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arXiv:2210.08725 · Physical review. B./Physical review. B

Engineering an imaginary Stark ladder in a dissipative lattice: Passive PT symmetry, K symmetry, and localized damping

Yu Zhang, Shu Chen

Published Jun 13, 2023• 9 citations
arXivHTMLPDF

Abstract

We study an imaginary Stark ladder model, and we propose a realization of the model in a dissipative chain with linearly increasing site-dependent dissipation strength. Due to the existence of $K$ symmetry and passive $\mathcal{PT}$ symmetry, the model exhibits quite a different feature from its Hermitian counterpart. With the increase of dissipation strength, the system first undergoes a passive $\mathcal{PT}$-symmetry breaking transition, with the shifted eigenvalues changing from real to complex, and then a $K$-symmetry restoring transition, characterized by the emergence of a pure imaginary spectrum with equal spacing. Accordingly, the eigenstates change from $\mathcal{PT}$-unbroken extended states to $\mathcal{PT}$-broken states, and finally to Stark-localized states. In the framework of the quantum open system governed by the Lindblad equation with linearly increasing site-dependent dissipation, we unveil that the dynamical evolution of a single-particle correlation function is governed by the Hamiltonian of the imaginary Stark ladder model. By studying the dynamical evolution of the density distribution under various initial states, we demonstrate that the damping dynamics displays distinct behaviors in different regions. A localized damping is observed in the strong dissipation limit.

DOI
10.1103/physrevb.107.224306

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We study an imaginary Stark ladder model, and we propose a realization of the model in a dissipative chain with linearly increasing site-dependent dissipation strength. Due to the existence of KKK symmetry and passive PT\mathcal{PT}PT symmetry, the model exhibits quite a different feature from its Hermitian counterpart. With the increase of dissipation strength, the system first undergoes a passive PT\mathcal{PT}PT-symmetry breaking transition, with the shifted eigenvalues changing from real to complex, and then a KKK-symmetry restoring transition, characterized by the emergence of a pure imaginary spectrum with equal spacing. Accordingly, the eigenstates change from PT\mathcal{PT}PT-unbroken extended states to PT\mathcal{PT}PT-broken states, and finally to Stark-localized states. In the framework of the quantum open system governed by the Lindblad equation with linearly increasing site-dependent dissipation, we unveil that the dynamical evolution of a single-particle correlation function is governed by the Hamiltonian of the imaginary Stark ladder model. By studying the dynamical evolution of the density distribution under various initial states, we demonstrate that the damping dynamics displays distinct behaviors in different regions. A localized damping is observed in the strong dissipation limit.
We investigate the dynamical properties of one-dimensional dissipative Fermi-Hubbard models, which are described by the Lindblad master equations with site-dependent jump operators. The corresponding non-Hermitian effective Hamiltonians with pure loss terms possess parity-time (PT\mathcal{PT}PT) symmetry if we compensate the system additionally with an overall gain term. By solving exactly the two-site Lindblad equation with fixed dissipation, we find that the dynamics of the rescaled density matrix shows an instability as the interaction increases over a threshold, which can be equivalently described in the scheme of non-Hermitian effective Hamiltonians. This instability is also observed in multisite systems and closely related to the PT\mathcal{PT}PT-symmetry breaking accompanied by appearance of complex eigenvalues of the effective Hamiltonian. Moreover, we unveil that the dynamical instability of the antiferromagnetic Mott phase comes from the PT\mathcal{PT}PT-symmetry breaking in highly excited bands, although the low-energy effective model of the non-Hermitian Hubbard model in the strongly interacting regime is always Hermitian. We also provide a quantitative estimation of the time for the observation of dynamical PT\mathcal{PT}PT-symmetry breaking which could be probed in experiments.
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 investigate a quantum dynamical phase transition induced by the competition between local unitary evolution and dissipation in a qubit chain with a strong, on-site Z2\mathbb{Z}_2Z2​ symmetry. While the steady-state of this evolution is always maximally-mixed, we show that the dynamical behavior of certain non-local observables on the approach to this steady-state is dictated by a quantum Ising model with a complex transverse-field (cTFIM). A crucial feature of our mapping to the cTFIM is an extensive number of local, weak, Z2\mathbb{Z}_2Z2​ symmetries possessed by the dynamics, allowing us to investigate these observables analytically and uncover a dynamical phase transition as the relative rate of unitary evolution and dissipation is tuned. We show that the weak-dissipation limit corresponds to a cTFIM with a large magnitude of the imaginary transverse-field, for which the many-body "ground-state" (with smallest real eigenvalue) is gapless, exhibiting quasi-long-range correlations of the local magnetization with a continuously-varying exponent. Correspondingly, the dynamics of the non-local observables show oscillatory behavior with an amplitude decaying exponentially in time. The strong-dissipation limit corresponds to a gapped ferromagnetic phase of the cTFIM, and non-local observables show exponential decay on the approach to equilibrium. This transition in (1+1)-dimensions has a peculiar, "two-sided" nature appearing as either first- or second-order depending on the phase from which the transition is approached, an analytic result which is corroborated by numerical studies. In higher dimensions, we present a field-theoretic understanding of the first-order nature of this transition, when approaching from the ferromagnetic phase of the cTFIM, though the nature of the phase with large imaginary transverse-field remains to be understood.
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.
The authors construct here an exactly solvable \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}}-symmetric non-Hermitian Heisenberg spin chain with complex-conjugate boundary Kondo couplings. They show that the model exhibits three distinct boundary phases: dissipative Kondo screening with unbroken \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}} symmetry, bound-mode phases featuring spontaneously broken \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}} symmetry and complex-conjugate eigenvalues, and an unscreened local-moment phase with a purely real spectrum. Benchmarking with non-Hermitian DMRG demonstrates quantitative agreement with Bethe ansatz, validating tensor-network methods for open quantum systems.
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