Fluctuating charge order in the cuprates: spatial anisotropy and feedback from superconductivity

We analyze the form of static charge susceptibility $\chi(q)$ in underdoped cuprates near axial momenta $(Q,0)$ and $(0,Q)$ at which short-range static charge order has been observed. We show that the momentum dependence of $\chi (q)$ is anisotropic, and the correlation length in the longitudinal direction is larger than in the transverse direction. We show that correlation lengths in both directions decrease once the system evolves into a superconductor, as a result of the competition between superconductivity and charge order. These results are in agreement with resonant x-ray scattering data [R. Comin et al., Science 347, 1335 (2015)]. We also argue that density and current components of the charge order parameter are affected differently by superconductivity --- the charge-density component is reduced less than the current component and hence extends deeper into the superconducting state. This gives rise to two distinct charge order transitions at zero temperature.