Superfluid stiffness of superconductors with delicate topology

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.

💡 Research Summary

The authors investigate superconductivity in two‑dimensional “delicate topological” bands—systems whose total Chern number vanishes but whose Brillouin zone can be partitioned into sub‑regions (Ω_i) each carrying a quantized Chern number C_i. By assuming that the occupied bands occupy a fixed set of orbitals on the boundaries of these regions, they define a gauge‑invariant integer C_i for each sub‑zone. In the presence of mirror symmetry, especially when the mirror eigenvalue is the same for all occupied bands along every high‑symmetry line (the “iso‑orbital” case), these C_i are basis‑independent.

The superconducting state is treated within mean‑field theory. The superfluid stiffness tensor D is split into a conventional part (intraband) and a geometric part (inter‑band) that involves matrix elements ⟨∂_μ m_k|n_k⟩⟨n_k|∂_ν m_k⟩. Because the quantum geometric tensor G_B(k)=M_B(k)−iF_B(k) is positive‑semidefinite, the determinant of D_geom can be bounded from below. The key result is

 det D ≥ (2π)^2 |Δ|^4