$N$-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 $d=3.\mathrm{}$ 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 $N>\mathrm{}{N}_c,$ $N_c=2.89\mathrm{}\left(4\right).$ 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 $\nu ).\mathrm{}$ Moreover, the irrelevant interaction bringing from the symmetric to the cubic fixed point gives rise to slowly decaying scaling corrections with exponent ${\omega }_2=0.010\mathrm{}\left(4\right).$ For $N=2,$ the isotropic fixed point is stable and the cubic interaction induces scaling corrections with exponent ${\omega }_2=0.103\mathrm{}\left(8\right).$ These conclusions are confirmed by a similar analysis of the five-loop $\epsilon$ expansion. A constrained analysis, which takes into account that $N_c=2\mathrm{}$ in two dimensions, gives $N_c=2.87\mathrm{}\left(5\right).$