Polar Kerr effect from chiral-nematic charge order

We analyze the polar Kerr effect in an itinerant electron system on a square lattice in the presence of a composite charge order proposed for the pseudogap state in underdoped cuprates. This composite charge order preserves discrete translational symmetries, and is "chiral-nematic" in the sense that it breaks time-reversal symmetry, mirror symmetries in $x$ and $y$ directions, and $C_4$ lattice rotation symmetry. The Kerr angle $\theta_K$ in $C_4$-symmetric system is proportional to the antisymmetric component of the anomalous Hall conductivity $\sigma_{xy}-\sigma_{yx}$. We show that this result holds when $C_4$ symmetry is broken. We show that in order for $\sigma_{xy}$ and $\sigma_{yx}$ to be non-zero the mirror symmetries in $x$ and $y$ directions have to be broken, and that for $\sigma_{xy}-\sigma_{yx}$ to be non-zero time-reversal symmetry has to be broken. The chiral-nematic charge order satisfies all these conditions, such that a non-zero signal in a polar Kerr effect experiment is symmetry allowed. We further show that to get a non-zero $\theta_K$ in a one-band spin-fluctuation scenario, in the absence of disorder, one has to extend the spin-mediated interaction to momenta away from $(\pi,\pi)$ and has to include particle-hole asymmetry. Alternatively, in the presence of disorder one can get a non-zero $\theta_K$ from impurity scattering: either due to skew scattering (with non-Gaussian disorder) or due to particle-hole asymmetry in case of Gaussian disorder. The impurity analysis in our case is similar to that in earlier works on Kerr effect in $p_x+ip_y$ superconductor, however in our case the magnitude of $\theta_K$ is enhanced by the flattening of the Fermi surface in the "hot" regions which mostly contribute to charge order.