Intrinsic Hall Effect in Multiband Superconductors : The Role of Quantum Geometry

Time-reversal symmetry (TRS) breaking in superconductors leads to novel electromagnetic responses, such as the polar Kerr effect, which is closely related to the anomalous Hall effect (AHE). While clean single-band superconductors generally exhibit no intrinsic Hall response, multiband systems can host a nonzero Hall effect in the presence of unconventional pairing, owing to a finite Berry curvature in the normal state. This thesis develops a microscopic framework for understanding intrinsic Hall effects in multiband superconductors, with particular emphasis on intraband contributions. The additional internal degrees of freedom (DOF) in these systems, beyond momentum and spin, referred to here as orbital DOF, introduce qualitatively new features compared to the single-band case. One such feature is that the order parameter can be nonunitary in the orbital channel, leading to an induced orbital polarization in the superconducting state. Previous work has shown that spin-triplet pairing, nonunitary in the spin DOF, induces spin polarization that can bias the normal-state spin Berry curvature, originating from spin-orbit coupling (SOC), resulting in a finite AHE even without interband pairing. Here, we explore how nonunitary pairing in the orbital channel can produce a similar effect. We show that superconducting states nonunitary in the orbital DOF can give rise to a finite AHE from purely intraband pairing, even if they remain unitary in the spin channel. In this scenario, spin polarization is induced indirectly through the interplay of SOC and orbital polarization. This effect biases the normal-state spin Hall effect, enabled by SOC, into a net Hall signal. Importantly, the pairing must occur between equal-spin states (i.e., equal-spin triplet pairing), as the Berry curvature contributions from opposite spins cancel. As a result, spin-singlet order parameters yield no Hall response from this mechanism. These conclusions are supported by numerical calculations using a tight-binding model on the honeycomb lattice, focusing on chiral spin-singlet and spin-triplet pairing states. The results offer new insight into the microscopic origin of the Kerr effect in TRS-breaking superconductors.