Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System

We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. By employing a density matrix renormalization group technique for the study of the stationary states of the boundary-driven Lindblad equation we are able to study extremely large systems (400 spins). We find both a diffusive and a subdiffusive phase depending on the strength of the disorder and on the anisotropy parameter of the Heisenberg chain. Studying finite-size effects, we show numerically and theoretically that a very large crossover length exists that controls the passage of a clean-system dominated dynamics to one observed in the thermodynamic limit. Such a large length scale, being larger than the sizes studied before, explains previous conflicting results. We also predict spatial profiles of magnetization in steady states of generic nondiffusive systems.

Figure 1

Phase diagram of a disordered anisotropic Heisenberg model in the high temperature ergodic phase. At a critical disorder strength $h_{c2}$ there is a transition from diffusive to subdiffusive spin transport. Black circles with error bars denote $h_{c2}$ determined from the steady-state current scaling $j\sim 1/L^{\gamma }$ in large systems ($L\approx 400$ sites, see Fig. 2). The underlying colors are for illustrative purposes and denote $\gamma$ obtained from small systems $L\le 7$, which, nevertheless, correctly depicts the two regimes, except close to $h=0$ and $\mathrm{\Delta }=0$.

Figure 2

Spin transport in the ergodic phase of the Heisenberg model. (a) Scaling of the average NESS spin current $j$ with system size, $j\sim 1/L^{\gamma }$, for $\mathrm{\Delta }=1$. Points are numerical data, lines are best fitting $1/L^{\gamma }$, with $\gamma =1$ for $0<h<h_{c2}\approx 0.55$ (black), and $\gamma >1$ for $h_{c2}<h<h_{c3}$ (red). For $h=1$, $h=0.5$ and $h=0.25$ the gray shading denotes standard deviation $\sigma (j)$ of current distribution (for $h=0.25$ it is barely visible as it is smaller than the size of square points). (b) Dependence of $\gamma$ on the disorder strength. At a critical disorder strength $h_{c2}$ one gets a transition from diffusive to subdiffusive spin transport, while at $h_{c1}=0$ there is a discontinuous transition from superdiffusive (for $h=0$) to diffusive for $h_{c1}<h<h_{c2}$. Inset: similar data for $\mathrm{\Delta }=1.2$ and $\mathrm{\Delta }=0.5$.

Figure 3

Scaling collapse of NESS current with system size for small $h$ and $\mathrm{\Delta }=1$ [additional data to that in Fig. 2 are plotted]. In the main plot we see that for $L>L_{*}\sim 1/h^{1.33}$ the scaling is diffusive, $j\sim 1/L$, while at shorter lengths it is asymptotically superdiffusive, $j\sim 1/L^{0.5}$, the same as in the clean model. Data for $L\le 300$ are shown ($L\le 400$ for $h=0.1$). The inset shows the scaling of the diffusion constant $D$ with $h$, with the straight (blue) line suggesting $D\sim 1/h^{0.66}$ divergence for small $h$.