Subdiffusive spin transport in disordered classical Heisenberg chains

We study the transport and equilibration properties of a classical Heisenberg chain, whose couplings are random variables drawn from a one-parameter family of power-law distributions. The absence of a scale in the couplings makes the system deviate substantially from the usual paradigm of diffusive spin hydrodynamics and exhibit a regime of subdiffusive transport with an exponent changing continuously with the parameter of the distribution. We propose a solvable phenomenological model that correctly yields the subdiffusive exponent, thereby linking local fluctuations in the coupling strengths to the long-time, large-distance behavior. It also yields the finite-time corrections to the asymptotic scaling, which can be important in fitting the numerical data. We show how such exponents undergo transitions as the distribution of the coupling gets wider, marking the passage from diffusion to a regime of slow diffusion, and finally to subdiffusion.

Figure 1

Sketch of the spin chain from Eq. (1). The spins ${\mathit{S}}_i$ live on the sphere $S^2$, and interact via broadly distributed nearest-neighbor couplings $J_i$, cf. Eqs. (2). We use periodic boundary conditions.

Figure 2

(a) Probability density function of the couplings $J_i$ for representative values of $\eta$. (b) Cumulative distribution function of the inverse couplings $R=J^{-1}$, showing the fat tails of the distribution for $\eta >0$. (c) Overview of the dynamical regimes as a function of $\eta$. The point at $\eta =0$ corresponds to logarithmically suppressed diffusion.

Figure 3

Spin dynamics in the slow diffusion regime, shown for $\eta =-0.5$. In each panel, the main figure shows the rescaled autocorrelator $\mathcal{A}\left(t\right)=\mathcal{C}(x=0,t)$ [cf. Eq. (6)]. The insets show the corresponding scaling collapse of the full correlation function $\mathcal{C}(x,t)$. (a) Autocorrelator and scaling collapse from Eqs. (9), i.e., assuming an asymptotic diffusive behavior with strong, anomalous corrections. (b) Diffusive scaling without finite-time corrections (i.e., setting $\lambda =0$, a one-parameter fit), showing that the corrections must be accounted for, at least up to the final time of the simulation, $t={10}^6$. (c) Numerical fit with an anomalous (subdiffusive) exponent, i.e., a direct two-parameter fit to Eq. (8). See main text for additional details.

Figure 4

Spin dynamics at the slow-diffusion/subdiffusion transition, $\eta =0$. (a) Rescaled autocorrelator (main panel) and scaling collapse of the correlation function (inset), as predicted by the logarithmic suppression of diffusion, i.e., a two-parameter fit to Eq. (11). (b) Same data as (a), but fitting with a subdiffusive exponent, i.e., a direct two-parameter fit to Eq. (8). Note that the scaling collapse in the tails of the correlations is slightly better using the logarithmic suppression, indicating that this is the correct picture.

Figure 5

Subdiffusive spin dynamics in the strongly disordered regime, shown for a representative $\eta =0.5$. Again, each main panel shows the autocorrelator, while the insets show the scaling collapse of the correlation function. (a) The autocorrelator and scaling collapse from Eq. (12), i.e., a two-parameter fit with the subdiffusive exponent $\alpha =(1-\eta )/(2-\eta )=1/3$, and with leading corrections included. (b) Subdiffusion with the same exponent as (a) but without the corrections, i.e., a one-parameter fit setting $\lambda =0$. (c) Scaling with an exponent obtained numerically without finite-time corrections, i.e., a direct two-parameter fit to Eq. (8). Again, note that the scaling collapse in the tails is better in (a) than in (c), indicating that the finite-time corrections provide the correct picture.

Figure 6

Bromwich contour for the inversion of the Mellin transform, Eq. (A11). The poles at integer values (black crosses) are responsible for a regular scaling of time and space, while the pole at $s=\eta -1$ (red dot) is responsible for subdiffusion when $\eta >0$, and for the anomalous corrections to diffusion when $-1<\eta <0$.

Figure 7

Slow diffusion in the effective model at $\eta =-0.5$, obtained by directly simulating Eq. (14). The leading and subleading terms predicted by the transfer matrix solution, and used to fit the spin correlations in the main text for the same value of $\eta$ (Fig. 3), provide an excellent fit to the data.

Figure 8

Subdiffusion in the effective model at $\eta =0.5$, obtained by directly simulating Eq. (14). Similarly to the case of slow-diffusion (Fig. 7), the transfer-matrix solution, which describes the spin correlations at the same $\eta$ (cf. Fig. 5), is in agreement with the effective model's numerics.

Figure 9

Logarithmically suppressed diffusion at $\eta =0$ in the effective model Eq. (14).