We study extended Hubbard and Peierls-Hubbard models with up to N =10 sites and half-filled bands with modified boundary conditions. For the extended Hubbard model, the finite-size dependence of the transition from the charge-density-wave regime (CDW) to the spin-density-wave (SDW) regime and of the condensation transition is critically examined, clarifying some previous results. For the Peierls-Hubbard model and N 10 with modified periodic boundary conditions, we always see a decrease of dimerization with U. Extrapolating these results to infinite N, we find that dimerization is enhanced by U for small electron-phonon coupling a &0.75, whereas it decreases with U for 0.75 (a~1. By extending this model by a nearest-neighbor repulsion V, the amplitude of dimerization reaches a maximum at the same value U 2V, where the CDW-to-SDW transition occurs in the extended Hubbard mode;l.

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The ground state of the two-dimensional (2D) t-J model is studied by exact diagonalization for seven diFerent unit cells with N =10, 16, 18, 20, 26, 32, and 36 sites on a square lattice. In the limit of low electron densities (up to six electrons) we calculate the lowest eigenvalues and classify the ground state according to wave vector K, total spin S, and point-group symmetry both in the limit J =0 (U=~Hubbard model) and for finite J )0. The transition to a completely phase-separated state for J)Jps(X) is examined by calculating the free energy as a function of the electron concentration. We find that Jps(X) increases with the lattice size X. An extrapolation of our finite-size results to the thermodynamic limit gives Jps(~)/t =4. 1 (+0. 1).Looking for a theoretical description of high-temperature superconductivity has raised interest in the two-dimensional (2D) t Jmodel. Up t-o now, in the physically important case of small deviation from half filling, i.e. , low doping with holes (by 5 -15 %), exact results for the 2D t-J model are available only by means of Lanczos diagonalizations of small systems with up to 26 sites. ' Today the ultimate goal of a correct and complete phase diagram of the infinite model (depending on the two parameters: hole doping x and antiferromagnetic exchange J) is far beyond reach. Unlike the case of low hole doping, the high doping regime (n =N, /N =1x « 1) has been studied only by a few authors, ' ' as it seems less important for the description of the essential physics in the high-T, copper oxides. Nevertheless, this limit is important for a more complete understanding of the t-J model as well. A detailed study of the low-density limit allows us to address some interesting physical questions, e.g. , the competition between weak-coupling (Fermi-liquid-like) and strongcoupling behavior.In this paper we also show that the problem of phase separation in the t-J model can be examined by a finitesize extrapolation of exact diagonalization results at low electron densities. Furthermore, we present a complete classification of the ground-state spin, momentum, and point-group symmetry for finite lattices with up to N =36 sites and few electrons (N, ) 6), both for J =0 (i.e. , the U = oo Hubbard model) and at finite J )0.We start from the 2D t-J Hamiltonian given in standard notation (cf. , Ref.3) as &=t g c t c +J g (S,S n, n~/4)-.

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