The stability of hole bound states in the $t\ensuremath{-}J$ model including short-range Coulomb interactions is analyzed using computational techniques on ladders with up to $2\ifmmode\times\else\texttimes\fi{}30$ sites. For a nearest-neighbor (NN) hole-hole repulsion, the two-holes bound state is surprisingly robust and breaks only when the repulsion is several times the exchange J. At $\ensuremath{\sim}10%$ hole doping the pairs break only for a NN repulsion as large as $V\ensuremath{\sim}4J.$ Pair-pair correlations remain robust in the regime of hole binding. The results support electronic hole-pairing mechanisms on ladders based on holes moving in spin-liquid backgrounds. Implications in two dimensions are also presented. The need for better estimations of the range and strength of the Coulomb interaction in copper oxides is remarked.
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 $2\ifmmode\times\else\texttimes\fi{}16$ and $\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}$ 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=2 in units of the lattice spacing, a counterintuitive result considering that the overall symmetry of the two-hole bound state is ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$. 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 $t\ensuremath{-}J$ 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 N\'eel 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 ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$. In particular, it is shown that they are present in an Ising spin background. The time retardation in the family of $t\ensuremath{-}J$ 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 ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ 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.
We study the structure of bound states appearing in systems governed by the Coulomb and short-range interactions. We analyze the binding energies and wave functions of the bound states generated by the Coulomb plus short-range potential. We demonstrate that Coulomb-induced shifts of the binding energy are closely correlated with the spatial distribution of the wave function. Furthermore, we show that the asymptotic behavior of wave functions of weakly bound states is qualitatively altered by Coulomb repulsion, leading to a modification of the near-threshold mass scaling that is otherwise universal for short-range interactions.