Checkerboard or stripes: Hard-core bosons on the checkerboard lattice as a model of charge ordering in planar cuprates

Abstract

The formation of density modulations observed in cuprates by means of scanning tunneling microscopy is analyzed in the framework of an effective model for hole-doped short-range two-dimensional antiferromagnets. That model has been derived from the $t\text{\ensuremath{-}}J$ model on the square lattice. It is defined in terms of hard-core bosons. The latter objects represent tightly bound hole pairs. Long-range Coulomb repulsion between them is assumed. The nature of the ground state for the effective model is determined by analyzing the density correlation function and the density structure factor. The exact diagonalization for four hard-core bosons in the 128-site cluster, which corresponds to eight holes in the $8\ifmmode\times\else\texttimes\fi{}8$ cluster of sites in the initial square lattice, is performed to derive these quantities. Signatures of the density modulations with the periodicity 4 spacings between nearest-neighbor sites in the initial square lattice in horizontal and vertical directions and with the full symmetry of that lattice may be seen both in the density correlation function and in the density structure factor. On the other hand, a strong tendency to form smectic order may be noticed in the results of the calculation. It turns out that the ground state is a symmetric combination of states representing stripes running either in the vertical or in the horizontal directions. In a real system, the same effect may have the formation of regions with short-range fluctuating stripe order oriented either vertically or horizontally.