The two-point correlation function of three-dimensional O(N) models: critical limit and anisotropy

TL;DR

This paper studies the low-momentum behavior of the two-point Green's function in three-dimensional O(N) models near criticality, revealing universal anisotropy decay characterized by a critical exponent close to 2 across all N.

Contribution

It introduces a comprehensive analysis of anisotropy decay in O(N) models using multiple theoretical approaches, establishing universality of the critical exponent rho around 2 for all N.

Findings

The critical exponent rho is approximately 2 for all N.

Non-Gaussian corrections to the Green's function are very small.

The decay of anisotropy is universal and related to the leading irrelevant operator.

Abstract

In three-dimensional O(N) models, we investigate the low-momentum behavior of the two-point Green's function G(x) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotational-invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice non-linear O(N) sigma models, 1/N-expansion, field-theoretical methods within the phi^4 continuum formulation. In non-rotational invariant physical systems with O(N)-invariant interactions, the vanishing of space-anisotropy approaching the rotational-invariant fixed point is described by a critical exponent rho, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N=\infty one finds rho=2. We show that, for all values of $N\geq 0$, $\rho\simeq 2$. Non-Gaussian corrections to the universal low-momentum behavior of G(x) are evaluated, and found to be very small.