Intrinsic Hall Effect in a Multiband Chiral Superconductor in the Absence of an External Magnetic Field

We identify an intrinsic Hall effect in multiband chiral superconductors in the absence of a magnetic field (i.e., an anomalous Hall effect). This effect arises from interband transitions involving time-reversal symmetry-breaking chiral Cooper pairs. We discuss the implications of this effect for the putative chiral $p$-wave superconductor, ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, and show that it can contribute significantly to Kerr rotation experiments. Since the magnitude of the effect depends on the structure of the order parameter across the bands, this result may be used to distinguish between different models proposed for the superconducting state of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

Contributions to the intrinsic Hall conductivity in a multiband chiral superconductor in the orbital basis, where $a$, $b$ label the orbitals and $i$, $j$ denote the photon polarization. Double lines denote the Green’s function (3). At one vertex, a photon of frequency $\nu$ induces a $b\to a$ interorbital transition. The time-reversed process on the right is subtracted to yield the Hall conductivity. As discussed in the text, the only nonzero contributions at $T=0$ result from the creation or annihilation of a $\pm$ quasiparticle pair. At finite $T$, scattering between (but not within) the $\pm$ quasiparticle branches also contribute.

Figure 2

Real (blue solid line) and imaginary (red dashed line) parts of the $T=0$ Hall conductivity in units of $e^2/\hslash$ as a function of $\hslash \omega$ for ${\Delta }_0=0.23\mathrm{meV}$ and $t=\mu =10t^{\prime }=0.4\mathrm{eV}$. Dotted line shows asymptotic high-frequency expression (see text).

Figure 3

Real (solid line) and imaginary (dashed line) parts of the $T=0$ Hall conductivity in units of $e^2/\hslash$ as a function of $\mu /t$ for ${\Delta }_0=0.23\mathrm{meV}$ and $t=\mu =10t^{\prime }=0.4\mathrm{eV}$.