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arXiv:2510.05224 · Physical review. B./Physical review. B

Superfluid weight in disordered flat-band superconductors as a competition between localization functionals

Kryštof Kolář, Tero Tapio Heikkilä, Päivi Törmä

Published Jun 18, 2026• 1 citation
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Abstract

According to Anderson's theorem, the gap of a time-reversal symmetric weak-coupling superconductor is unaffected by non-magnetic disorder. However, the superfluid weight (stiffness) is reduced in the disordered limit by a factor of $Δτ$, a product of the scattering time $τ$ and the superconducting order parameter $Δ$. Here we show that the opposite holds true in flat-band superconductors. While non-magnetic disorder does reduce the order parameter, we find that its direct effect on superfluid weight is mostly negligible. We show analytically that the effect of disorder is to lowest order given in terms of the difference between the intraband and interband parts of the localization functional of impurity wavefunctions, finding it to be typically vanishing.

DOI
10.1103/yxdx-92r8

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According to Anderson's theorem, the gap of a time-reversal symmetric weak-coupling superconductor is unaffected by non-magnetic disorder. However, the superfluid weight (stiffness) is reduced in the disordered limit by a factor of ΔτΔτΔτ, a product of the scattering time τττ and the superconducting order parameter ΔΔΔ. Here we show that the opposite holds true in flat-band superconductors. While non-magnetic disorder does reduce the order parameter, we find that its direct effect on superfluid weight is mostly negligible. We show analytically that the effect of disorder is to lowest order given in terms of the difference between the intraband and interband parts of the localization functional of impurity wavefunctions, finding it to be typically vanishing.
Universal suppression of superfluid weight by non-magnetic disorder in sss-wave superconductors independent of quantum geometry and band dispersion
Motivated by the experimental progress in controlling the properties of the energy bands in superconductors, significant theoretical efforts have been devoted to study the effect of the quantum geometry and the flatness of the dispersion on the superfluid weight. In conventional superconductors, where the energy bands are wide and the Fermi energy is large, the contribution due to the quantum geometry is negligible, but in the opposite limit of flat-band superconductors the superfluid weight originates purely from the quantum geometry of Bloch wave functions. Here, we study how the energy band dispersion and the quantum geometry affect the disorder-induced suppression of the superfluid weight. In particular, we consider non-magnetic disorder and s-wave superconductivity. Surprisingly, we find that the disorder-dependence of the superfluid weight is universal across a variety of models, and independent of the quantum geometry and the flatness of the dispersion. Our results suggest that a flat-band superconductor is as resilient to disorder as a conventional superconductor.
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We investigate the disordered flat-band superconductivity within the attractive Hubbard model on the kagome lattice by contrasting the flat-band-preserving disorder [Phys. Rev. B 98, 235109 (2018)] with the random hopping disorder that breaks the flat-band degeneracy. Through Bogoliubov-de Gennes mean-field calculations, we find that the superfluid weight is much more robust under the flat-band-preserving disorder, while the system eventually undergoes a transition to an insulator as disorder becomes strong enough. The almost linear interaction-dependence of the superfluid weight in the weak coupling limit found with the flat-band-preserving disorder confirms the persistent flat-band signature, whereas the exponential behavior of a dispersive-band character arises with the random hopping counterpart. In addition, in the exact diagonalization of the one-particle density matrix, we identify an occupation spectrum structure attributed to the flat-band states, demonstrating the connection between the resilient flat band and the enhanced robustness of superconductivity.
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