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

TL;DR

This paper calculates the critical equation of state for three-dimensional O(N) models with N=5, 6, 32, 64, relevant for high-T_c superconductivity and QCD, using perturbative series and polynomial approximations.

Contribution

It introduces a systematic approximation scheme for the critical equation of state that incorporates known analytical properties and Goldstone singularities, extending previous methods.

Findings

Accurate critical exponents and effective potential expansions for N=5, 6, 32, 64.

Universal amplitude ratios determined with high precision.

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

Abstract

We determine the scaling equation of state of the three-dimensional O(N) universality class, for N=5, 6, 32, 64. 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 obtain the critical exponents and the small-field, high-temperature, expansion of the effective potential (Helmholtz free energy) by analyzing the available perturbative series, in both fixed-dimension and epsilon-expansion schemes. Then, we determine the critical equation of state by using a systematic approximation scheme, based on polynomial representations valid in the whole critical region, which satisfy the known analytical properties of the equation of state, take into account the Goldstone singularities at the coexistence curve and match the small-field, high-temperature, expansion of the effective potential. This allows us also to determine several 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.