The scaling equation of state of the three-dimensional O(N) universality class: N >= 4

TL;DR

This paper determines the critical equation of state for the three-dimensional O(N) universality class for N=4, 5, 6, 32, 64, providing insights relevant to various physical theories and employing a systematic approximation scheme.

Contribution

It introduces a polynomial parametric representation method to accurately model the equation of state across the critical regime for multiple N values.

Findings

Good agreement with large-N expansion for N ≥ 32.

Provides estimates of universal amplitude ratios.

Models critical behavior relevant to QCD and high-T_c superconductivity.

Abstract

We determine the critical equation of state of the three-dimensional O(N) universality class, for N=4, 5, 6, 32, 64. The N=4 is relevant for the chiral phase transition in QCD with two flavors, the N=5 model is relevant for the SO(5) theory of high-T_c superconductivity, while the N=6 model is relevant for the chiral phase transition in two-color QCD with two flavors. We first consider the small-field expansion of the effective potential (Helmholtz free energy). Then, we apply a systematic approximation scheme based on polynomial parametric representations that are valid in the whole critical regime, satisfy the correct analytic properties (Griffiths' analyticity), take into account the Goldstone singularities at the coexistence curve, and match the small-field expansion of the effective potential. From the approximate representations of the equation of state, we obtain estimates of universal amplitude ratios. We also compare our approximate solutions with those obtained in the large-N expansion, up to order 1/N, finding good agreement for N \gtrsim 32.