Spin-$1/2$ XXZ chain coupled to two Lindblad baths: Constructing nonequilibrium steady states from equilibrium correlation functions

State-of-the-art approaches to extract transport coefficients of many-body quantum systems broadly fall into two categories: (i) they target the linear-response regime in terms of equilibrium correlation functions of the closed system; or (ii) they consider an open-system situation typically modeled by a Lindblad equation, where a nonequilibrium steady state emerges from driving the system at its boundaries. While quantitative agreement between (i) and (ii) has been found for selected model and parameter choices, also disagreement has been pointed out in the literature. Studying magnetization transport in the spin-$1/2$ XXZ chain, we here demonstrate that at weak driving, the nonequilibrium steady state in an open system, including its buildup in time, can remarkably be constructed just on the basis of correlation functions in the closed system. We numerically illustrate this direct correspondence of closed-system and open-system dynamics, and show that it allows the treatment of comparatively large open systems, usually only accessible to matrix product state simulations. We also point out potential pitfalls when extracting transport coefficients from nonequilibrium steady states in finite systems.

Figure 1

(a) Infinite-temperature correlation function ${\langle S_r^z\left(t\right)S_{N/2}^z\left(0\right)\rangle }_{\text{eq}}$ for $N=36$ and $\mathrm{\Delta }=1.5$. The dashed curves indicate Gaussians, see also Ref. [59]. (b) Diffusive growth of root-mean-squared displacement $\mathrm{\Sigma }\left(t\right)\propto \sqrt{t}$. A diffusion constant $D/J\approx 0.6$ can be extracted from the approximately constant plateau of $2D\left(t\right)=\frac{d}{dt}{\mathrm{\Sigma }}^2\left(t\right)$ at $tJ\lesssim 10$. For longer times, finite-size effects become relevant.

Figure 2

Open-system dynamics for the spin-$1/2$ XXZ chain coupled to two Lindblad baths, as obtained for anisotropy $\mathrm{\Delta }=1.5$, $N=20$ sites (with periodic boundary conditions), small coupling $\gamma /J=0.1$, and weak driving $\mu =0.1$. Numerical results from the full stochastic unraveling (data) are compared to the prediction based on closed-system correlation functions [cf. Eq. (20)]. (a) Time evolution of the local magnetization $\langle S_r^z\left(t\right)\rangle$ for different sites $r$. (b) Site dependence of the steady state at $tJ=50$.

Figure 3

[(a) and (b)] Analogous data as in Fig. 2, but now for $N=36$ sites, for which stochastic unraveling is unfeasible. (c) Magnetization injected by the first bath as a function of time, see also [65]. A diffusion constant $D/J\approx 0.99$ [76] can be extracted from the slopes in (b) and (c).

Figure 4

Diffusion coefficient $D\left(t\right)={\partial }_t\langle S_r^z\left(t\right)\rangle /{\nabla }^2\langle S_r^z\left(t\right)\rangle$ based on our prediction for the dynamics of the weakly driven open system in Fig. 3. Data is obtained at lattice site $r=12$ and the approximately constant plateau for $tJ\lesssim 25$ is consistent with the transport behavior of the isolated XXZ chain in Fig. 1. Similar data for predictions in smaller systems with $N=20$ and $N=28$ are shown for comparison.