Dynamical Scarring from Scrambling in Two Dimensional Topological Materials

TL;DR

This work uses out-of-time-ordered correlators (OTOCs) to investigate information scrambling in two-dimensional topological materials, contrasting bulk dynamics with edge phenomena. By analyzing two non-interacting models—the Kitaev lattice with chiral edge modes and the Kane–Mele model with helical edge modes—the authors show that the bulk scrambling exhibits direction-dependent butterfly velocities $v_b$, while edge modes induce dynamical scars that propagate along the boundary with velocity $v_F$ and do not scramble. The scars are robust to edge disorder and even pass through collisions or corners, behaving as carriers of initial-state information along topological channels. These results provide a dynamical probe of edge topology and deepen the understanding of information spreading in topological phases, with potential extensions to disorder, higher Chern numbers, and multi-edge systems.

Abstract

Out-of-time ordered correlators are a probe of how the information of an initial perturbation is effectively scrambled under unitary time evolution, widely used to study quantum chaos. They have also been used to demonstrate that information is trapped in the zero dimensional edge modes of topological insulators and superconductors, and does not become scrambled. Here we study scrambling in two dimensional topological models. In the bulk the butterfly velocity, the speed at which the out-of-time ordered correlator spreads, gains a directional dependence from the underlying lattice. Furthermore when there are chiral or helical edge modes present these cause a form of dynamical scarring. The information about an initial perturbation on the boundary of the system travels around the edge, carried by the edge modes, but is not scrambled over very long time scales. The direction and speed of the scars are given by the velocities of the linearly dispersing edge modes. We further show that these scars do not interact, passing through each other. We back up these results with analytical and numerical calculations on exemplary models.

Figures (14)

• Figure 1: The OTOC $C_{j,j_0}(t)$ at different time steps following a perturbation in the center, at site $\vec{r}_0=(11,11)a$, in the topologically trivial phase with $\nu=0$, see main text for details. Each point is a lattice site. Note that the normalization of the color scheme is for each time step, $C_{\rm max}(t)=\textrm{max}[C_{j,j_0}(t)]$. Here we check the bulk scrambling which spreads throughout the system with a characteristic velocity,before scattering from the edges.
• Figure 2: The OTOC $C_{j,j_0}(t)$ at different time steps following a perturbation in the center, at site $\vec{r}_0=(11,11)a$, in the topologically non-trivial phase with $\nu=1$, see main text and fig. \ref{['fig:otocm0']} for more details. After a time $t\approx 6/J$ the correlations have hit the edge of the system and scatter back, resulting in a quickly scrambled system.
• Figure 3: The OTOC $C_{j,j_0}(t)$ as a function of $t$ for different sites $\vec{r}$ following a perturbation in the center at site $\vec{r}_0=(11,11)a$. This is for the topologically non-trivial phase with $\nu=1$. In the panel (a) one can see that as we move further from the perturbation along the $x$-direction the onset of the non-zero OTOC moves later in time, as expected. In panel (b) we compare several different points, two are an equal distance from the perturbation along different directions, one is further away along the diagonal.
• Figure 4: The OTOC $C_{j,j_0}(t)$ at different time steps following a perturbation on the edge, at site $\vec{r}_0=(1,11)a$, in the topologically non-trivial phase with $\nu=1$, see main text and fig. \ref{['fig:otocm0']} for more details. In the $\nu=1$ phase the chiral edge modes propagate clockwise, as does the scar in the scrambling visible as a dark purple region on the boundary.
• Figure 5: The OTOC $C_{j,j_0}(t)$ at different time steps following a perturbation on the edge, at site $\vec{r}_0=(1,11)a$, in the topologically non-trivial phase with $\nu=-1$, see main text and fig. \ref{['fig:otocm0']} for more details. In the $\nu=-1$ phase the chiral edge modes propagate counter-clockwise, as does the scar in the scrambling visible as a dark purple region on the boundary.