Origin of cuprate charge order and its relation to pairing
In plain words
Below roughly 150 K, underdoped cuprates develop a weak, short-range periodic ripple in electron density (charge order) with a period of 3 to 4 lattice spacings; whether it competes with superconductivity, grows out of a modulated superconductor, or causes the pseudogap is unsettled.
Precise statement
In hole-doped YBa2Cu3O6+x, Bi2Sr2CaCu2O8+d and HgBa2CuO4+d at $0.08 < p < 0.16$, resonant x-ray scattering finds incommensurate bond-centered charge order with wavevector $q_{\mathrm{CDW}} \sim 0.25\text{ to }0.33\,\text{reciprocal lattice units}$, decreasing with $p$, short correlation length at zero field, and long-range three-dimensional order above $\sim 150\,\mathrm{kG}$. Determine (a) whether $q_{\mathrm{CDW}}(p)$ is set by Fermi-surface features (antinodal states) or by strong-coupling local physics, and (b) whether the charge order is an independent competitor of uniform d-wave superconductivity or a subsidiary of a parent pair density wave. An answer is a theory that predicts $q_{\mathrm{CDW}}(p)$, the d-symmetry form factor, the onset temperature and the field-induced 3D order in at least two families.
What would settle it
A model that reproduces $q_{\mathrm{CDW}}(p)$ and the form factor across families, tested by experiments that tune the balance between charge order and superconductivity (field, uniaxial strain) and check the predicted response of pair-density-wave correlations.
Status in the literature
Charge order in YBa2Cu3O6+x was found by resonant x-ray scattering in 2012 and becomes three-dimensional and long range in fields above $\sim 150\ \mathrm{kG}$ (2015); its origin and role remain disputed.