Quantum Liang Information Flow vs. Out-of-Time-Order Correlators as Chaos Diagnostics in the Mixed-Field Ising Chain

Abstract

We systematically compare Quantum Liang Information Flow (QLIF) a recently proposed causal information measure with the out-of-time-order correlator (OTOC) as diagnostics of quantum chaos in the one-dimensional mixed-field Ising chain. Using exact diagonalization and MPS-TEBD, we show that the early-time power-law growth and wavefront propagation velocity of QLIF are identical for integrable and chaotic parameters, being controlled solely by the local Hamiltonian structure. The QLIF signal strength depends sensitively on the initial state, spanning four orders of magnitude across product states, ground state eigenstate evolution, and quantum quench protocols. We identify the time-integrated QLIF as a late-time chaos diagnostic: it grows linearly (monotonically) in chaotic systems, reflecting irreversible thermalization, while saturating or oscillating in integrable systems, reflecting reversible quasiparticle dynamics. These findings establish QLIF as a complementary probe to OTOC, with distinct optimal operating regimes.

Figures (6)

• Figure 1: Overview of $|T_d(t)|$ across distances. Initial state: Néel state ($|\!\uparrow\downarrow\cdots\rangle$); MPS-TEBD, $\chi=128$, $dt=0.05$, $B=0.8$. Integrable: $h_z=0$; Chaotic: $h_z=0.5$. Top row: $L=30$, $d=4, 7$; Bottom row: $L=30$, $d=10$ and $L=50$, $d=17$. Each panel marks $t_{\max}=d/v_{\max}$ (black dashed), $t_{LR}=d/v_{LR}$ (green dash-dotted), and $t_B=d/v_B$ (purple dashed). In the early-time regime ($t < t_{\max}$), the integrable and chaotic curves are completely indistinguishable.
• Figure 2: $(t,d)$ heatmap of $|T_d(t)|$. Initial state: Néel state; $L=30$, frozen$=10$, obs$=11\sim20$, $d=1\sim10$, $B=0.8$, $\chi=128$, $dt=0.05$, $t_{\max}=10$. Left: integrable ($h_z=0$); Right: chaotic ($h_z=0.5$). White dashed line: $v_{\max}=1.6$; cyan dash-dotted line: $v_{LR}=5.44$. The light cone front in both systems propagates at $v_{\max}$.
• Figure 3: Comparison of $|T_d(t)|$ for five initial-state--evolution combinations. $L=30$, frozen$=10$, obs$=20$, $d=10$, $B=0.8$, $\chi=128$, $dt=0.05$, $t_{\max}=10$. N (Néel $\to$ integrable/chaotic $H$, blue) has the strongest signal; A (integrable GS $\to$ integrable $H$, green) is next; C (chaotic GS $\to$ chaotic $H$, orange) and B (integrable GS $\to$ chaotic $H$, red) have the weakest signals. Ground states were computed by DMRG. Green dash-dotted line: $t_{LR}=d/v_{LR}=1.8$ (Lieb--Robinson upper bound); black dashed line: $t_{\max}=d/v_{\max}=6.2$ (wavefront arrival time).
• Figure 4: Raw $T_d(t)$ (with sign) for each combination. Parameters same as Fig. \ref{['fig:initial_state_log']}. (a) Néel $\to$ integrable/chaotic $H$; (b) integrable GS $\to$ integrable $H$ (eigenstate local quench); (c) chaotic GS $\to$ chaotic $H$ (eigenstate local quench); (d) integrable GS $\to$ chaotic $H$ (global quench). Note the vastly different vertical scales across panels. Green dash-dotted line: $t_{LR}=d/v_{LR}=1.8$; black dashed line: $t_{\max}=d/v_{\max}=6.2$. Velocity definitions are given in §\ref{['sec:velocity']}.
• Figure 5: Late-time QLIF starting from the Néel state. $L=20$, frozen$=8$, obs$=12$, $d=4$, $B=0.8$, $\chi=128$, $dt=0.1$, $t_{\max}=40$. Integrable: $h_z=0$; Chaotic: $h_z=0.5$. (a) $|T_d|$ semilogy; (b) raw $T_d$ (with sign); (c) time integral $\int_0^t T_d\,dt'$. Purple dash-dotted line: $t_{LR}=d/v_{LR}=0.7$ (Lieb--Robinson upper bound); black dashed line: $t_{\max}=d/v_{\max}=2.5$ (wavefront arrival time); green dash-dotted line: $t_{\rm scr}=L/v_{\max}\approx 12.5$ (scrambling time, when signal traverses the entire chain). $v_{\max}=2\min(J,B)=1.6$ is the maximum group velocity in the integrable limit; $v_{LR}=2eJ=5.4$.