Boundary-driven magnetization transport in the spin-$1/2$ XXZ chain: Role of the system-bath coupling strength and timescales

TL;DR

This work probes the relationship between transport coefficients obtained from closed-system linear response and open-system boundary-driven dynamics in the spin-$1/2$ XXZ chain. Using stochastic unraveling and TEBD-based MPS methods, the authors show that the dc diffusion constant $\mathcal{D}_{dc}$ from open-system dynamics strongly depends on the system–bath coupling $\gamma$ and does not converge to the Kubo value in the thermodynamic limit, indicating a limitation of boundary-driven approaches for extracting dc transport. By analyzing the time-dependent diffusion constant $\mathcal{D}(t)$, they identify an initial plateau whose width scales as $t_{fs}J \propto L$, consistent with finite-size effects, while the approach to the nonequilibrium steady state scales as $t^*J \propto L^{2}$, highlighting a noncommuting order of limits $\lim_{L\to\infty}\lim_{t\to\infty}$ versus $\lim_{t\to\infty}\lim_{L\to\infty}$. The results hold for both integrable and nonintegrable XXZ variants, underscoring fundamental limitations in using open-system dc coefficients for transport predictions, though boundary-driven methods remain powerful for time-dependent diffusion and finite-size scaling analyses with potential for broader applications.

Abstract

Understanding the transport properties of quantum many-body systems is a central challenge in condensed matter and statistical physics. Theoretical studies usually rely on two main approaches: Dynamics of linear-response functions in closed systems and boundary-driven dynamics governed by Markovian master equations for open systems. While the equivalence of their dynamical behavior has been explored in recent studies, a systematic comparison of the transport coefficients obtained from these two classes of approaches remains a largely open question. Here, we address this gap by comparing and contrasting the dc diffusion constant $\mathcal{D}_{\text{dc}}$ according to the two approaches, focusing on the specific example of magnetization transport in the spin-$1/2$ XXZ chain. Using exact numerical simulations for finite system sizes, we find (i) a clear mismatch between the two $\mathcal{D}_{\text{dc}}$ and (ii) a strong dependence of $\mathcal{D}_{\text{dc}}$ on the system-bath coupling strength for the open system, where neither (i) nor (ii) tend to vanish in the thermodynamic limit. These findings suggest limitations of the open-system approach to transport coefficients. To gain insight into the origin of (i) and (ii), we go beyond $\mathcal{D}_{\text{dc}}$ and analyze the full time dependence of the diffusion coefficient $D(t)$ in the open system. In this way, we find that both (i) and (ii) vanish up to a finite time scale. While this time scale gradually increases with system size and tends to be macroscopic in the thermodynamic limit, this increase is still slow compared to the increase of the time to reach the steady state, where (i) and (ii) do not vanish. This observation can be seen as a wrong, yet unavoidable order of limits of long times first and large system sizes afterwards.

Figures (9)

• Figure 1: Sketch summarizing our key findings for the time-dependent diffusion constant $\mathcal{D}(t)$ from the open-system approach (purple). The closed-system Kubo-based approach Karrasch2014kraft2024Wang2024 is indicated (orange). An early-time plateau shows a (nearly) $\gamma$-independent agreement up to a system-size–dependent scale $t_{\mathrm{fs}}J \propto L$. At long times, the NESS-based dc values disagree with Kubo and strongly depend on the system–bath coupling $\gamma$.
• Figure 2: Sketch of the open quantum system scenario, where a spin-$1/2$ XXZ chain with open boundaries is coupled at its ends to two Lindblad baths. Thus, transport is induced and at sufficient long times the nonequilibrium steady state is reached featuring a constant current $j_{\text{ness}}$ and a characteristic density profile. Since we consider an integrable and nonintegrable scenario, corresponding nearest- and next-to-nearest neighbor interactions are indicated.
• Figure 3: Time evolution in the open system for the integrable spin-1/2 XXZ chain with anisotropy $\Delta = 1.5$ and $L = 20$ sites, as resulting for strong system-bath coupling $\gamma/J = 1.0$ and weak driving $\mu = 0.1$. Numerical data are obtained from exact stochastic unraveling (SU) with $N_{\text{traj}} = 50000$ trajectories and time-evolving block decimation (TEBD) with bond dimension $\chi = 150$. (a) Local densities, (b) local currents, and (c) steady-state profile at time $t J = 200$. In (b) the average over the time window $tJ=100\dots 200$ for the shown bulk sites of the SU data is indicated by the yellow line. Panel (c) depicts the linear function $f(r)$ from fitting over bulk sites $r=5..15$ of SU data at times $tJ=200$.
• Figure 4: Similar data as the one in Fig. \ref{['Fig2']}, but now for a small system-bath coupling strength $\gamma/J = 0.1$.
• Figure 5: Open-system diffusion constant $\mathcal{D}_{\text{dc}}$ versus system-bath coupling strength $\gamma$ for the integrable spin-1/2 XXZ chain with anisotropy (a) $\Delta = 1.5$ and (b) $\Delta = 3.0$, as resulting for weak driving $\mu = 0.1$. Analogously, in (c) diffusion constants for the nonintegrable system with $\Delta'=0.5$ and $\Delta =1.5$ are depicted. Numerical data from exact stochastic unraveling (SU) are shown for two system sizes $L = 10, 20$. Numerical data from TEBD calculations up to system sizes $L=50$ are also indicated. For comparison, numerical data for the closed-system diffusion constant according to the Kubo formula are depicted in (a) Karrasch2014, (b) kraft2024, and (c) Wang2024. Further, open-system diffusion constants $D_r(t)$ extracted at (representative) finite times $tJ=20$ from Figs. \ref{['Fig5']}, \ref{['Fig6']} are indicated in purple.