Comprehensive numerical and analytical study of two holes doped into the two-dimensionalt−Jmodel

We report on a detailed examination of numerical results and analytical calculations devoted to a study of two holes doped into a two-dimensional, square lattice described by the $t-J$ model. Our exact diagonalization numerical results represent the first solution of the exact ground state of two holes in a 32-site lattice. Using this wave function, we have calculated several important correlation functions, notably the electron momentum distribution function and the hole-hole spatial correlation function. Further, by studying similar quantities on smaller lattices, we have managed to perform a finite-size scaling analysis. We have augmented this work by endeavouring to compare these results to the predictions of analytical work for two holes moving in an infinite lattice. This analysis relies on the canonical transformation approach formulated recently for the $t-J$ model. From this comparison we find excellent correspondence between our numerical data and our analytical calculations. We believe that this agreement is an important step helping to justify the quasiparticle Hamiltonian, and, in particular, the quasiparticle interactions that result from the canonical transformation approach. Also, the analytical work allows us to critique the finite-size scaling ansatzes used in our analysis of the numerical data. One important feature that we can infer from this successful comparison involves the role of higher harmonics in the two-particle, d-wave symmetry bound state—the conventional $\left[\cos {(k}_x\right)-\cos {(k}_y\left)\right]$ term is only one of many important contributions to the d-wave symmetry pair wave function.