Charged fixed point in the Ginzburg-Landau superconductor and the role of the Ginzburg parameter kappa

We present a semi-perturbative approach which yields an infrared-stable fixed point in the Ginzburg-Landau for N=2, where $N/2$ is the number of complex components. The calculations are done in $d=3$ dimensions and below $T_c$, where the renormalization group functions can be expressed directly as functions of the Ginzburg parameter $\kappa$ which is the ratio between the two fundamental scales of the problem, the penetration depth $\lambda$ and the correlation length $\xi$. We find a charged fixed point for $\kappa>1/\sqrt{2}$, that is, in the type II regime, where $\Delta\kappa\equiv\kappa-1/\sqrt{2}$ is shown to be a natural expansion parameter. This parameter controls a momentum space instability in the two-point correlation function of the order field. This instability appears at a nonzero wave-vector ${\bf p}_0$ whose magnitude scales like $\sim\Delta\kappa^{\bar{\beta}}$, with a critical exponent $\bar{\beta}=1/2$ in the one-loop approximation, a behavior known from magnetic systems with a Lifshitz point in the phase diagram. This momentum space instability is argued to be the origin of the negative $\eta$-exponent of the order field.