the spectrum in a fixed symmetry sector changes from a broken pseudo-Hermitian phase with complex conjugate eigenvalues to a pseudo-Hermitian phase with real eigenvalues, passing through a series of exceptional points and associated dissipative quantum phase transitions. The homogeneous limit supports a nontrivial steady state, and away from this limit this state gives rise to a slow logarithmic growth of the decay rate (spectral gap) with system size. Using the exact solution, it is furthermore shown how at large coupling strengths the ratio of the imaginary to the real part of the eigenvalues becomes approximately quantized in the remaining symmetry sectors.
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.
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.
The ground-state degeneracy of the quantum spin system is a characteristic of nontrivial topology when it is gapped and robust against disordered perturbation. The corresponding quantum phase transition (QPT) is usually driven by a real parameter. We study a non-Hermitian Ising chain with two transverse fields, one real and another imaginary, based on the exact solution and numerical simulation. We show that topological degeneracy still exists and can be obtained by an imaginary transverse field from a topologically trivial phase of a Hermitian system. The topological degeneracy is robust against the random imaginary field and therefore expected to be immune to disordered dissipation from the spontaneous decay in experiment. The underlying mechanism is the nonlocal symmetry, which emerges only in thermodynamic limit and relates two categories of QPTs in the quantum spin system, rooted from topological order and symmetry breaking, respectively.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
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.