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

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 ${\varphi }_3=0.601(10)$ 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.