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 $N=2\mathrm{}$ $(XY$ model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the $O\left(2\right)\mathrm{}$ fixed points. Along this line the exponent $\eta$ has the constant value $1/4,$ while the exponent $\nu$ runs in a continuous and monotonic way from 1 to $\infty$ [from Ising to $O\left(2\right)\mathrm{}].$ In the four-loop approximation, for $N>~3$ we find a cubic fixed point in the region $u,v>~0.$

• Received 19 November 2001

DOI:https://doi.org/10.1103/PhysRevB.66.184410