Observation of the polar Kerr effect in $\mathrm{Sr_2RuO_4}$ [1], a layered material considered to realize the chiral $p_x+ip_y$ superconducting state, has lead to extensive theoretical investigations of the anomalous Hall response $\sigma_{xy}(\omega)$ in $p_x+ip_y$ superconductors. These studies consider either multi-band superconductor models or effects of potential disorder caused by weak impurities.

This work generalizes existing theories of disorder-induced Hall response [2-5] to the case of strong impurities. We consider a low concentration of strong short-range potential impurities characterized by a scattering phase $\delta$. We show that such impurities in the $p$-wave superconductor lead to sub-gap bound states at energy $\Delta\cos\delta$ similar to Yu-Shiba-Rusinov states hosted by magnetic impurities in $s$-wave superconductors. These states form an impurity band which also governs the Hall response. We calculate $\sigma_{xy}(\omega)$ as function of temperature and frequency. It exhibits rich behaviour and sharp threshold features at frequencies $\omega=\Delta\pm\Delta\cos\delta$ which we identify with particular transition processes between the condensate, the impurity band and the continuous spectrum of the $p_x+ip_y$ superconductor.

[1] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006).

       [2] J.Goryo, Phys. Rev. B 78, 060501(R) (2008).

       [3] R. M. Lutchyn, P. Nagornykh, V. M.Yakovenko, Phys. Rev. B 80, 104508 (2009).

       [4] S. Li, A. V. Andreev, and B. Z.Spivak, Phys. Rev.B 92, 100506 (2015).

       [5] E. J. König, A. Levchenko, Phys. Rev. Lett.118, 027001 (2017).