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Investigating the time evolution of complexity in quantum systems entailsevaluating the spreading of the system's state across a defined basis in itscorresponding Hilbert space. Recently, the Krylov basis has been identified asthe one that minimizes this spreading. In this study, we develop a numericalexploration of the Krylov complexity in quantum states following a quench inthe Lipkin-Meshkov-Glick model. Our results reveal that the long-term averagedKrylov complexity acts as an order parameter when the quench originates from azero magnetic field. It effectively discriminates between the two dynamicphases induced by the quench, sharing a critical point with the conventionalorder parameter. Additionally, we examine the inverse participation ratio andShannon entropy in both the Krylov basis and the energy basis. A matchingdynamic behavior is observed in both bases when the initial state possesses aspecific symmetry. This behavior is analytically elucidated by establishing theequivalence between the Krylov basis and the pre-quench energy eigenbasis.

Krylov Complexity and Dynamical Phase Transition in the quenched LMG model