Critical behavior of the two-dimensionalN-component Landau-Ginzburg Hamiltonian with cubic anisotropy

We study the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy. We compute and analyze the fixed-dimension perturbative expansion of the renormalization-group functions to four loops. The relations of these models with N-color Ashkin-Teller models, discrete cubic models, the planar model with fourth-order anisotropy, and the structural phase transition in adsorbed monolayers are discussed. Our results for $N=2$ $(XY$ model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the $O(2)$ fixed points. Along this line the exponent η has the constant value $1/4,$ while the exponent ν runs in a continuous and monotonic way from 1 to ∞ [from Ising to $O(2)].$ In the four-loop approximation, for $N>~3$ we find a cubic fixed point in the region $u,v>~0.$