Gauge-invariant electromagnetic response of a chiral $p_x+i{p}_y$ superconductor

We present a gauge-invariant theory of the electromagnetic response of a chiral $p_x+i{p}_y$ superconductor in the clean limit. Due to the spontaneously broken time-reversal symmetry, the effective action of the system contains an anomalous term not present in conventional superconductors. As a result, the electromagnetic charge and current responses contain anomalous terms, which explicitly depend on the chirality of the superconducting order parameter. These terms lead to a number of unusual effects, such as coupling of the transverse currents to the collective plasma oscillations and a possibility of inducing the charge density by the magnetic field perpendicular to the conducting planes. We calculate the antisymmetric part of the conductivity tensor (the intrinsic Hall conductivity) and show that it depends on the wave vector of the electromagnetic field. We also show that the Mermin–Muzikar magnetization current and the Hall conductivity are strongly suppressed at high frequencies. Finally, we discuss the implications of the theory to the experiments in ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$.

Figure 1

(Color online) Schematic picture of the experimental setup in Ref. 3. The incoming laser beam is incident normally to the conducting planes of the layered superconductor ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$. The beam is focused by the lens to a spot of the diameter $l$ at the surface of the sample. The solid (red) line shows the Gaussian profile of the polarized electric field $E_x$ in the beam (Ref. 7). The interlayer distance in ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$ is $d$.

Figure 2

Renormalization of the polarization functions (33) due to the Coulomb interaction $V\left(\mathit{q}\right)$ in the RPA. The wavy line denotes $V\left(\mathit{q}\right)$, which couples only to the charge vertices represented by the Pauli matrices ${\tau }_3$.

Figure 3

(Color online) Frequency dependence of the function $I\left(\omega \right)$ given by Eq. (73). This function determines the frequency dependence of the ac Hall conductivity (104) of a chiral $p_x+i{p}_y$ superconductor in the dynamic limit. The solid (red) and dashed (blue) lines show the real and imaginary parts of $I\left(\omega \right)$.

Figure 4

(Color online) Schematic setup of a proposed experiment to measure the electric charge induced by a magnetic field perpendicular to the layers in ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$, as given by Eq. (106). A current in the solenoid or the coil produces a magnetic field $B_z$ inducing an electric charge at the center, which is carried by the radial currents shown by the arrows. The induced charge is detected by the AFM tip.