Complex dynamics approach to dynamical quantum phase transitions: the Potts model

TL;DR

This paper applies complex dynamics and renormalization group methods to analyze dynamical quantum phase transitions in the 3-state Potts model, revealing fractal structures and boundary condition effects.

Contribution

It introduces a novel complex dynamics approach using RG transformations to study quantum phase transitions in the Potts model, including in higher dimensions.

Findings

First-order phase transitions in 1D and 2D systems.

Boundary conditions can change transition nature.

Fractal structures in the Julia set influence phase behavior.

Abstract

This paper introduces complex dynamics methods to study dynamical quantum phase transitions in the one- and two-dimensional quantum 3-state Potts model. The quench involves switching off an infinite transverse field. The time-dependent Loschmidt echo is evaluated by an exact renormalization group (RG) transformation in the complex plane where the thermal Boltzmann factor is along the positive real axis, and the quantum time evolution is along the unit circle. One of the characteristics of the complex dynamics constituted by repeated applications of RG is the Julia set, which determines the phase transitions. We show that special boundary conditions can alter the nature of the transitions, and verify the claim for the one-dimensional system by transfer matrix calculations. In two dimensions, there are alternating symmetry-breaking and restoring transitions, both of which are first-order, despite the criticality of the Curie point. In addition, there are finer structures because of the fractal nature of the Julia set. Our approach can be extended to multi-variable problems, higher dimensions, and approximate RG transformations expressed as rational functions.