Third-harmonic exponent in three-dimensional N-vector models

TL;DR

This paper calculates the crossover exponent for the spin-3 operator in 3D O(N) models using six-loop field theory, providing results that align well with experimental data for N=2 systems.

Contribution

It presents a new six-loop calculation of the crossover exponent in three-dimensional O(N) models, specifically for the spin-3 operator, with implications for experimental systems.

Findings

Calculated $eta_3 = 1.413(10)$ for N=2.

Good agreement with experimental estimates in materials undergoing structural transitions.

Provides a theoretical benchmark for the crossover exponent in 3D O(N) models.

Abstract

We compute the crossover exponent associated with the spin-3 operator in three-dimensional O(N) models. A six-loop field-theoretical calculation in the fixed-dimension approach gives $\phi_3 = 0.601(10)$ for the experimentally relevant case N=2 (XY model). The corresponding exponent $\beta_3 = 1.413(10)$ is compared with the experimental estimates obtained in materials undergoing a normal-incommensurate structural transition and in liquid crystals at the smectic-A--hexatic-B phase transition, finding good agreement.