Spectral functions and dynamic critical behavior of relativistic $Z_2$ theories

TL;DR

This paper studies the dynamic critical behavior of a relativistic $Z_2$ scalar field theory using real-time lattice simulations, revealing relativistic quasi-particle peaks, soft modes near criticality, and determining the dynamic critical exponent for different universality classes.

Contribution

It provides first-principles spectral function calculations near criticality for relativistic $Z_2$ theories, identifying dynamic scaling and critical exponents for Models A and C.

Findings

Spectral functions show relativistic quasi-particle peaks above $T_c$.

Strong infrared contributions appear near $T_c$.

Soft modes indicate collective excitations in the ordered phase.

Abstract

We investigate the dynamic critical behaviour of a relativistic scalar field theory with $Z_2$ symmetry by calculating spectral functions of the order parameter at zero and non-vanishing momenta from first-principles classical-statistical lattice simulations in real-time. We find that at temperatures above the critical point $(T > T_c)$, the spectral functions are well described by relativistic quasi-particle peaks. Close to the transition temperature $(T \sim T_c)$, we observe strong infrared contributions building up. In the ordered phase at low temperatures $(T < T_c)$, in addition to the quasi-particle peak, we observe a soft mode with a dispersion relation indicative of collective excitations. Investigating the spectral functions close to $T_c$, we demonstrate that the behavior in the vicinity of the critical point is controlled by dynamic scaling functions and the dynamic critical exponent $z$, which we determine from our simulations. By considering the equations of motion for a closed system and a system coupled to a heat bath, we extract the dynamic critical behavior for two different dynamic universality classes (Models A & C) in two and three spatial dimensions.