Light-Cone Spreading of Perturbations and the Butterfly Effect in a Classical Spin Chain

We find that the effects of a localized perturbation in a chaotic classical many-body system—the classical Heisenberg chain at infinite temperature–spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterized by the Lyapunov exponents and the butterfly speed, respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analyzing the scaling forms, we relate the growth, spread, and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang equation.

Figure 1

Simultaneous growth and ballistic spread of a perturbation in a classical Heisenberg spin chain whose spin dynamics (Fig. 3) is diffusive at $T=\infty$. The speed of spreading obtained from the classical OTOC, $D(x,t)$ (see text), defines a “light cone.” The results are shown for a perturbation at time $t=0$ of size $ϵ={10}^{-3}$ at the center of a system of size $L=2048$.

Figure 2

The inset plots $D(x,t)$ as a function of $x$, at different times ($t=40,50,\dots ,100$), showing growth and ballistic propagation of the perturbation front. The scaled data (main panel) show that the front is fit well by Eq. (6) with $\mu =0.494$ and $v_b=1.642$ near $x\sim v_{b}t$. Here $\epsilon ={10}^{-8}$ and averaging was done over $2\times {10}^4$ realizations.

Figure 3

The spatial profile of $C(x,t)$ [Eq. (5)] at different times $t$ for a system of size $L=512$ at $T=\infty$ with averaging over $10^5$ initial conditions. The left inset shows a collapse of the data after a diffusive scaling of $x/\sqrt{t}$ while the right inset shows the resultant $t^{-1/2}$ scaling of the autocorrelation.

Figure 4

The main panel shows $t_{D_0}$ (defined in the main text) as a function of $x$, for $D_0=100\epsilon =0.1$ and different initial spin configurations (gray scatter). The mean (black connected data points) over $10^4$ configurations is also shown and has a slope $\approx 1/[1.6417(2)]$. The upper inset shows the distribution of $t_{D_0}$ at space points $x=100,200,\dots ,700$, while the lower inset shows collapse of the distributions with a width scaling as $\sim x^{1/3}$. The dotted curve in the lower inset is the Gaussian fit to the fluctuations at $x=600$.

Figure 5

(a) Plot of $\ln [D(x,t)/{ϵ}^2]/(2t)$ versus $t$ at $x=0$ (black), 32,64,96,128(magenta), for $ϵ={10}^{-4}$ (thick dotted lines), $ϵ={10}^{-6}$ (dashed lines) and $ϵ={10}^{-8}$ (thin dotted lines), for $L=1024$ from solving the nonlinear Equation of motion [Eq. (2)]. The solid lines are results from the linearized dynamics and correspond to the limit $\epsilon \to 0$ and hence gives ${\lambda }_D$ (see text). The dashed orange line corresponds to $⟨\ln [\delta {\mathbf{S}}^2(x,t)/2{ϵ}^2]⟩/(2t)$, obtained from the linearized dynamics for $x=0$ and we see the slightly different saturation value corresponding to ${\lambda }_L$ (see text). (b) Inset plots the results for the linearized dynamics for $x=32$, 64, 96, 128 on scaling the time axis by $x$. The collapsed data approximately fits the solid line corresponding to the scaling form Eq. (6) with $\mu \approx 0.494$, $v_b\approx 1.64$.

Figure 6

Distribution of the “height” variable $h(x,t)=\log [\delta {\mathbf{S}}^2(x,t)/{\epsilon }^2]/2$ at $x=0$. The inset shows the distribution of $h(0,t)$ at different times while the main plot shows the collpase of data obtained after a $t^{1/3}$ scaling.