Superfluid stiffness of superconductors with delicate topology

TL;DR

The paper investigates how delicate topology in two-dimensional bands, where the total Chern number vanishes yet sub-BZ Chern numbers are nonzero, influences superconductivity. It develops a basis-invariant bound on the geometric part of the superfluid weight in terms of the sub-BZ Chern numbers, and demonstrates this bound via Chern dartboard insulators with iso- and aniso-orbital symmetry. In iso-orbital cases the bound scales linearly with the number of mirrors, offering a route to particularly robust superconductivity, especially in flat bands; in dispersive bands the bound provides a conservative estimate for the geometric contribution. The findings highlight the significance of quantum geometry and delicate topology for stabilizing superconductivity and point to experimental platforms and broader transport phenomena where these effects may be observed.

Abstract

We consider superconductivity in two-dimensional delicate topological bands, where the total Chern number vanishes but the Brillouin zone can be divided into subregions with a quantized nontrivial Chern number. We formulate a lower bound on the geometric contribution to the superfluid weight in terms of the sum of the absolute values of these sub-Brillouin zone Chern numbers. We verify this bound in Chern dartboard insulators, where the delicate topology is protected by mirror symmetry. In iso-orbital models, where the mirror representation is the same along all high-symmetry lines, the lower bound increases linearly with the number of mirror planes. This work points to delicate bands as promising candidates for particularly stable superconductivity, especially in narrow bands where the kinetic energy is suppressed due to lattice effects.

Figures (5)

• Figure 1: Berry curvature $\mathcal{F}^{\mathcal{B}}_{xy}$ in the first Brillouin zone for (a) the model with one mirror symmetry, and (b) the model with two mirrors (see supplemental material supplemental for Hamiltonians). Positive values of $\mathcal{F}^{\mathcal{B}}_{xy}$ are shown in red, and negative in blue. Dashed lines indicate mirror high-symmetry lines that partition the Brillouin zone into subregions with well defined quantized Chern numbers.
• Figure 2: Zero-temperature superfluid weight (blue) in flat-band models with (a) $n_{\rm M}=1$, (b) $n_{\rm M}=2$, (c) $n_{\rm M}=3$ and (d) $n_{\rm M}=4$ mirrors, as a function of the interaction $U$. The red dashed lines indicate the bound derived in terms of the Chern number in the irreducible Brillouin zone [Eq. \ref{['eq:flat_band_bound']}]. The slope of the bound grows linearly with $n_{\rm M}$. The bound is derived for an isolated flat band with uniform pairing, and holds for low interactions. The pairing is not perfectly uniform for all models, but the bound still holds. The difference in on-site energies of the two orbitals is set to $m=0.25$, a value at which the pairing is almost uniform for $n_{\rm M}=1$ and $n_{\rm M}=3$ (see Fig. \ref{['fig:flatband_param']}). The lowest flat band is half-filled.
• Figure 3: Upper panels: Superfluid weight (blue) for flat-band models with (a) $n_{\rm M}=1$, (b) $n_{\rm M}=2$, (c) $n_{\rm M}=3$ and (d) $n_{\rm M}=4$ mirror symmetries, together with the bound from Eq. \ref{['eq:flat_band_bound']} (red dashed line), as a function of $m$. At $m=0$ and $m=2$, the gap closes in the single-particle spectrum, and $|C|=1$ in the irreducible BZ in between these values. Lower panels: Order parameters $\Delta_1$ and $\Delta_2$. The UPC is fulfilled when $\boldsymbol{\Delta} \propto \mathbb{1}$, i.e. when the lines intersect, which occurs only for $n_{\rm M}=1,3$ close to $m\approx0.25$. In all panels, the filling fraction of the lowest band is set to $\nu=1/2$, and the interaction to $U=-1$.
• Figure 4: Superfluid weight computed for dispersive models with (a) $n_{\rm M}=1$, (b) $n_{\rm M}=2$, (c) $n_{\rm M}=3$ and (d) $n_{\rm M}=4$, together with the bound from Eq.\ref{['eq:final_bound']} (red dashed line). The blue line corresponds to the geometric part $D^{\rm geom}$, and the green line to the total superfluid weight including also the conventional contribution $D^{\rm conv}$. The ratio $\Delta_{1}/\Delta_2$ is evaluated in the $U\to0$ limit to evaluate how well the UPC is satisfied, $\Delta_1/\Delta_2=1$ indicating uniform pairing. The UPC is fulfilled only for $n_{\rm M}=1$, for other models the bound serves only as an approximation. The filling fraction is set to $\nu=1/2$, and $m=0.25$.
• Figure 5: The trace and determinant of the integrated quantum metric for a) the aniso-orbital and b) the iso-orbital model. The minimum of the blue line corresponds to the minimal quantum metric. In the aniso-orbital case, the trace of integrated metric is finite, while the determinant vanishes, implying that superconductivity is unstable in the flat-band limit. For this particular model, it is unstable even without flattening due to the band dispersion being independent of $x$.