We study charge ordered $d$-wave resonating valence bond states $\left(d\mathrm{RVB}\right)$ in the doped cuprates, and estimate the energies of these states in a generalized $t\text{−}J$ model by using a renormalized mean-field theory. The long-range Coulomb potential tends to modulate the charge density in favor of the charge ordered RVB state. The possible relevance to the recently observed $4\times 4$ checkerboard patterns in tunneling conductance in high-$T_c$ cuprates is discussed.

• Received 24 August 2004

DOI:https://doi.org/10.1103/PhysRevB.71.184514