N-component Ginzburg-Laudau 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 less than or similar to 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)