Energy transport in disordered classical spin chains

We present a numerical study of the diffusion of energy at high temperature in strongly disordered chains of interacting classical spins evolving deterministically. We find that quenched randomness strongly suppresses transport with the diffusion constant becoming reduced by several orders of magnitude upon the introduction of moderate disorder. We have also looked for but not found signs of a classical many-body localization transition at any nonzero strength of the spin-spin interactions.

Figure 1

(Color online) Disorder-averaged energy diffusion constant $D$ as a function of the spin-spin interaction $J$. The line has slope 8 on this log-log plot.

Figure 2

(Color online) Short-time behavior of $C\left(t\right)$ for $J=0.32$ (red, noticeably different trace) and $J=0.08,0.12,0.16$ (these are almost identical data in this plot). Note rescaling of the vertical axis by $J^2$.

Figure 3

(Color online) Current autocorrelations on medium time scales $\sim 1/J$ for $J=0.32,0.16,0.12,0.08$, from top (red) to bottom (green) trace at $tJ=2$. Note the rescaling of both the vertical and time axes. The inset shows near cancellation between short and medium times.

Figure 4

(Color online) Top panel: long-time tail in the current autocorrelation function for $J=0.20,0.24,0.28,0.32,0.40$ shown from bottom to top in the order of $J$ listed. Bottom panel: to estimate the exponent we multiply the data by $t^{5/4}$ (and also display lines with slope $\pm 0.05$). Although these data do not exclude an exponent that varies with $J$, we interpret these results as supportive of a single exponent $x\approx 0.25$ at asymptotically long times but with a more pronounced short-time transient at smaller $J$.

Figure 5

(Color online) Long time tails as seen from $\eta \left(t\right)$ for $J=0.20,0.24,0.28,0.32,0.36,0.40,0.48$ (bottom to top). Black line is a guide to the eye with slope $-1/4$. Note that the short-time transients are stronger here, as compared to the autocorrelation data in Fig. 4.

Figure 6

(Color online) Variation in $\kappa \left(t\right)$ for of $J=0.64,0.56,0.48,0.40,0.36,0.32,0.28,0.24,0.20,0.18,0.16,0.14,0.12$ plotted vs $s={\left(1+Jt\right)}^{-0.25}$ (larger $J$ have larger $\kappa$ as $s\to 0$). Lines are merely guides to the eye and statistical errors are too small to be seen on most of these points. This figure is used for obtaining $J\ge 0.36$ entries in Table .

Figure 7

(Color online) Same data as in Fig. 6 but now scaled and displayed in a way that allows one to see the extrapolations to $s=0$ $\left(t\to \infty \right)$ for small $J$. Because of the rescaling of the plotted in the previous figure. Note the different rescaling schemes used in preceding and current plots to focus on collapse of short-time data (previous plot) vs long-time extrapolations (present plot). As before black lines are drawn through the data for guiding the eyes. Colored lines are results of polynomial fits, as explained in the text. This figure is used for obtaining entries in Table for $J\le 0.32$.

Figure 8

(Color online) Finite-size effects for $J=0.16,0.32,0.40$ (bottom to top near $s=0$): 100 vs 20 spins for $J=0.40$, 10, and 20 vs 100 spins for $J=0.32$ and 20 vs 100 spins for $J=0.16$. Red color is used to indicate the data influenced by finite-size effects according to Eq. (15) with $C_D=10$. Inset: $J=0.16$ data replotted.

Figure 9

(Color online) Simulated roundoff effects at $J=0.14$: low frequency, long-time conductivity is largest for $P=1$, smaller for $P=0.1$, but essentially indistinguishable between $P=0$ and $P=0.01$. Recall that intrinsic roundoff noise in our double-precision runs is $P_{intr}\lesssim {10}^{-15}$. Inset: data with $P=0.1,0.01,0$.