# Complex semiclassical analysis of the Loschmidt amplitude and dynamical quantum phase transitions

## TL;DR

This paper introduces a complex semiclassical method to analyze the Loschmidt amplitude in spin systems, revealing dynamical quantum phase transitions through theoretical and numerical approaches.

## Contribution

It develops a new semiclassical computational technique for the Loschmidt amplitude applicable to generic spin systems, enabling analysis of dynamical phase transitions.

## Key findings

- Dynamical phase diagrams identified for the infinite-range transverse Ising model.
- Dynamical transitions depend on initial conditions and quench parameters.
- Semiclassical results are validated by numerical experiments.

## Abstract

We propose a new computational method of the Loschmidt amplitude in a generic spin system on the basis of the complex semiclassical analysis on the spin-coherent state path integral. We demonstrate how the dynamical transitions emerge in the time evolution of the Loschmidt amplitude for the infinite-range transverse Ising model with a longitudinal field, exposed by a quantum quench of the transverse field $\Gamma$ from $\infty$ or $0$. For both initial conditions, we obtain the dynamical phase diagrams that show the presence or absence of the dynamical transition in the plane of transverse field after a quantum quench and the longitudinal field. The results of semiclassical analysis are verified by numerical experiments. Experimental observation of our findings on the dynamical transition is also discussed.