Functional approach to superfluid stiffness: Role of quantum geometry in unconventional superconductivity

TL;DR

This work addresses how quantum geometry shapes superconducting stiffness in multiband systems with unconventional order. It develops a mean-field BCS framework that yields a general expression for the superfluid weight, decomposed into a conventional term, a geometrical term from the quantum metric, and a novel functional term arising from the momentum-dependent gap and non-Abelian Wilczek-Zee geometry. In the isolated-narrow-band regime, the functional contribution is controlled by the Wilczek-Zee connection and the two-point fidelity magnitude and is not generally reducible to the minimal quantum metric. The framework is illustrated on an extended Kane-Mele model, comparing conventional $s$-wave and chiral $d$-wave pairing, showing the functional term can be small but nonzero and that topology enhances the geometric component. The results provide a quantitative route to analyze superfluid stiffness in van der Waals and moiré superconductors with unconventional pairing.

Abstract

Nontrivial quantum geometry of electronic bands has been argued to facilitate superconductivity even for the case of flat dispersions where the conventional contribution to the superfluid weight is suppressed by the large effective mass. However, most previous work focused on the case of conventional superconductivity while many contemporary superconducting quantum materials are expected to host unconventional pairing. Here, we derive a generalized expression for the superfluid weight employing mean-field BCS theory for systems with time-reversal symmetry in the normal state and arbitrary unconventional superconducting order with zero-momentum intraband pairing. Our derivation reveals the necessity of incorporating functional derivatives of the grand potential with respect to the superconducting gap function. Through perturbative analysis in the isolated narrow-bands limit, we demonstrate that this contribution arises from quantum geometrical effects, specifically due to a nontrivial Wilczek-Zee connection. Utilizing the newly obtained expressions for the superfluid weight, we apply our framework to an extended Kane-Mele model, contrasting conventional $s$-wave superconductivity with chiral $d$-wave superconductivity.

Figures (6)

• Figure 1: Principal structure of the second derivative of the free energy, comprising the tree-level contribution from the dispersion, and the two possible contractions of the pairing interaction, which contains information about the multiorbital nature of the band structure. Here, $M^{-1}$ depends on the pairing interaction, the Bogoliubov coefficients, and the quasiparticle eigenvalues.
• Figure 2: (a) Sketch of the extended Kane-Mele model on a honeycomb lattice with sublattice $A$ (filled circles) and sublattice $B$ (empty circles) with red arrows indicating the hopping amplitudes, similar to the illustration in Ref. lau2022universal. (b) Energy bands for $t_2 = 0.349t$, $t_3 = -0.264t$, $t_4=0.026t$, $\varphi = 1.377$, and $M = 0$. In this case, the Kane-Mele model hosts a nearly dispersionless band.
• Figure 3: Kane-Mele Hamiltonian for fixed $t_2 = 0.349t$, $t_3 = -0.264t$, and $t_4 = 0.026t$ with $s$-wave superconducting order parameter whereas the Brillouin zone has been discretized by a $20 \times 20$ grid. In both plots, the numerical errors are of order $\le 1\%$. (a) For fixed interaction strength $U_0 = 3t$, the blue, red, and green curves represent the conventional, geometrical, and functional superfluid weight along the $M$-axis for fixed $\varphi = 1.377$. While the solid colored lines represent the diagonal elements, the dashed colored lines (which are zero) represent the off-diagonal elements. The black dashed vertical line indicates the change of Chern number. (b) Geometrical superfluid weight for fixed interaction strength $U_0 = 3t$ plotted against the hopping phase and the staggered onsite potential.
• Figure 4: Real and imaginary components of the chiral $d$-wave superconducting gap function, corresponding to the form factors given in Eqs. \ref{['eq:basis-functions-chiral-d-wave-hex-1']} (left) and \ref{['eq:basis-functions-chiral-d-wave-hex-2']} (right), respectively black2014chiral. The black lines indicate the hexagonal Brillouin zone.
• Figure 5: Kane-Mele Hamiltonian for fixed $t_2 = 0.349t$, $t_3 = -0.264t$, and $t_4 = 0.026t$ with chiral $d$-wave superconducting order parameter whereas the Brillouin zone has been discretized by a $20 \times 20$ grid. In all of these plots, the numerical errors are of order $\le 1\%$. (a) Logarithmic plot of the conventional (blue), geometrical (red) and functional (green) contributions to the superfluid weight along the $M$-axis for fixed $\varphi = 1.377$ with an interaction strength of $U_0 = 3t$. Note that the functional superfluid weight has a negative contribution to the total superfluid weight. (b) Geometrical and (c) functional superfluid weight for fixed interaction strength $U_0 = 3t$ plotted against the hopping phase and the staggered onsite potential.