Anomalous dynamics and equilibration in the classical Heisenberg chain

The search for departures from standard hydrodynamics in many-body systems has yielded a number of promising leads, especially in low dimensions. Here, we study one of the simplest classical interacting lattice models, the nearest-neighbor Heisenberg chain, with temperature as the tuning parameter. Our numerics expose strikingly different spin dynamics between the antiferromagnet, where it is largely diffusive, and the ferromagnet, where we observe strong evidence either of spin superdiffusion or an extremely slow crossover to diffusion. This difference also governs the equilibration after a quench, and, remarkably, is apparent even at very high temperatures.

Figure 1

Anomalous hydrodynamics in the FM: Power-law scaling of the autocorrelator ${\mathcal{A}}^S\left(t\right)$, and extracted exponents and crossover scales for the FM (blue, $+$) and AFM (orange, $\times$). (a) and (c) show ${\mathcal{A}}^S\left(t\right)$ for $\mathcal{E}=-0.3$ and $\mathcal{E}=-0.7$, respectively. The dotted lines show the power-law fit (7), and the dashed lines show the finite-time corrected fit (9). (b) shows the estimated anomalous exponents, while (d) shows the diffusion crossover times estimated from (9)—the inset zooms in on the points $\mathcal{E}=-0.4$ to $\mathcal{E}=0$.

Figure 2

Scaling collapses of the spin correlations ${\mathcal{C}}^S(x,t)$. Colors correspond to different fixed times from $t=2000$ up to $t={10}^5$ [$t=8\times {10}^4$ in (d)]. (a) and (b) show the diffusive collapse in the AFM at $\mathcal{E}=-0.3$ and $\mathcal{E}=-0.7$, respectively; (c) and (d) show anomalous collapses at $\mathcal{E}=-0.3$ and $\mathcal{E}=-0.7$ in the FM, with exponents $\alpha =0.532$ and $\alpha =0.648$, respectively. Dotted lines show a Gaussian scaling function as a guide to the eye.

Figure 3

Anomalous scaling in the FM. (a) shows the the autocorrelator ${\mathcal{A}}^S\left(t\right)$, vertically offset for clarity, for, in descending order, $\mathcal{E}=-0.9$ to $\mathcal{E}=0$ in steps of 0.1. The power-law fits have the exponents of Fig. 1, except the black lines at $\mathcal{E}=-0.8$ and $\mathcal{E}=-0.9$, which are fit with the KPZ exponent ${\alpha }_{\mathrm{KPZ}}=2/3$. (b) and (c) show the scaling collapse using the extracted exponent comparing to the KPZ scaling function (dashed) and a Gaussian (dotted) at $\mathcal{E}=-0.8$ and $\mathcal{E}=-0.9$, respectively.

Figure 4

Equilibration dynamics of ${\mathcal{O}}^z\left(t\right)-{\mathcal{O}}_{\mathrm{eq}}^z$ (blue) and ${\mathcal{O}}^{\left|\right|}\left(t\right)-{\mathcal{O}}_{\mathrm{eq}}^{\left|\right|}$ (orange), where ${\mathcal{O}}^{\left|\right|}=({\mathcal{O}}^x+{\mathcal{O}}^y)/2$ is the average of the in-plane components. (a)–(c) show the equilibration of $Q$, $E$, and $C$ at $\mathcal{E}=-0.5$ in the FM; (d) shows the equilibration of $E$ at $\mathcal{E}=-0.5$ in the AFM. $Q$ and $E$ appear to equilibrate with an anomalous exponent of $\alpha \approx 0.6$ in the FM, though the data are equally well described by a combination of power laws. The energy fluctuations (heat capacity) and the AFM equilibrate diffusively.