Equilibrium and dynamical phase transitions in fully connected quantum Ising model: Approximate energy eigenstates and critical time

TL;DR

This paper investigates the equilibrium and dynamical phase transitions in the finite-size fully connected quantum Ising model, introducing approximate eigenstates and analyzing critical times and their scaling behavior.

Contribution

It provides approximate energy eigenstates with high overlap to exact states and analyzes the divergence of critical times in dynamical phase transitions.

Findings

Good match in energy gap and geometric entanglement between approximate and exact states.

Concurrence agrees well in the paramagnetic phase for large systems.

Critical times diverge logarithmically or as a power law depending on initial conditions and parameters.

Abstract

We study equilibrium as well as dynamical properties of the finite-size fully connected Ising model with a transverse field at the zero temperature. In relation to the equilibrium, we present approximate ground and first excited states that have large overlap -- except near the phase transition point -- with the exact energy eigenstates. For both the approximate and exact eigenstates, we compute the energy gap, concurrence, and geometric measure of quantum entanglement. We observe a good match in the case of energy gap and geometric entanglement between the approximate and exact eigenstates. Whereas, when the system size is large, the concurrence shows a nice agreement only in the paramagnetic phase. In a quench dynamics, we study the time period and the first critical time, which play important roles in the dynamical phase transitions, based on a dynamical order parameter and the Loschmidt rate, respectively. When all the spins are initially polarized in the direction of their mutual interaction, both the time period and critical time diverges logarithmically with the system size at the dynamical critical point. When all the spins are initially in the direction of transverse field, both the time period and critical time exhibit logarithmic or power-law divergences depending on the final field strength. In the case of convergence, we provide estimates for the finite-size scaling and converged value.