Information spreading and scrambling in disorder-free multiple-spin-interaction models

Tripartite mutual information (TMI) is an efficient observable to quantify the ability of a scrambler for the unitary time-evolution operator with a quenched many-body Hamiltonian. In this paper we give numerical demonstrations of the TMI in disorder-free (translationally invariant) spin models with three-body and four-body multiple-spin interactions. The dynamical behavior of the TMI of these models does not exhibit a linear light cone for sufficiently strong interactions. In early-time evolution, the TMI displays a distinct negative increase fitted by a logarithmiclike function. This is in contrast to the conventional linear light-cone behavior present in the $XXZ$ model and its near-integrable vicinity. The late-time evolution of the TMI in finite-size systems is also numerically investigated. The multiple-spin interactions make the system nearly integrable and weakly suppress the spread of information and scrambling. The observation of the late-time value of the TMI indicates that the scrambling nature of the system changes by interactions and this change can be characterized by a phase-transition-like behavior of the TMI, reflecting the system's integrability and violating the eigenstate thermalization hypothesis.

Figure 1

Schematic image of the time evolution of the state with doubled Hilbert space. The spatial partitioning of the system is represented where four subsystems $A, B, C$, and $D$ are introduced. Each part is $L/2$ lattice sites.

Figure 2

Level spacing ratio for (a) the three-body model and (b) the four-range model [(b)]. We set $J_1=0.3, v=0.2$, and (a) $L=12$–18 and (b) $L=12$–20. The red and blue dashed lines represent the WD value 0.53 and the Poisson value 0.39, respectively.

Figure 3

Histograms of the local magnetization in the three-body and four-range models (we employ all eigenstates for $L=10$–16): (a) $J_0=0.5$, three-body model case, where the model is nonintegrable as for the LSA; (b) $J_0=2.0$, three-body model case; (c) $t_0=0.5$, four-range model case, where the model is nonintegrable as for the LSA; and (d) $J_0=2.0$, four-range model case.

Figure 4

Early-time evolution of three-body model $H_{3\mathrm{B}}$. The logarithmic fitting lines are $-{\tilde{I}}_3=0.0639{log}_{2}t+0.184, 0.0575{log}_{2}t+0.1624$, and $0.0507{log}_{2}t+0.1435$ for $L=10$, 12, and 14, respectively. We set $J_1=0.3, J_0=2$, and $v=0.2$. In the fitting, we use the data points within $t\in [0.1,8]$. The unit of time is $2\hslash /v$.

Figure 5

Early-time evolution of the $XXZ$ model. The linear fitting lines are $-{\tilde{I}}_3=0.0549t-0.0231, 0.0459t-0.0199$, and $0.0395t-0.0174$ for $L=10$, 12, and 14, respectively. We set $v=0.2$ and $J_1=0.3$. In the fitting, we use the data points within $t\in [0.28,6.94]$. The unit of time is $2\hslash /v$.

Figure 6

Late-time evolution of the three-body model. The fitting lines are $-{\tilde{I}}_3=0.0457{log}_{2}t+0.2748, 0.0465{log}_{2}t+0.2508$, and $0.0560{log}_{2}t+0.1958$ for $L=10$, 12, and 14, respectively. We set $J_1=0.3, J_0=2$, and $v=0.2$. The unit of time is $2\hslash /v$. In the fitting, we use the data points within $t\in [2.5,200]$.

Figure 7

Late-time evolution of the $XXZ$ model We set $v=0.2$ and $J_1=0.3$. The unit of time is $2\hslash /v$.

Figure 8

Early-time evolution of the four-range model. The fitting lines are $-{\tilde{I}}_3=0.0494{log}_{2}t+0.2246, 0.0829{log}_{2}t+0.1737$, and $0.0402t-0.0260$ for $t_0=2$, 1, and 0.05, respectively. We use the data points within $t\in [0.25,8.5]$ and $t\in [0.7,5.7]$ for the logarithmic fitting and linear fitting, respectively. For the data at $t_0=2$ and the timescale $t\sim 1/t_0=0.5$, the curvature of the behavior of the TMI takes a peak. Here $L=12$ and the unit of time is $2\hslash /v$.

