Dynamical quantum phase transition, metastable state, and dimensionality reduction: Krylov analysis of fully connected spin models

We study quenched dynamics of fully-connected spin models. The system is prepared in a ground state of the initial Hamiltonian and the Hamiltonian is suddenly changed to a different form. We apply the Krylov subspace method to map the system onto an effective tridiagonal Hamiltonian. The state is confined in a potential well and is time-evolved by nonuniform hoppings. The dynamical singularities for the survival probability can occur when the state is reflected from a potential barrier. Although we do not observe any singularity in the spread complexity, we find that the entropy exhibits small dips at the singular times. We find that the presence of metastable state affects long-time behavior of the spread complexity, and physical observables. We also observe a reduction of the state-space dimension when the Hamiltonian reduces to a classical form.