Analytic solution of the domain-wall nonequilibrium stationary state

We consider the out-of-equilibrium dynamics generated by joining two domains with arbitrary opposite magnetizations. We study the stationary state which emerges by the unitary evolution via the spin-1/2 $XXZ$ Hamiltonian, in the gapless regime, where the system develops a stationary spin current. Using the generalized hydrodynamic approach, we present a simple formula for the space-time profile of the spin current and the magnetization exact in the limit of long times. As a remarkable effect, we show that the stationary state has a strongly discontinuous dependence on the strength of interaction as confirmed by the exact analytic expression of the Drude weight that we compute. These features allow us to give a qualitative estimation for the transient behavior of the current which is in good agreement with numerical simulations. Moreover, we analyze the behavior around the edge of the magnetization profile, and we argue that, unlike the $XX$ free-fermionic point, interactions always prevent the emergence of a Tracy-Widom scaling.

Figure 1

(Main) Spin current at the junction, i.e., ${\langle {\mathit{j}}_{{\mathit{s}}^z}\rangle }_{\zeta =0}$ for a quench from the DW with $\gamma =\pi /\phi$. The time-dependent density-matrix renormalization-group (DMRG) data for the spin current between lattice sites (0,1) and (1,2) (the thin black lines) and their average (the thick black line) are compared with the stationary values associated with the rational approximation of the golden ratio (the horizontal dashed lines). The dotted vertical lines represent the typical time scale at which the current passes from one rational approximation to the next one. (The inset) The analytic stationary profile for the golden ratio is compared with different rational approximations.

Figure 2

Scaling of the particle density profile at the edge of the light cone for different values of the interactions and DW initial condition. As expected, in the free-fermion case the scaling of the data is governed by the Airy kernel [97]. However, when interactions are turned on, the behavior becomes purely diffusive preventing a Tracy-Widom-like scaling.