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Relaxation to a Parity-Time Symmetric Generalized Gibbs Ensemble after a Quantum Quench in a Driven-Dissipative Kitaev Chain

Elias Starchl and Lukas M. Sieberer
Phys. Rev. Lett. 129, 220602 – Published 23 November 2022
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  • Introduction.—
  • Model.—
  • PT-symmetric GGE.—
  • Relaxation of subsystem parity.—
  • Directional parity pumping.—
  • Evolution of subsystem entropy.—
  • Discussion.—
  • ACKNOWLEDGMENTS
    • Supplemental Material
    References

    Abstract

    The construction of the generalized Gibbs ensemble, to which isolated integrable quantum many-body systems relax after a quantum quench, is based upon the principle of maximum entropy. In contrast, there are no universal and model-independent laws that govern the relaxation dynamics and stationary states of open quantum systems, which are subjected to Markovian drive and dissipation. Yet, as we show, relaxation of driven-dissipative systems after a quantum quench can, in fact, be determined by a maximum entropy ensemble, if the Liouvillian that generates the dynamics of the system has parity-time symmetry. Focusing on the specific example of a driven-dissipative Kitaev chain, we show that, similar to isolated integrable systems, the approach to a parity-time symmetric generalized Gibbs ensemble becomes manifest in the relaxation of local observables and the dynamics of subsystem entropies. In contrast, the directional pumping of fermion parity, which is induced by nontrivial non-Hermitian topology of the Kitaev chain, represents a phenomenon that is unique to relaxation dynamics in driven-dissipative systems. Upon increasing the strength of dissipation, parity-time symmetry is broken at a finite critical value, which thus constitutes a sharp dynamical transition that delimits the applicability of the principle of maximum entropy. We show that these results, which we obtain for the specific example of the Kitaev chain, apply to broad classes of noninteracting fermionic models, and we discuss their generalization to a noninteracting bosonic model and an interacting spin chain.

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    • Received 28 March 2022
    • Accepted 31 October 2022

    DOI:https://doi.org/10.1103/PhysRevLett.129.220602

    © 2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Open quantum systems & decoherenceQuantum quench
    1. Physical Systems
    1-dimensional systems
    1. Properties
    PT-symmetry
    1. Techniques
    Quantum master equation
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Elias Starchl and Lukas M. Sieberer*

    • Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria

    • *lukas.sieberer@uibk.ac.at

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    Issue

    Vol. 129, Iss. 22 — 23 November 2022

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    Images

    • Figure 1
      Figure 1

      Subsystem parity after quenches to the trivial (green, μ=−4J) and topological (blue, μ=−J) PT-symmetric phases for γ=0.3J, δ=0, and ℓ=20. The solid lines are obtained from Eqs. (9) and (10), where we set α+=0.08 and α−=0.11 to achieve best agreement with the numerical data shown as dashed lines. Straight vertical and horizontal lines indicate t=tF and the PTGGE predictions for the stationary values, respectively. In all figures, L is chosen large enough to avoid finite-size effects.

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    • Figure 2
      Figure 2

      Deviation from the PTGGE due to δ≠0 for μ=−2.5J, γ=0.1J, δ=10−7J, and ℓ=20. The rescaled subsystem parity (dashed line) follows Eq. (9) (solid line) up to the crossover timescale t×≈5.7tF defined as |⟨Pℓ(t×)⟩−⟨Pℓ(t×)⟩PTGGE|=⟨Pℓ(t×)⟩PTGGE. Inset: t× diverges logarithmically for δ→0. The numerical data are in good agreement with an analytical estimate [53].

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    • Figure 3
      Figure 3

      Directional pumping of subsystem parity for a quench to the topological PT-symmetric phase with μ=−0.5J, γ=0.3J, δ=0, and ℓ=30. For periodic boundary conditions (PBC), the subsystem parity [black line, numerics; blue shading, sign of numerical data; red line, Eq. (10) with α+=α−=0.09] crosses zero at multiples of both ts,+ and ts,−. In contrast, for open boundary conditions (OBC), zero crossings occur only at multiples of ts,− and ts,+ for subsystems, respectively, Lℓ (violet line) and Rℓ (blue line). Factors e±2γt compensate for additional exponential decay (left end) and growth (right end) due to edge modes [53].

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    • Figure 4
      Figure 4

      Quasiparticle-pair contribution to the subsystem entropy after quenches to the trivial (green, μ=−4J) and topological (blue, μ=−J) PT-symmetric phases for γ=0.3J, δ=0, and ℓ=20. The numerical data (dashed lines) are close to Eq. (11) (solid lines). Inset: For the trivial quench at t=2tF, the difference between the numerical data and Eq. (11) (blue dots) vanishes as 1/ℓ (orange line).

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