Symmetry-breaking dynamics of the finite-size Lipkin-Meshkov-Glick model near ground state

Abstract: We study the dynamics of the Lipkin-Meshkov-Glick (LMG) model with finite number of spins. In the thermodynamic limit, the ground state of the LMG model with isotropic Hamiltonian in broken phase breaks to a mean-field ground state with a certain direction. However, when the spins number $N$ is finite, the exact ground state is always unique and is not given by a classical mean-field ground state. Here we prove that for $N$ is large but finite, through a tiny external perturbation, a localized state which is close to a mean-field ground state can be prepared, which mimics a spontaneous symmetry breaking (SSB). Besides, we find the localized in-plane spin polarization oscillates with two different frequencies $\sim O(1/N)$, and the lifetime of the localized state is long enough to exhibit this oscillation. We numerically test the analytical results and find that they agree with each other very well. Finally, we link the phenomena to quantum time crystals and quasicrystals.