Detailed Lanczos study of one- and two-hole band structure and finite-size effects in thet-Jmodel

We present accurate numerical results for low-lying one- and two-hole states in the t-J model on a 4×4 lattice. We find six level crossings in the one-hole ground state for 0t/Jt/J values of these crossings and the associated ground-state quantum numbers are given. A degeneracy of k=(0,0) S=1/2, and S=3/2 one-hole levels at t/J=1/2 is noted, which is consistent with a recent analytical result. For small t/J, the S=1/2 one-hole and S=0 two-hole bandwidths on the 4×4 lattice are ${\mathit{W}}_{\mathit{h}}$=[1.190 445 7(1)]t and ${\mathit{W}}_{\mathrm{hh}}$=[2.575(4)]${\mathit{t}}^2$/J, respectively. The origin of these qualitatively different behaviors is discussed, and a simple relation is found between the small-(t/J) one-hole bandwidth and a static-hole ground-state matrix element. The linear-t term in ${\mathit{W}}_{\mathit{h}}$ is apparently a finite-lattice artifact. As a measure of finite-size effects we determined the rms hole-hole separation in the two-hole ground states; we find evidence of important finite-size effects for t/J≳1, for which the rms hole-hole separation is clearly constrained by the 4×4 lattice. Intermediate-(t/J) hole separations and binding energies for 0.3≲t/J≲1, however, scale approximately as powers of t/J, and can be used to give bulk-limit estimates for t/J=3. In particular, we estimate that the bulk-limit ground-state rms hole-hole separation at t/J=3 is ≊1.8${\mathit{a}}_0$, corresponding to 7 Å in the high-temperature superconductors. The similarity to the observed in-plane coherence length of ${\xi }_{\mathit{a}\mathit{b}}$≊14 Å supports the identification of t-J model hole pairs with the Cooper pairs of high-temperature superconductivity.