Quantum geometric anomalous Hall response in orbitally nonunitary superconductors

Abstract

We investigate the anomalous Hall response (AHR) in a multiband superconductor at optical frequencies, a phenomenon intimately related to the polar Kerr effect, a key probe of time-reversal symmetry breaking in superconductors. In translationally invariant multiband systems with purely intraband pairing, Galilean invariance decouples center-of-mass and relative motion of Cooper pairs, leading to the widespread expectation that a finite AHR requires either disorder or finite interband pairing amplitudes. However, this restriction can be lifted by the quantum geometric effects inherent to multiband Bloch states. Using a honeycomb lattice tight-binding model with Kane-Mele spin-orbit coupling, we analyze the AHR for the time-reversal symmetry broken chiral $d$-wave spin-singlet and chiral $p$-wave equal-spin-triplet pairing states with intraband pairing only. We demonstrate, through both analytical and numerical calculations, that the spin-singlet state yields a vanishing AHR, even with its broken time-reversal symmetry, whereas the equal-spin triplet state exhibits a finite AHR, even when it is spin-unitary. We attribute the latter to orbital nonunitarity, which, in the presence of spin-orbit coupling, generates the spin-polarized Bogoliubov quasiparticle states required for a finite AHR. The response is mediated by interband velocity matrix elements governed by the quantum geometry. This finding establishes that spin-unitary, but orbitally nonunitary pairing, can generate a finite AHR even without interband pairing and thereby revises the criteria for Kerr signals in superconductors.