[Paper Review] RG analysis and Magnetic instability in gapless superconductors

Deog Ki Hong|ArXiv.org|Jun 9, 2005

Physics of Superconductivity and Magnetism3 citations

TL;DR

This paper investigates magnetic instability in gapless superconductors using renormalization group (RG) analysis and effective field theory. It shows that when the Fermi surface mismatch is close to the superconducting gap, infrared divergences trigger a secondary gap formation via a power-suppressed four-Fermi interaction, stabilizing the system. For larger mismatches, the system instead forms a supercurrent condensate, restoring a positive Meissner mass and eliminating magnetic instability.

ABSTRACT

We study the magnetic instability of gapless superconductors. The instability arises due to the infrared divergence associated with the gapless modes of the superconductor. When the Fermi-surface mismatch between pairing fermions is close to the gap, the gapless modes have a quadratic energy dispersion relation at low energy and open a secondary gap at the Fermi surface, which is only power-suppressed by the coupling. On the other hand, for a large mismatch, we find the gapless superconductor does not open a secondary gap, but instead makes transition to a new phase by forming the condensate of supercurrents. We calculate the condensate of supercurrents by minimizing the effective potential. In the new phase, the Meissner mass is positive for the magnetic fields orthogonal to the direction of the condensate but zero in the parallel direction.

Motivation & Objective

• To understand the origin of magnetic instability in gapless superconductors arising from infrared divergences in gapless modes.
• To determine whether gapless superconductors remain stable or undergo phase transitions under strong Fermi surface mismatch.
• To analyze the role of effective interactions and scaling behavior of gapless modes near the Fermi surface.
• To derive the conditions under which a secondary gap opens or a supercurrent condensate forms.
• To establish the stability of the system by computing the Meissner mass and effective potential in both phases.

Proposed method

• Construct a minimal model of gapless superconductivity with two flavors and a neutralizing electron fluid, using a four-Fermi interaction from integrated-out gluons.
• Apply mean-field theory to compute the free energy and identify unstable solutions when δμ > Δ.
• Perform renormalization group (RG) analysis on the low-energy effective Lagrangian of gapless modes, revealing anomalous scaling behavior (Ψ → s^−1/4Ψ).
• Integrate out gapped modes to generate an effective attractive four-Fermi interaction for gapless modes, with coupling scaling as s^−1/2.
• Solve the gap equation for the secondary gap Δs using one-loop self-energy diagrams, yielding Δs ∝ (G_s^2 ν_*/v_*) δμ.
• Analyze the Meissner mass in both the secondary gap and supercurrent condensate phases to assess magnetic stability.

Experimental results

Research questions

• RQ1 What causes the magnetic instability in gapless superconductors, and how is it related to infrared divergences in gapless modes?
• RQ2 Under what conditions does a gapless superconductor form a secondary gap versus a supercurrent condensate?
• RQ3 How does the scaling behavior of gapless modes affect the relevance of the four-Fermi interaction in the infrared?
• RQ4 What is the quantitative form of the secondary gap, and how does it depend on coupling constants and mismatch?
• RQ5 How does the Meissner mass behave in the two competing phases, and what does this imply for magnetic stability?

Key findings

• The magnetic instability in gapless superconductors originates from infrared divergences in the gapless modes, particularly when δμ ≈ Δ.
• When δμ ≈ Δ, the system develops a secondary gap Δs ≈ 4.2 (G_s/G)^2 g^4 δμ, which is power-suppressed in coupling and stabilizes the system.
• For large δμ > Δ, the system avoids gap formation and instead forms a supercurrent condensate, breaking rotational symmetry.
• In the supercurrent condensate phase, the Meissner mass is positive for magnetic fields perpendicular to the condensate direction but zero for parallel fields.
• The effective four-Fermi interaction for gapless modes becomes relevant due to anomalous scaling (Ψ ∝ s^−1/4), driving the instability unless a gap opens.
• The secondary gap is not exponentially suppressed but scales as δμ times a power of the coupling, which may have implications for neutron stars and ultracold atomic gases.