Enhanced Kohn-Luttinger topological superconductivity in bands with nontrivial geometry

TL;DR

This work investigates how the electronic wavefunction geometry, encoded in the form factor Λ(k,q), influences Kohn-Luttinger superconductivity in spin- and valley-polarized metals. By projecting to bands near the Fermi surface and incorporating band geometry into the gap equation via Λ, the authors show that the geometry can both suppress and enhance Tc, with dramatic exponential Tc boosts in a Lowest Landau Level (LLL) form factor and resonances tied to the Berry flux enclosed by the Fermi surface. They derive analytical insights (e.g., λ_l ≈ J_l(β B k_f^2) for odd l in the large-mass limit) and demonstrate that intravalley pairing is generally favored over intervalley pairing under these geometric effects. Applying the framework to rhombohedral graphene multilayers and a two-band model, they illustrate substantial Tc enhancements with increasing layer number and displacement field, suggesting band-geometry engineering as a viable path to high-Tc topological superconductivity in graphene-based systems.

Abstract

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.

Figures (6)

• Figure 1: (a) Linearized gap equation including the effects of band geometry as captured by the form factor $\Lambda(\bm k, \bm q)$. (b) The difference between intervalley pairing and intravalley pairing. The form factor adds a correction to the pairing vertex that is real because of time-reversal symmetry for intervalley pairing. However in a valley and spin-polarized metals, time-reversal symmetry is broken in the normal state, and the form factors correction is generally complex.
• Figure 1: The renormalized interactions under the random phase approximation. We modify every vertex to include the effects of band projection.
• Figure 2: Effect of band geometry on the bubble (a) and renormalized interactions (b). Panel (c) shows the Fourier transform of the renormalized interactions for different angular momentum channels for the cases with trivial band geometry $|\mathcal{W}|=1$ (squares), and non-trivial band geometry (circles). Non-trivial band geometry gives rise to an attractive channel that is otherwise absent. In (d) we plot $T_c$ for the case of intravalley pairing in units of $W$, the energy cutoff, for various values of $\beta$. $\beta = 1$ corresponds to the ideal band geometry limit, which provides the highest $T_c$. The corresponding superconducting order parameters at the Fermi surface are displayed with arrows representing the phase of the order parameter. Panel (e) compares the critical temperatures of intervalley pairing with intravalley pairing, and the intravalley pairing is significantly favored. We can analytically show in the case of large electron mass that the coupling constant for any odd pairing channel is given by the Bessel function, and panel (f) shows that it agrees well with the numerical results of pannel (d).
• Figure 2: Results of the effects of band projection for the $\zeta$-lattice defined in Ref. Hofmann_2022. $T^{\text{intra}}_c$ is the critical temperature in a valley polarized setting and thus intravalley pairing, while $T^{\text{inter}}_c$ is the critical temperature for intervalley pairing. $T^{\text{inter}}_{c0}$ represents the critical temperature for intervalley coupling with all the vectors are polarized in the same direction on the Fermi surface.
• Figure 3: Superconducting critical temperature using quartic dispersion. The top panels of (a) and (b) show the critical temperature of intravalley pairing $T^{\text{intra}}_{c}/W$ and intervalley $T^{\text{inter}}_{c}/W$ respectively as a function of $B$. Intravalley pairing has a higher critical temperature for all values of $B$. The bottom panels show the same quantities but as a ratio of $T^{\text{inter}}_{c0}$ which is the critical temperature of intervalley pairing with trivial band geometry $|\mathcal{W}|=1$. The case of intravalley pairing with the LLL band geometry always leads to an enhanced $T_c$. However, for the intervalley pairing, the effect of band geometry can suppress $T_c$. Top row displays the corresponding superconducting order parameter at Fermi surface.