Quantum Geometry Induced Kekul\'{e} Superconductivity in Haldane phases

Chiral two-dimensional electron gases, which capture the electronic properties of graphene and rhombohedral graphene systems, exhibit singular momentum-space vortices and are susceptible to interaction-induced topological Haldane phases. Here, we investigate pairing interactions in these inversion-symmetric Haldane phases of chiral two-dimensional electron gases. We demonstrate that the nontrivial band topology of the Haldane phases enhances intra-valley ($\Qq = \pm 2 \Kk_D$) pair susceptibility relative to inter-valley ($\Qq = 0$) pair susceptibility, favoring the emergence of a lattice-scale pair-density wave order at low densities. When longitudinal acoustic phonons mediate the pairing interaction, the superconducting phase diagram is determined by the parity of the chiral index $J$. For even values of $J$, the system supports a chiral Kekul\`{e} superconducting order below a critical electron density $n_e$ with $n_e < n_{e,2}$, which transitions to an s-wave Kekul\`{e} superconducting order. For odd values of $J$, a $\pi$-phase chiral Kekul\`{e} superconducting order appears below a critical density $n_e < n_{e,1}$, while in the density regime $n_{e,2} n_{e,1}$. Our findings are relevant to superconductivity in rhombohedral graphene and Kagome metals.