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 and a five-loop calculation in epsilon expansion give phi(3)=0.600(10) for the experimentally relevant case N=2 (XY model). The corresponding exponent beta(3)=1.414(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.