Enhanced Kohn-Luttinger topological superconductivity in bands with nontrivial geometry

We study the effect of the electron wavefunction on Kohn-Luttinger superconductivity. The role of the wavefunction is encoded in a complex form factor describing the topology and geometry of the bands. We show that the electron wavefunction significantly impacts the superconducting transition temperature and superconducting order parameter. We illustrate this using the lowest Landau level form factor and find exponential enhancement of Tc for the resulting topological superconductor. We find that the ideal band geometry, which favors a fractional Chern insulator in the flat band limit, has an optimal Tc. Finally, we apply this understanding to a model relevant to rhombohedral graphene multilayers and unravel the importance of the band geometry for achieving robust superconductivity.

💡 Research Summary

The manuscript investigates how the internal structure of Bloch wavefunctions—encoded in a complex form factor—modifies the Kohn‑Luttinger (KL) mechanism of superconductivity. In the conventional KL picture, purely repulsive Coulomb interactions generate an effective attraction only at high angular momentum, leading to extremely low critical temperatures (Tc). The authors project the interacting Hamiltonian onto the low‑energy bands and introduce the form factor Λτ(k,q)=⟨uτ(k)|uτ(k+q)⟩, where |uτ(k)⟩ is the periodic part of the Bloch state and τ labels valley (or spin) degrees of freedom. This factor appears both in the screened interaction (via the polarization bubble) and directly in the pairing vertex.

They decompose Λ into a magnitude |W|≤1 and a phase e^{iF}. The magnitude measures the quantum distance between Bloch states (the Fubini‑Study metric) and always reduces the effective pairing strength because it multiplies the interaction by a factor ≤1. The phase, however, carries the Berry connection and can break the degeneracy between opposite angular‑momentum channels (l and –l). When time‑reversal symmetry is broken (as in a spin‑ and valley‑polarized metal), the phase becomes non‑trivial, enhancing one channel while suppressing the other, which can raise Tc dramatically.

A concrete illustration uses the lowest Landau level (LLL) form factor
ΛLLL(k,q)=exp