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

Abstract

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 $κ$ which is the ratio between the two fundamental scales of the problem, the penetration depth $λ$ and the correlation length $ξ$. We find a charged fixed point for $κ>1/\sqrt{2}$, that is, in the type II regime, where $Δκ\equivκ-1/\sqrt{2}$ 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 $\simΔκ^{\barβ}$, with a critical exponent $\barβ=1/2$ 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 $η$-exponent of the order field.