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 $\left({\mathrm{Sr}}_2{\mathrm{RuO}}_4\right)$ whose Hall conductivity has been studied extensively. We study the effects of intraband and interband pairing and find there is qualitatively larger change in the intrinsic Hall conductivity when there is interband pairing, with the change in magnitude linear in the pairing gap. We argue that interband 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.

Figure 1

Contribution to intrinsic Hall conductivity in the band basis, where $s,\overline{s},s^{\prime }$ label bands. Filled and dotted curves denote the normal and anomalous Green function, respectively. In both the normal state and the superconducting state with purely intraband pairing, there is a contribution only from the diagram (a), which originates from the propagation of the electron in the upper band and the hole in the lower band (or vice versa for the superconducting state). However, the diagram (b) shows that, with a nonzero interband pairing, there is a possibility of interband propagation; this particular diagram involves the interband normal propagation coming from the combination of the intraband normal and anomalous propagation with interband pairing, giving rise to additional processes contributing to the Hall conductivity.

Figure 2

Comparison between the integrand of the Hall conductivity of the normal state in Eq. (7) with that of (a) the purely intraband pairing superconductivity in Eq. (20) and (b) the purely $s$-wave superconductivity in Eq. (29). We have defined $k_F\equiv \sqrt{2\mu m^{*}},R_K=2\pi /e^2$ (“the effective Berry curvature” ${\mathcal{F}}^{xy}$ normalized to give the normal state value for $\left|\Delta \right|=0$) and set for both (a) and (b) $m^{*}\alpha /k_F=0.1,h/\alpha k_F=0.2$ and $\left|\Delta \right|/\alpha k_F=0.05$.