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

We analyze the form of static charge susceptibility $\chi \left(q\right)$ 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 $\chi \left(q\right)$ 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.

Figure 1

The Fermi surface (black) with the occupied states shown in green. The location of the hot spots for CO [the points on the Fermi surface separated by $Q_y=(0,Q)$ or $Q_x=(Q,0)]$ are marked as $1,2,-1,-2,\cdots$ with the corresponding directions of Fermi velocities, $v_F$. The notations $\pm k_0$ and $\pm k_{\pi }$ are for the center-of-mass momentum of charge order parameter ${\mathrm{\Delta }}_k^Q=c_{k+Q/2,\alpha }^{†}{\delta }_{\alpha \beta }{c}_{k-Q/2,\beta }$.

Figure 2

The Feynman diagrams for (a) ${\mathrm{\Pi }}_{k_0}\left(Q\right)$ and ${\mathrm{\Pi }}_{k_{\pi }}\left(Q\right)$ in the normal state, and (b) normal and anomalous contributions to ${\mathrm{\Pi }}_{k_0}\left(Q\right)$ in the superconducting state. (c) Four-point diagrams contributing to ${\beta }_{11}$ and ${\beta }_{12}$ in the free-energy [Eq. (2)] in the normal state.

Figure 3

The phase diagram in variables $T$ and $x$ (adapted from Ref. [18]), with more details near $T=0$ based on our results (the inset). N and TRSB stand for nematic and time-reversal-symmetry breaking in the pseudogap state, respectively, and CDW is the phase with broken translational symmetry (in a clean system).