Recent low-energy Scanning Tunneling Spectroscopy (STS) of a lightly hole-doped copper oxide Ca_{2-x}Na_xCuO_2Cl_2 has revealed intense spatial modulations of four unit-cell “checkerboard” periodicity [1]. Using the t-J model (large-U Hubbard model) and the “plain vanilla” renormalized mean-field theory [2] stability of such inhomogeneous superstructures around hole doping 1/8 is investigated. We find a non-uniform 4a x 4a bond order wave involving simultaneously small inhomogeneous staggered plaquette currents as well as a small charge density modulation similar to pair density wave order [3]. The stability of a supersolid phase (involving both translation symmetry breaking and superconducting order) is also investigated. Relevance of these phases to the phase diagram of the underdoped high-Tc materials is discussed.

[1] T. Hanaguri et al.; Nature 430, 1001 (2004).
[2] F.C. Zhang, C. Gros, T.M. Rice and H. Shiba; Supercond. Sci. Technol. 1, 36 (1988).
[3] D.P.; Phys. Rev. B 72, 060508(R) (2005).

Toulouse