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

TL;DR

This work analyzes how non-magnetic disorder affects the superfluid weight \\mathcal{D}_s in flat-band superconductors, revealing that the change is governed by a competition between interband and intraband localization functionals of impurity-bound states. By deriving analytic expressions for the disorder-induced changes and validating them with self-consistent numerics on the Creutz ladder and Lieb lattice, the authors show that these two localization functionals typically have comparable magnitudes, making the flat-band superfluid weight robust to disorder up to disorder strengths on the order of the gap, \\Delta. The key insight is that quantum geometry, encoded in the gauge-invariant \\Omega_I and intraband \\tilde{\\Omega} functionals, controls the balance between suppression of interband geometric contributions and disorder-enabled intraband currents. The results offer a general perspective on the resilience of flat-band superconductivity to disorder and suggest a geometric mechanism by which supercurrents persist even as the order parameter is weakened, with potential extensions to dispersive regimes and other rank-1 disorder models. Practical implications include experimental probes of stability against disorder in moiré- and Wannier-based flat-band platforms and a framework for understanding disorder-induced transport in quantum geometric materials.

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.

Figures (4)

• Figure 1: (a) Schematic illustrating the system considered. Here $W$ is the disorder strength in units of the largest hopping strength, and $|a\rangle$ and $|r\rangle$ are the states in the band of interest and the other bands, respectively. (b) Illustration of the principal result of this work. (c) Mean-field superfluid weight as a function of $W$ for the one-dimensional (1D) Creutz ladder (red) and two-dimensional (2D) Lieb lattice (teal). The interaction strength $|U|=1$. See Supplementary Material (SM), Section \ref{['suppl:numericsdetails']} for details of the calculation. (d) Same as (c) but for the averaged mean-field gap. (e) Ratio of the mean-field superfluid weight to the mean-field gap as a function of $W/\overline \Delta$. Our analytical results predict $\delta \mathcal{D}_s|_{\hat{\Delta}} \approx 0$ up to $W/\overline \Delta=1$.
• Figure 2: (a) Superfluid weight as a function of disorder for the 1D Creutz ladder. The full weight is shown in red, and the value without intraband terms in teal. (b) The intraband contribution to the superfluid weight (full red), obtained as the difference between the full and the value without the intraband terms. The analytical expression is shown in dashed teal. (c) Same as (a) but for the 2D Lieb lattice. (d) Same as (b) but for the 2D Lieb lattice. The interaction used was $U=1$ in units of hopping. A gap was opened in the Lieb lattice by asymmetric hoppings. See SM \ref{['suppl:numericsdetails']} for parameters and numerical procedures.
• Figure S1: Schematic of the Lieb (a) and Creutz (b) lattices used. The unit cell is enclosed in a grey frame.
• Figure S2: The localization functionals $\Omega_I(\ket{\mathrm{BS}})$ and $\tilde{\Omega}(\ket{\mathrm{BS}})$ as a function of orbital positions parametrized by $x$ in the unit cell. Here $x=0$ corresponds to all orbitals being at the same point, while $x=1$ corresponds to the orbitals arranged as in Fig. \ref{['fig:figlattices']}a.