From quantum to classical dynamics: The relativistic $O(N)$ model in the framework of the real-time functional renormalization group

TL;DR

This paper studies how the relativistic $O(N)$ model transitions from quantum to classical behavior using the real-time functional renormalization group, focusing on dynamic critical exponents and universality classes at finite temperature.

Contribution

It introduces a nonperturbative real-time RG approach to analyze the thermal dynamics and universality class of the relativistic $O(N)$ model across different temperature regimes.

Findings

Quantifies the dynamic critical exponent $z$ at large temperatures.

Identifies the effective universality class for the model at finite temperature.

Contrasts finite temperature behavior with zero temperature expectations.

Abstract

We investigate the transition from unitary to dissipative dynamics in the relativistic $O(N)$ vector model with the $\lambda (\varphi^{2})^{2}$ interaction using the nonperturbative functional renormalization group in the real-time formalism. In thermal equilibrium, the theory is characterized by two scales, the interaction range for coherent scattering of particles and the mean free path determined by the rate of incoherent collisions with excitations in the thermal medium. Their competition determines the renormalization group flow and the effective dynamics of the model. Here we quantify the dynamic properties of the model in terms of the scale-dependent dynamic critical exponent $z$ in the limit of large temperatures and in $2 \leq d \leq 4$ spatial dimensions. We contrast our results to the behavior expected at vanishing temperature and address the question of the appropriate dynamic universality class for the given microscopic theory.