Electromagnetic and spin Hall responses in chiral topological superconductors

We present a fully quantum mechanical derivation of the electromagnetic and spin response of a chiral topological superconductor that allows us to include fluctuations of the order parameter self consistently. By rewriting the electromagnetic and spin transport response kernels in terms of dressed vertices, we extract the Hall parts of the charge-charge, spin-charge and spin-spin responses. We show that although the spin-spin Hall response is quantized when the superconducting pairing preserves spin rotation symmetry, the spin-charge and the charge-charge responses are not quantized. We also show that none of the three responses is quantized when the superconducting pairing spontaneously breaks spin rotation symmetry. To understand the physical significance of these results, we identify an operator whose projected spectrum determines the leading contribution to the nonquantized spin-charge response. We conclude by applying our results to recently introduced models of superconductivity in twisted TMDs.