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

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.

Figure 1

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

Figure 2

Deviation from the PTGGE due to $\delta \ne 0$ for $\mu =-2.5J$, $\gamma =0.1J$, $\delta ={10}^{-7}J$, and $\ell =20$. The rescaled subsystem parity (dashed line) follows Eq. (9) (solid line) up to the crossover timescale $t_{\times }\approx 5.7t_F$ defined as $|⟨P_{\ell }(t_{\times })⟩-⟨P_{\ell }(t_{\times }){⟩}_{\mathrm{PTGGE}}|=⟨P_{\ell }(t_{\times }){⟩}_{\mathrm{PTGGE}}$. Inset: $t_{\times }$ diverges logarithmically for $\delta \to 0$. The numerical data are in good agreement with an analytical estimate [53].

Figure 3

Directional pumping of subsystem parity for a quench to the topological PT-symmetric phase with $\mu =-0.5J$, $\gamma =0.3J$, $\delta =0$, and $\ell =30$. For periodic boundary conditions (PBC), the subsystem parity [black line, numerics; blue shading, sign of numerical data; red line, Eq. (10) with ${\alpha }_{+}={\alpha }_{-}=0.09$] crosses zero at multiples of both $t_{s,+}$ and $t_{s,-}$. In contrast, for open boundary conditions (OBC), zero crossings occur only at multiples of $t_{s,-}$ and $t_{s,+}$ for subsystems, respectively, $L_{\ell }$ (violet line) and $R_{\ell }$ (blue line). Factors $e^{\pm 2\gamma t}$ compensate for additional exponential decay (left end) and growth (right end) due to edge modes [53].

Figure 4

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