Polar Kerr effect from a time-reversal symmetry breaking unidirectional charge density wave

We analyze the Hall conductivity $\sigma_{xy}(\omega)$ of a charge ordered state with momentum $\mathbf{Q}=(0,2Q)$ and calculate the intrinsic contribution to the Kerr angle $\Theta_K$ using the fully reconstructed tight-binding band structure for layered cuprates beyond the low energy hot spots model and particle hole symmetry. We show that such a unidirectional charge density wave (CDW), which breaks time reversal symmetry as recently put forward by Wang and Chubukov [Phys. Rev. B {\bf 90}, 035149 (2014)], leads to a nonzero polar Kerr effect as observed experimentally. In addition, we model a fluctuating CDW via a large quasiparticle damping of the order of the CDW gap and discuss possible implications for the pseudogap phase. We can qualitatively reproduce previous measurements of underdoped cuprates but making quantitative connections to experiments is hampered by the sensitivity of the polar Kerr effect with respect to the complex refractive index $n(\omega)$.