Quasi-Hole Properties in Two-Dimensional Strongly Correlated Systems

The Hubbard and t-J models are widely believed to hold the key to understanding the high-temperature superconductors. This thesis studies the effects of small doping in these models, largely by numerical means. The main results are the following. Chapter 2 uses the Hartree approximation to examine the stability of bag solutions of the two-dimensional, one-band Hubbard model, at intermediate values of the on -site electron interaction, U, with nearest-neighbour (NN, t) and next-nearest-neighbour (NNN, t^' ) hopping terms. For U = 3t, even weak NNN hopping destabilizes the polarons. Thus Hartree theory suggests a constraint on spin-bag theory: the NNN hopping integral must be small unless both U is moderately large and t ^' > 0. The effect of NNN hopping is also investigated in the t-J model using exact diagonalization. Results for the phase diagram and the band structure are presented. Chapter 3 investigates the t-J model and its variants using the string and hopping bases. For the string basis, states to order t^{10} are treated exactly, so that the Green function is obtained to order t^{20}; the tree approximation is used for the remainder. The hopping basis, which relaxes constraints imposed in the string basis, contains all states generated by up to 8 hops. Properties studied include the ground-state energy, the effective mass, the bandwidth, the spectral function, the self energy, and the density of states. The Ising limit of the t-J model is treated well by the string basis, but the Heisenberg limit J_| = J_ {z} requires the hopping basis, which gives apparently good results for all J > 0.1t. Chapter 4 presents accurate numerical results for low-lying one- and two-hole states in the t-J model on a 4 x 4 lattice. A previously unremarked degeneracy of vec k = (0,0) S = 1/2 and S = 3/2 one-hole levels at t/J = 1/2 is noted; the degeneracy 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 x 4 lattice are W_{h} = 1.1904457(1)t and W_{hh} = 2.575(4)t^{2}/J respectively. 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 x 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 at the more physical t/J = 3.