Post-Quench Evolution of Complexity and Entanglement in a Topological System

TL;DR

The paper studies how complexity and entanglement evolve after a quantum quench in a one-dimensional topological system, the Su-Schrieffer-Heeger model. It compares circuit complexity and Fubini-Study distance against entanglement entropy and spectra, using momentum-space quench dynamics and correlation matrices. The authors show that circuit complexity saturates rapidly, with revivals tied to system size, and that the Fubini-Study approach can detect some phase transitions while topological order is better captured by entanglement measures. The results challenge holographic expectations about relative saturation times and suggest that complexity-based probes can be practical for diagnosing phase structure in topological systems, including potential experimental relevance.

Abstract

We investigate the evolution of complexity and entanglement following a quench in a one-dimensional topological system, namely the Su-Schrieffer-Heeger model. We demonstrate that complexity can detect quantum phase transitions and shows signatures of revivals; this observation provides a practical advantage in information processing. We also show that the complexity saturates much faster than the entanglement entropy in this system, and we provide a physical argument for this. Finally, we demonstrate that complexity is a less sensitive probe of topological order, compared with measures of entanglement.

Figures (6)

• Figure 1: Evolution of EE for $N$=1000, $N$=1500, $N$=2000 and subsystem of $l$=100, $l$=150, $l$=200 respectively for quenching from (a) the critical point to a massive phase (b) a massive phase to the critical point (c) between massive phases. Insets: EE vs time scaled by the partition size, $S(t)/l$ vs $t/l$.
• Figure 2: Evolution of entanglement spectrum $N$=1500 and subsystem of $l$=150 for quenching from (a) the non-topological to the topological phase (b) the topological to the non-topological phase.
• Figure 3: Motion on the Bloch sphere for transition from the non-TI to QCP phase ($k=(2\pi/N)(m+1/2)$): (a) $m$ = 470 (b) $m$ = 505 (c) $m$ = 870 for $N$=1000.
• Figure 4: The circuit complexity for $N$=1500. Quenching (a) from a massive phase to the critical point (b) to a massive phase. Insets: Negative logarithm of the fidelity: $-\ln {\cal F}_{12}$.
• Figure 5: The complexity from the Fubini-Study line element for $N$=1500. Quenching (a) from a massive phase to the critical point (b) to a massive phase.