[Paper Review] Charged Fixed Point Found in Superconductor Below Tc

H. Kleinert, Flavio S. Nogueira|arXiv (Cornell University)|Apr 30, 2001

Theoretical and Computational Physics4 references3 citations

TL;DR

This paper presents a perturbative renormalization group analysis in the Ginzburg-Landau model below Tc in three dimensions, identifying an infrared-stable fixed point for κ > 1/√2 in the type II superconductor regime. The fixed point arises from a momentum-space instability at nonzero wavevector p₀ ∼ ∆κ¯β with critical exponent ¯β = 1/2, explaining the negative η-exponent of the order field via a Lifshitz-type instability.

ABSTRACT

We present a perturbative approach which yields, for the first time, an infrared-stable fixed point in the Ginzburg-Landau (GL) model at the oneloop level. The calculations are done in d = 3 dimensions and below Tc, 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 / √ 2, that is, in the type II regime, where ∆κ ≡ κ − 1 / √ 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 p0 whose magnitude scales like ∼ ∆κ ¯ β, with a critical exponent ¯ β = 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.

Motivation & Objective

• To identify a stable fixed point in the Ginzburg-Landau model below Tc using perturbative renormalization group methods.
• To determine the conditions under which a charged infrared-stable fixed point emerges in the superconducting phase.
• To relate the emergence of the fixed point to a momentum-space instability in the two-point correlation function of the order field.
• To explain the origin of the negative η-exponent in the order field correlation function through a Lifshitz-type instability.
• To establish ∆κ = κ − 1/√2 as a natural expansion parameter in the type II regime.

Proposed method

• A one-loop renormalization group analysis is performed in d = 3 dimensions for the Ginzburg-Landau model below Tc.
• The renormalization group functions are expressed directly in terms of the Ginzburg parameter κ = λ/ξ, the ratio of penetration depth to correlation length.
• The two-point correlation function of the order field is analyzed to detect momentum-space instabilities at nonzero wavevector p₀.
• The critical exponent ¯β = 1/2 is derived in the one-loop approximation, governing the scaling of p₀ ∼ ∆κ¯β with ∆κ = κ − 1/√2.
• The analysis identifies a charged fixed point that is infrared-stable for κ > 1/√2, corresponding to the type II superconductor regime.
• The formalism connects the instability to the negative η-exponent via the structure of the correlation function in momentum space.

Experimental results

Research questions

• RQ1 Does a charged infrared-stable fixed point exist in the Ginzburg-Landau model at the one-loop level below Tc?
• RQ2 What is the role of the Ginzburg parameter κ in determining the stability and nature of fixed points in the superconducting phase?
• RQ3 How does a momentum-space instability in the order field correlation function arise, and what controls its wavevector p₀?
• RQ4 Why does the order field exhibit a negative η-exponent, and how is this related to the instability at nonzero momentum?
• RQ5 Is ∆κ = κ − 1/√2 a natural expansion parameter in the type II regime, and how does it govern the instability scale?

Key findings

• An infrared-stable fixed point is found in the Ginzburg-Landau model at the one-loop level below Tc for κ > 1/√2, corresponding to the type II superconductor regime.
• A momentum-space instability emerges in the two-point correlation function of the order field at a nonzero wavevector p₀ ∼ ∆κ¯β with ¯β = 1/2 in the one-loop approximation.
• The instability is tied to the emergence of a charged fixed point, which is stable in the infrared for κ > 1/√2.
• The parameter ∆κ = κ − 1/√2 is identified as the natural expansion parameter controlling the instability scale and the critical behavior.
• The momentum-space instability provides a physical mechanism for the negative η-exponent of the order field, consistent with Lifshitz point behavior in magnetic systems.
• The results establish a direct link between the Ginzburg parameter, momentum-space instabilities, and critical exponents in type II superconductors at the one-loop level.