Two holes in a locally antiferromagnetic background: The role of retardation and Coulomb repulsion effects

The problem of two holes in the presence of strong antiferromagnetic fluctuations is revisited using computational techniques. Two-dimensional clusters and two-leg ladders are studied with the Lanczos and truncated Lanczos algorithms. Lattices with up to $2\times 16$ and $\sqrt{32}\times \sqrt{32}$ sites are studied. The motivation of the paper is the recently discussed spatial distribution of holes in ladders where the maximum probability for the hole-hole distance is obtained at $d=\sqrt{2}$ in units of the lattice spacing, a counterintuitive result considering that the overall symmetry of the two-hole bound state is $d_{x^2-y^2}$. Here this effect is shown to appear in small ladder clusters that can be addressed exactly, and also in planes. The probability distribution of hole distances $d$ was found to be broad with several distances contributing appreciably to the wave function. The existence of holes in the same sublattice is argued to be a consequence of non-negligible retardation effects in the $t-J$ model. Effective models with instantaneous interactions nevertheless capture the essence of the hole pairing process in the presence of short-range antiferromagnetic fluctuations (especially regarding the symmetry properties of the condensate), similarly as the (nonretarded) BCS model contains the basic features of the more complicated electron-phonon problem in low-temperature superconductors. The existence of strong spin singlets in the region where the two-hole bound state is located is here confirmed, and a simple explanation for its origin in the case of planes is proposed using the Néel state as a background, complementing previous explanations based on a spin-liquid undoped state. It is predicted that these strong singlets should appear regardless of the long distance properties of the spin system under consideration, as long as the bound state is $d_{x^2-y^2}$. In particular, it is shown that they are present in an Ising spin background. The time retardation in the family of $t-J$ models leads naturally to low-energy hole states with nonzero momentum and spin one, providing a possible explanation for apparent SO(5)-symmetric features observed recently in this context. Finally, the influence of a short-range Coulombic repulsion is analyzed. Rough estimations suggest that at a distance of one lattice spacing this repulsion is larger than the exchange $J$. The hole distribution in the $d_{x^2-y^2}$ bound state is reanalyzed in the presence of such repulsion. Very short hole-hole distances lose their relevance in the presence of a realistic hole-hole interaction.