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

Abstract

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 $\ensuremath{\lesssim}1%$ in the case of $\ensuremath{\gamma}$ and $\ensuremath{\nu}).$ Moreover, the irrelevant interaction bringing from the symmetric to the cubic fixed point gives rise to slowly decaying scaling corrections with exponent ${\ensuremath{\omega}}_{2}=0.010(4).$ For $N=2,$ the isotropic fixed point is stable and the cubic interaction induces scaling corrections with exponent ${\ensuremath{\omega}}_{2}=0.103(8).$ These conclusions are confirmed by a similar analysis of the five-loop $\ensuremath{\epsilon}$ expansion. A constrained analysis, which takes into account that ${N}_{c}=2$ in two dimensions, gives ${N}_{c}=2.87(5).$