Open quantum spin chains with non-reciprocity: a theoretical approach based on the time-dependent generalized Gibbs ensemble

TL;DR

This work develops a time-dependent generalized Gibbs ensemble (t-GGE) framework to describe the dissipative, non-reciprocal dynamics of an open XX spin chain. By focusing on the rapidity distribution, the authors derive a closed set of evolution equations that capture magnetization and current dynamics beyond non-interacting fermion analyses, and they validate the approach against tensor-network and quantum-trajectory simulations. The study reveals how non-reciprocity shapes the rapidity distribution, induces a magnetization current, and yields algebraic-like late-time decays whose exponents depend on initial states and dissipation strength, while also highlighting possible logarithmic corrections at long times. These results demonstrate that t-GGE can provide quantitative, scalable insight into the behavior of weakly dissipative, integrable quantum systems under reservoir engineering, with potential extensions to interacting integrable models.

Abstract

We study an open quantum spin chain with non-reciprocal dissipation using a theoretical approach known as time-dependent generalized Gibbs ensemble. In the regime of weak dissipation the system is fully characterized by its rapidity distribution and we derive a closed set of coupled differential equations governing their time evolution. We check the accuracy of this theory by benchmarking the results against numerical simulations. Using this framework we are able to compute both the magnetization density and current dynamics, identifying some relations between the two. The problem of the anomalous power-law exponents identified in a previous work is discussed. Our work constitutes a theoretical approach that is able to describe the physics of non-reciprocal open quantum spin chains beyond analyses based on non-interacting fermions.

Figures (6)

• Figure 1: Time evolution of the local magnetization $n_j$ defined in Eq. \ref{['Eq:n:Spin']}, for the two initial states $\ket{\Uparrow}$ and $\ket{+}$, in both the reciprocal ($\phi = 0$) and maximally non-reciprocal ($\phi = -\pi/2$) cases. The wide lines are the results found in Ref. Begg2024 via tensor-network (TN) simulations for open boundary conditions and $J/\kappa = 1$ (courtesy of the authors S.E. Begg and R. Hanai for sharing the data). For the purple curve on panel (a), the total number of lattice sites is $L=500$ and the considered site $j=450$; for both the green curve of panel (a) and the blue curve of panel (b), $L=250$ and $j=L/2$. The thin lines are obtained via the t-GGE equation \ref{['eq:GGEeqNum']}.
• Figure 2: Time evolution of the local fermion density $n$ for the model \ref{['eq:ff']} for several non-reciprocity angles $\phi=0,-\pi/4,-\pi/2$. Different panels correspond to different initial states, parametrized by $\theta$ according to the expression in Eq. \ref{['Eq:Initial:State']}: panel (a), $\theta=\pi/3$; panel (b), $\theta=\pi/4$ corresponding to $\ket{\Psi_0} = \ket{+}$; panel (c), $\theta=\pi/6$; panel (d), $\theta=0$ corresponding to $\ket{\Psi_0} = \ket{\Uparrow}$. The colored continuous curves are obtained by numerically evaluating the integral \ref{['eq:n_ff']}. The dotted curves are guides to the eye for the late-time power-law scaling. In panel (d), all the curves overlap perfectly; the curves for $\phi=0,-\pi/4$ have been slightly shifted to improve visibility.
• Figure 3: Time evolution of the rapidity distribution $\varrho(k,t)$. Different panels refer to different initial states and angles $\phi$: (a) $\ket{\Uparrow}$ and $\phi=0$; (b) $\ket{\Uparrow}$ and $\phi=-\pi/2$; (c) $\ket{+}$ and $\phi=0$; (d) $\ket{+}$ and $\phi=-\pi/2$. For all panels, the continuous curves are obtained by solving numerically the Eq. \ref{['eq:GGEeqNum']}. On the other hand, the crosses correspond to a finite lattice of $L=14$ sites where computations are performed via a quantum trajectory stochastic algorithm, averaging over 5000 trajectories, with the parameters $J=1$ and $\kappa =0.02$, and periodic boundary conditions.
• Figure 4: Short time evolution of the local magnetization $n$, for the two initial states $\ket{\Uparrow}$ and $\ket{+}$, in both the reciprocal ($\phi = 0$) and maximally non-reciprocal ($\phi = -\pi/2$) cases. The colored curves are obtained via a stochastic quantum trajectory algorithm with 5000 trajectories for $L=14$ lattice sites, with periodic boundary conditions. The parameters are $J=1$, $\kappa=0.02$ (continuous blue) or $\kappa=1$ (dashed red). The dash-dotted black curves are obtained by solving Eq. \ref{['eq:GGEeqNum']} and integrating the resulting rapidity distribution numerically.
• Figure 5: First and second logarithmic derivatives of $n$, for various initial states parameterized by $\theta$ and various strength of non-reciprocity $\phi$. The continuous curves are obtained via the t-GGE approach in Eq. \ref{['eq:GGEeqNum']} describing the dynamics of the spin model. The dashed curves in panels (d-f) are obtained via the dynamical Eqs. \ref{['eq:rhok_ff']} describing the time-evolution of the free fermionic model \ref{['eq:ff']}.