Krylov Complexity and Dynamical Phase Transition in the quenched LMG model

TL;DR

This paper explores how Krylov complexity evolves in the quenched Lipkin-Meshkov-Glick model, revealing it as an order parameter that distinguishes dynamical phases and correlates with traditional measures.

Contribution

It introduces a numerical analysis of Krylov complexity as a dynamical phase indicator in the LMG model, linking it to energy basis properties and symmetry considerations.

Findings

Krylov complexity acts as an order parameter for dynamical phases.

Long-term Krylov complexity discriminates between phases induced by a quench.

Matching behavior observed in Krylov and energy bases under specific symmetry conditions.

Abstract

Investigating the time evolution of complexity in quantum systems entails evaluating the spreading of the system's state across a defined basis in its corresponding Hilbert space. Recently, the Krylov basis has been identified as the one that minimizes this spreading. In this study, we develop a numerical exploration of the Krylov complexity in quantum states following a quench in the Lipkin-Meshkov-Glick model. Our results reveal that the long-term averaged Krylov complexity acts as an order parameter for this model. It effectively discriminates between the two dynamic phases induced by the quench, sharing a critical point with the conventional order parameter. Additionally, we examine the inverse participation ratio and the Shannon entropy in both the Krylov basis and the energy basis. A matching dynamic behavior is observed in both bases when the initial state possesses a specific symmetry. This behavior is analytically explained by establishing the equivalence between the Krylov basis and the pre-quench energy eigenbasis.