Spin diffusion from an inhomogeneous quench in an integrable system

Abstract

Generalized hydrodynamics predicts universal ballistic transport in integrable lattice systems when prepared in generic inhomogeneous initial states. However, the ballistic contribution to transport can vanish in systems with additional discrete symmetries. Here we perform large scale numerical simulations of spin dynamics in the anisotropic Heisenberg XXZ spin 1/2 chain starting from an inhomogeneous mixed initial state which is symmetric with respect to a combination of spin reversal and spatial reflection. In the isotropic and easy-axis regimes we find non-ballistic spin transport which we analyse in detail in terms of scaling exponents of the transported magnetization and scaling profiles of the spin density. While in the easy-axis regime we find accurate evidence of normal diffusion, the spin transport in the isotropic case is clearly super-diffusive, with the scaling exponent very close to 2/3, but with universal scaling dynamics which obeys the diffusion equation in nonlinearly scaled time.

Figure 1. Dynamics of spin and current densities.

Time evolution of spin density (a,b) and current (c,d) profile j(x, t)=tr(ρ(t)j_x) for the isotropic point Δ=1 (a,c), and Δ=2 (b,d), following an inhomogeneous quench. One can see that the spreading is much faster for Δ=1, in both cases though it is slower than ballistic. Dashed green curves guide the eye towards scaling x∼t^2/3 in a, and x∼t^1/2 in (b). Data are shown for n=320 and small initial polarization μ=π/1,800.

Figure 2. Scaling exponents of magnetization spreading.

(a,b) Local exponent α(t) calculated as a numerical log-derivative d log Δs(t)/d log t for Δ=1 (a) and Δ=2 (b) (dashed lines indicate exponents 2/3 and 1/2, respectively, while dashed lines in the insets show best power-law fits to Δs(t)—red curve), both for μ=π/1,800. (c) Conjecture for the dependence α(Δ) at high temperatures and small μ. The inset shows the diffusion constant obtained from Fick’s law for various values of Δ in the diffusive regime, converging to a finite value at large Δ (agreeing with ref. 28). (d) Dependence on μ for Δ=1 shows a small but significant change in the behaviour: for μ≈1 it is closer to α=3/5 while for small μ it becomes close to α=2/3 (dashed). The blue (circles) and red (crosses) symbols represent wave function and density operator evolutions respectively. We average over samples of 10–130 random initial wave-functions for each blue data point. For intermediate μ the error-bars (denoting the estimated s.d.) are larger since the simulation is less efficient in that regime (Methods section).

Figure 3. Scaling profiles.

Scaling of density and current profiles with x/t^α. In (a,b) we show the scaling of magnetization profiles, (a) for Δ=1 using α=2/3, and (b) for Δ=2 and using α=1/2 (note that the points for different times overlap almost perfectly; the insets show the convergence of the relative root-mean-square difference (in %) between data s(x, t) and scaled erf-profiles (see text) as a function of time). Frames (c,d) show the emergence of Fick’s law at late times (shown at t=160), comparing current profiles (red) to gradients of spin density (blue)—both indistinguishable from Gaussians, for Δ=1 in (c) and Δ=2 in (d). In all plots the system size is n=320.

Figure 4. Simulation complexity.

(a) Von Neumann entanglement entropy S for the fully polarized initial state (μ=1) at the isotropic point Δ=1. (b) Operator space entanglement entropy S^# for Δ=1 (blue) and Δ=2 (green), both for μ=π/1,800. Bipartition into two equal halves and a system size of n=320 are used.