Dynamic Critical Exponent from One- and Two-Particle Irreducible 1/N Expansions of Effective and Microscopic Theories

TL;DR

This paper compares effective and microscopic theories for the dynamic critical exponent using 1/N expansions, revealing that at NLO of the 2PI expansion, both theories agree and diffusive modes dominate, regardless of kinematics.

Contribution

It demonstrates that the NLO of the 2PI 1/N expansion aligns microscopic and effective theories, clarifying the role of diffusive modes and kinematic effects on the dynamic critical exponent.

Findings

At NLO of 1PI expansion, microscopic and effective theories differ significantly.

At NLO of 2PI expansion, the theories become equivalent and diffusive modes dominate.

The improved calculation yields a slightly smaller and more stable critical exponent.

Abstract

We study the dynamic critical exponent from effective and microscopic theories. We employ a simple TDGL model, or model A in the classification of Hohenberg and Halperin, as an effective theory and the imaginary time formalism of the finite-temperature filed theory as a microscopic theory. Taking an O(N) scalar model as an example and carrying out the 1/N expansion up to the NLO in the 1PI and 2PI effective actions, we compare the low-energy and low-momentum behavior of the response function in the effective theory and of the retarded Green's function in the microscopic theory. At the NLO of the 1PI 1/N expansion the low-energy and low-momentum behavior of the two-point function is very much different in the microscopic and effective theories: in the field theory it is dominated by the propagating mode while in model A it is dominated by the diffusive mode. Also, in the microscopic theory the dynamic critical exponent, z, depends on whether the kinematics is relativistic or nonrelativistic. In contrast, at the NLO of the 2PI 1/N expansion the microscopic and effective theories are equivalent. They satisfy exactly the same Kadanoff-Baym equation. Also, whether the kinematics is relativistic or nonrelativistic in the microscopic theory becomes irrelevant. This implies that the diffusive mode with z = 2 + O(1/N) is dominant at low energies and momenta even in the microscopic theory at the NLO of the 2PI 1/N expansion, though we do not explicitly solve the Kadanoff-Baym equation. We also try to improve the calculation of the dynamic critical exponent of model A by incorporating the static 2PI NLO correlations. The obtained critical exponent is slightly smaller than the previous result and its N dependence is also milder than the previous one.