Superconductivity in a Chern band: effect of time-reversal-symmetry breaking on superconductivity

Time-reversal-symmetry breaking is generally understood to be detrimental for superconductivity. However, recent experiments found superconductivity emerging out of a normal state showing a finite anomalous Hall effect, indicative of time-reversal-symmetry breaking, in diverse systems from kagome metals, $1T'$-WS$_2$, to twisted MoTe$_2$ and rhombohedral graphene. Motivated by these findings, we study the stability of superconducting orders and the mechanisms that suppress superconductivity in the prototypical anomalous Hall system, the Haldane model, where complex hopping parameters result in loop-current order with a compensated flux pattern. We find that neither spin-singlet nor spin-triplet states are generically suppressed, but the real-space sublattice structure plays a crucial role in the stability of the orders. Interestingly, the nearest-neighbor chiral states of $d\pm id$ or $p\pm i p$ symmetry couple linearly to the flux, such that the two otherwise degenerate chiralities split under finite flux. As an experimental probe to distinguish the various orders in this system, we study the anomalous thermal Hall effect, $κ_{xy} / T$, which vanishes at zero temperature for topologically trivial superconducting states, but reaches a finite value corresponding to the Chern number in a topologically non-trivial superconducting state. Our results illustrate that broken time-reversal symmetry through a finite flux is neither generically destructive for superconductivity, nor does it imply non-trivial topological order of the emerging superconducting state. However, in the case of multiple competing pairing channels, the loop-current order can favor a chiral superconducting state.