Quantum Geometry Induced Kekulé Superconductivity in Haldane phases

TL;DR

Problem addressed: identifying how band topology in inversion-symmetric Haldane phases of chiral 2DEGs shapes superconducting instabilities. Approach: band-projected pairing analysis in Chern bands with Chern number $J$, showing coherence factors yield enhanced intra-valley pairing at ${\bf Q}= \pm 2 {\bf K_D}$ and suppressed inter-valley pairing at ${\bf Q}=0$, governed by $m/\mu$. Key findings: longitudinal acoustic phonon–mediated pairing supports a chiral Kekulé superconducting state; for even $J$, intra-valley Kekulé dominates at low densities with possible $s$-wave Kekulé at higher densities, while for odd $J$ a $\pi$-phase Kekulé state appears at low density and other phases arise as density increases. Significance: the Kekulé order stems from the quantum geometry of the Haldane phase, is robust to trigonal warping, and may be realized in rhombohedral graphene and Kagome metals, offering a topologically protected route to lattice-scale PDW superconductivity.

Abstract

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 (${\bf Q} = \pm 2 {\bf K_D}$) pair susceptibility relative to inter-valley (${\bf Q} = 0$) pair susceptibility, favoring the emergence of a lattice-scale pair-density wave order. When longitudinal acoustic phonons mediate the pairing interaction, the system supports a chiral Kekulè superconducting order. Our findings are relevant to superconductivity in rhombohedral graphene and Kagome metals.

Figures (7)

• Figure 1: Hamiltonian pseudo-spinor field $\hat{{\bf h}}$ on the Bloch sphere for a) intra-valley pairing and b) inter-valley pairing in the Haldane phase, with the Fermi surface projection indicated by the dotted red (blue) circles for ${\bf K}_D\,(-{\bf K}_D)$ valleys. The projections of $\hat{{\bf h}}$ onto the $\hat{z}$-direction for intra-valley pairs ${\bf p}$ and $-{\bf p}$ are parallel, whereas they are anti-parallel for inter-valley pairs ${\bf p}$ and $-{\bf p}$. This spinor structure of the Haldane phase results in the suppression of the coherence factors of the inter-valley pair susceptibility (see text for details).
• Figure 2: Feynman diagrams for the a) intra- and b) inter-valley pair susceptibility of chiral 2DEGs. The solid (dotted) lines denote the band projected electron propagator of the ${\bf K}_D(-{\bf K}_D)$ valleys.
• Figure 3: a) Feynman diagrams for inter-valley scattering interaction $G_1$ and inter-valley exchange interaction $G_2$, and b) Feynman diagrams for intra-valley scattering interaction $G_4$ and pair tunnelling $G_3$. The solid (dotted) lines correspond to the electron propagator in the valley ${\bf K}_D$$(-{\bf K}_D)$. The colors correspond to the microscopic interactions on the circular Fermi surfaces for both inter-valley and intra-valley pairing.
• Figure 4: a) The ratio of the inter-valley critical temperature ($T_c^D$) to the intra-valley critical temperature ($T^S_c$) as a function of $x$. b) Phase diagram for the even chirality C2DEG model of $J$-layer RG with $J=4,6$ and $8$ in increasing order. The electron density $n_{e}$ is in units of cm$^{-2}$ and the Haldane mass is in meV. At low densities, $n_e < n_{e,2}$ the chiral-$J$ Kekulé superconducting channel dominates. c) Phase diagram for the odd chirality C2DEG model of $J$-layer RG with $J=3,5$ and $7$ in increasing order, with the same axis label as in b). The chiral-$J$ Kekulé superconducting channel dominates for $n_e < n_{e,2}$, giving way to ${\bf Q} =0$ valley triplet chiral superconductivity for $n_{e,1} < n_e < n_{e,2}$, with the phase boundary indicated by the solid line. Conventional valley singlet s-wave superconductivity dominates for $n_e >n_{e,2}$, with the dashed line indicating the phase boundary. d) Schematic of the Kekulé superconductor on the projected honeycomb lattice, with the superconducting unit cell indicated by the dotted lines.
• Figure 5: Tree-level Feynmann diagram corresponding to the self-consistent solution to the gap equation for the mean field $\hat{\Delta}$.