Figure 9

(a) Schematic image of the partitioning of the $A$ and $D$ subsystems with distance $r$. Subsystems $A$ and $D$ include two sites. We impose open boundary conditions. (b) Spreading of information in the $XXZ$ model from subsystem $A$ to subsystem $D$ with distance $r$ along the time evolution. The plotted value is the TMI $-{\tilde{I}}_3$. For the result, we set $L=12$ and $1\le r\le 8$. The dashed lines are guides to the eye for approximated level lines. The unit of time on the vertical axis is $2\hslash /v$.

Figure 10

Spreading of information from subsystem $A$ to subsystem $D$ with distance $r$ along the time evolution of (a) and (b) the three-body spin model and (c) and (d) the four-range model. The data are for (a) the chaotic case with $J_0=0.5$, (b) the nearly integrable case with $J_0=2$, (c) the chaotic case with $t_0=0.5$, and (d) the nearly integrable case with $t_0=2$. The plotted value is the TMI $-{\tilde{I}}_3$. For all results, we set $L=12$ and $1\le r\le 8$. The dashed lines are guides to the eye for approximated level curves. The unit of time on the vertical axis is $2\hslash /v$.

Figure 11

(a) The TMI at $t={10}^5$ vs $J_0$ for the three-body case. The data lines of three different system sizes seem to cross at $J_0\sim 2.5$. (b) The TMI vs system size $L$ for the three-body case for each $J_0$. We set $v=0.2$ and $J_1=0.3$. (c) The TMI at $t={10}^5$ vs $t_0$ for the four-range case. The data lines of three different system sizes cross at $t_0\sim 1$. (d) The TMI vs system size $L$ for the four-body case for each $t_0$. We set $v=0.2$.

Figure 12

(a) Early-time evolution of mutual information $I(A,C)$ of the (a) three-body and (b) four-range models. We use OBC. The fitting lines are (a) $I(A,C)=-0.1606t+10.0032, -0.2090t+9.9318$, and $-0.2118t+9.9214$ for $J_0=0\left(XXZ\right)$, 0.05, and 0.1, respectively, and (b) $I(A,C)=-0.1825t+10.0157$ and $-0.2250t+10.0247$ for $t_0=0.05$ and 0.1, respectively. The unit of time is $2\hslash /v$.

Figure 13

Additional data in open boundary condition. (a) Early-time evolution of the three-body model $H_{3\mathrm{B}}$. The logarithmic fitting lines are $-{\tilde{I}}_3=0.0373{log}_{2}t+0.0970, 0.0311{log}_{2}t+0.0840$, and $0.0267{log}_{2}t+0.0735$ for $L=10$, 12, and 14, respectively. We set $J_1=0.3, J_0=2$, and $v=0.2$. In the fitting, we use the data points within $t\in [0.1,8]$. (b) Early-time evolution of the $XXZ$ model. The linear fitting lines are $-{\tilde{I}}_3=0.0295t-0.0145, 0.0246t-0.0121$, and $0.0211t-0.0105$ for $L=10$, 12, and 14, respectively. We set $v=0.2$ and $J_1=0.3$. In the fitting, we use the data points within $t\in [0.28,6.94]$. (c) Late-time evolution of the three-body model. The fitting lines are $-{\tilde{I}}_3=0.0269{log}_{2}t+0.1889, 0.0384{log}_{2}t+0.1226$, and $0.0504{log}_{2}t+0.0535$ for $L=10$, 12, and 14, respectively. We set $J_1=0.3, J_0=2$, and $v=0.2$. In the fitting, we use the data points within $t\in [2.5,200]$. (d) Late-time evolution of the $XXZ$ model. We set $v=0.2$ and $J_1=0.3$. The unit of time is $2\hslash /v$.

Figure 14

Early-time evolution of the four-range model with OBC. The fitting lines are $-{\tilde{I}}_3=0.0527{log}_{2}t+0.1344, 0.0606{log}_{2}t+0.0945$, and $0.0217t-0.0158$ for $t_0=2$, 1, and 0.05, respectively. We use the data points within $t\in [0.25,8.5]$ and $t\in [0.7,5.7]$ for the logarithmic fitting and linear fitting, respectively. For the data $t_0=2$, at the timescale $t\sim 1/t_0=0.5$ and the curvature of the behavior of the TMI takes a peak. Here $L=12$.