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[Paper Review] Phase stiffness in flat-band superconductors with nodal pairing

A. Yu. Zyuzin, A. Yu. Zyuzin|arXiv (Cornell University)|Mar 12, 2026
Iron-based superconductors research0 citations
TL;DR

The paper analyzes a two-band model with momentum-dependent interband hybridization between a dispersive band and a flat band, showing a parabolic node in the flat-band quasiparticle spectrum that yields a quadratic temperature dependence of the superconducting phase stiffness at low T, and discusses disorder effects.

ABSTRACT

We study Bogoliubov quasiparticle spectrum in a two-band system with momentum-dependent hybridization between a dispersive band and a flat band. The interplay between the interband mixing and intraband Cooper pairing may give rise to a parabolic node in the spectrum of flat band quasiparticles, resulting in a quadratic temperature dependence of the superconducting phase stiffness at low temperatures. We also comment that nonmagnetic disorder induces Machida-Shibata deep subgap resonances suggesting the sensitivity of flat-band superconductivity to disorder.

Motivation & Objective

  • Motivate understanding of superconductivity when pairing and phase coherence are at different scales in flat-band systems.
  • Develop a minimal two-band model with momentum-dependent interband hybridization that produces a flat band separated by a gap from a dispersive band.
  • Analyze the quasiparticle spectrum and identify conditions leading to a parabolic nodal structure in the flat-band sector.
  • Compute the superconducting phase stiffness and its temperature dependence in the presence of asymmetric gap functions.

Proposed method

  • Diagonalize the two-band Hamiltonian with momentum-dependent interband hybridization to obtain the band dispersions.
  • Formulate the BdG Hamiltonian with intraband pairing gaps Δ1 and Δ2 in the spin-singlet, band-triplet channel.
  • Derive the quasiparticle spectrum at charge neutrality and in the asymmetric gap case, highlighting the parabolic node.
  • Compute the superconducting current and phase stiffness D from the Matsubara Green function in Nambu space.
  • Obtain analytic expressions for D as a function of T and Δ2, and discuss limits where D shows quadratic T-dependence.
Figure 1: Plot of the band dispersion $\epsilon_{{\bm{k}}}$ in Eq. ( 4 ), normalized by $\eta$ for different values of the parameter $2mv^{2}/\eta=0,0.1,0.9$ . As this parameter increases, the lower band flattens.
Figure 1: Plot of the band dispersion $\epsilon_{{\bm{k}}}$ in Eq. ( 4 ), normalized by $\eta$ for different values of the parameter $2mv^{2}/\eta=0,0.1,0.9$ . As this parameter increases, the lower band flattens.

Experimental results

Research questions

  • RQ1Does momentum-dependent interband hybridization between a flat band and a dispersive band generate nodes in the flat-band quasiparticle spectrum?
  • RQ2How does the phase stiffness D(T) behave at low temperatures in a nearly flat-band superconductor with nodal pairing?
  • RQ3What is the impact of gap asymmetry (Δ1 ≠ Δ2) on the flat-band quasiparticle spectrum and phase stiffness?
  • RQ4How do nonmagnetic impurities affect subgap states and the stability of superconductivity in this flat-band model?

Key findings

  • A parabolic nodal structure can arise in the flat-band quasiparticle spectrum due to interband mixing and pairing.
  • The phase stiffness D exhibits a quadratic temperature dependence at low T when the lower-flat-band spectrum has a parabolic node.
  • In the symmetric Δ1 = Δ2 case the flat band is fully gapped and D is exponentially suppressed at T ≪ |Δ2|.
  • In the asymmetric Δ1 ≠ 0, Δ2 = 0 (or vice versa) limit the node appears and D scales with T in a way that reflects the node.
  • For small T and Δ2, D ≈ |Δ2|^2/(24πT) and D ≈ |Δ2|/8π at very low T with a correction term, indicating a linear-in-gap scaling of phase stiffness in the appropriate regime.
  • Using a BKT-inspired bound, T_BKT is estimated as (π/2)D with D evaluated at T_BKT, yielding T_BKT/|Δ2| ≈ 1/16 as an upper bound.
Figure 2: Plot of the lower band dispersion in Eq. ( 22 ), $E_{{\bm{k}},-}$ , normalized by $\eta$ as a function of $\lambda k$ for different values of the parameters: dashed curves correspond to the $\pi$ -shift case $\Delta_{1}=-\Delta_{2}=0.3\eta$ ; the curves that vanish at the origin correspond
Figure 2: Plot of the lower band dispersion in Eq. ( 22 ), $E_{{\bm{k}},-}$ , normalized by $\eta$ as a function of $\lambda k$ for different values of the parameters: dashed curves correspond to the $\pi$ -shift case $\Delta_{1}=-\Delta_{2}=0.3\eta$ ; the curves that vanish at the origin correspond

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This review was created by AI and reviewed by human editors.