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Hall conductivity in the normal and superconducting phases of the Rashba system with Zeeman field

We study the intrinsic Hall conductivity of the ordinary and topological superconducting phases of a Rashba metal in a perpendicular Zeeman field. In this system the normal metal breaks time reversal symmetry while the superconducting order parameter does not, in contrast to the chiral p-wave superconducting state predicted in the monolayer strontium ruthenate (Sr$_2$RuO$_4$) whose Hall conductivity has been studied extensively. We study the effects of intra-band and inter-band pairing and find there is qualitatively larger change in the intrinsic Hall conductivity when there is inter-band pairing, with the change in magnitude linear in the pairing gap. We argue that inter-band pairing leads in general to higher energy costs for the topological phase compared to the topologically trivial phase and thus that the qualitative behavior of the intrinsic Hall conductivity with superconductivity in these systems could provide important clues about the nature of pairing in the superconducting phase and even some hints of whether it is topological or not.