Dynamical Quantum Phase Transitions: A Geometric Picture

TL;DR

This paper demonstrates that a semiclassical approximation can accurately describe dynamical quantum phase transitions in large many-body systems, providing a geometric interpretation of the Loschmidt echo and order parameter dynamics.

Contribution

It introduces a simple semiclassical method for analyzing dynamical quantum phase transitions and offers a geometric perspective linking fidelity return rate and order parameter dynamics.

Findings

Semiclassical approximation agrees well with exact quantum calculations in the FC-TFIM.

The method provides an intuitive geometric interpretation of the Loschmidt echo.

The approach captures the dynamical phase diagram of the model.

Abstract

The Loschmidt echo (LE) is a purely quantum-mechanical quantity whose determination for large quantum many-body systems requires an exceptionally precise knowledge of all eigenstates and eigenenergies. One might therefore be tempted to dismiss the applicability of any approximations to the underlying time evolution as hopeless. However, using the fully connected transverse-field Ising model (FC-TFIM) as an example, we show that this indeed is not the case, and that a simple semiclassical approximation to systems well described by mean-field theory (MFT) is in fact in good quantitative agreement with the exact quantum-mechanical calculation. Beyond the potential to capture the entire dynamical phase diagram of these models, the method presented here also allows for an intuitive geometric interpretation of the fidelity return rate at any temperature, thereby connecting the order parameter dynamics and the Loschmidt echo in a common framework. Videos of the post-quench dynamics provided in the supplemental material visualize this new point of view.