Information propagation in isolated quantum systems

Entanglement growth and out-of-time-order correlators (OTOC) are used to assess the propagation of information in isolated quantum systems. In this work, using large scale exact time evolution we show that for weakly disordered nonintegrable systems information propagates behind a ballistically moving front, and the entanglement entropy growths linearly in time. For stronger disorder the motion of the information front is algebraic and subballistic and is characterized by an exponent, which depends on the strength of the disorder, similarly to the sublinear growth of the entanglement entropy. We show that the dynamical exponent associated with the information front coincides with the exponent of the growth of the entanglement entropy for both weak and strong disorder. We also demonstrate that the temporal dependence of the OTOC is characterized by a fast nonexponential growth, followed by a slow saturation after the passage of the information front. Finally, we discuss the implications of this behavioral change on the growth of the entanglement entropy.

Figure 1

OTOC [see Eq. (3)] for the $L=31$ random Heisenberg chain in the $S_z=\frac{1}{2}$ sector at weak disorder $W=0.3$ (left) and intermediate disorder $W=1.8$ (right). At weak disorder, a linear light cone is visible (contour lines at three thresholds indicated on the color bar), which changes to a power-law light cone at intermediate disorder, with considerably slower information spreading. Here, we average over a modest number of disorder realizations ($n=10$ for $W=0.3$ and $n=45$ for $W=1.8$), but symmetrize the OTOC, effectively doubling the number of realizations.

Figure 2

OTOC at fixed distances (indicated by numbers) from the initial excitation in the middle of the lattice as function of time, for two disorder strengths $W=0.3$ $[n=10]$ (bottom left) and 1.8 $[n=45]$ (bottom right). Darker colors represent longer distances, $x=3-15$. Upper panels show the logarithmic derivative of the corresponding bottom panel. System size is $L=31$.

Figure 3

OTOC at fixed times as function of site index, for two disorder strengths $W=0.3$ $[n=10]$ (bottom left) and 1.8 $[n=45]$ (bottom right). Darker colors represent later times on a linear grid, $t=0.6,\cdots ,12$, and the lines are shifted for clarity. Upper panels show the semi-logarithmic derivative of the corresponding bottom panel. System size is $L=31$.

Figure 4

(Left column) Extraction of the dynamical exponents from the shape of the OTOC “light cones” for two disorder strengths $W=0.9$ and 1.8 $[n>100$, except for $L=31$, where we used the same realizations as shown in Fig. 1], two thresholds ($3.24\times {10}^{-4}$ and 0.0186, corresponding to the two distinctive groups of colored lines on each panel) and various system sizes, $L=17,21,25,$ and 31 (larger sizes correspond to more intense color). The dashed black lines are power law fits to the contour lines. The dependence of the extracted dynamical exponent on the threshold is plotted on the right column, for same disorder strengths and system sizes. The dashed black line here marks the thresholds used for the data on the left column, the orange solid line represents the final selection of the dynamical exponent, which does not depend on the size of the system asymptotically. Error bars represent the statistical errors in the extraction of the contours or the exponents, correspondingly.

Figure 5

Dynamical exponent relating space and time as extracted from the shape of the OTOC “light cones” for a system size of $L=25$ (blue line), and the spread of the entanglement entropy (black line). The dynamical exponent for the entanglement entropy was taken from Ref. [36] for $L=28$. Error bars indicate the statistical uncertainty and do not include systematic errors. As the exponent depends on the choice of the threshold, we show the maximal exponent as well as the exponent of the front propagation for large threshold ($\eta =0.02$).