The N-component Ginzburg-Landau Hamiltonian with cubic anisotropy: a six-loop study

We consider the Ginzburg-Landau Hamiltonian with a cubic-symmetric quartic interaction and compute the renormalization-group functions to six-loop order in d=3. We analyze the stability of the fixed points using a Borel transformation and a conformal mapping that takes into account the singularities of the Borel transform. We find that the cubic fixed point is stable for N>N_c, N_c = 2.89(4). Therefore, the critical properties of cubic ferromagnets are not described by the Heisenberg isotropic Hamiltonian, but instead by the cubic model at the cubic fixed point. For N=3, the critical exponents at the cubic and symmetric fixed points differ very little (less than the precision of our results, which is $\lesssim 1%$ in the case of $\gamma$ and $\nu$). Moreover, the irrelevant interaction bringing from the symmetric to the cubic fixed point gives rise to slowly-decaying scaling corrections with exponent $\omega_2=0.010(4)$. For N=2, the isotropic fixed point is stable and the cubic interaction induces scaling corrections with exponent $\omega_2 = 0.103(8)$. These conclusions are confirmed by a similar analysis of the five-loop $\epsilon$-expansion. A constrained analysis which takes into account that $N_c = 2$ in two dimensions gives $N_c = 2.87(5)$.