The real-space renormalization-group method is applied to the one-dimensional extended Hubbard model. It is shown that in the half-filled-band case, the phase diagram consists of two phases and that the transition from the spin-ordered phase to the charge-oxdered phase is continuous, a result which differs from that obtained previously from the broken-symmetry Hartree-Fock approximation.

We present an analysis based on the extrapolation of finite-size results to the infinite onedimensional spin-2 dimerized isotropic Heisenberg system for the whole range of the dimerization parameter 5 (I 5 I &1) at zero temperature. This system undergoes a transition at 5=0, and a gap opens in the spectrum of elementary excitations. The exponent v, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N =18). We investigate two finite-size-scaling hypotheses, assuming a pure power-1aw behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate v, we use the derivative cf the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), v=0. 71+0.01, in agreement with previous estimates, and in case (2), v=0. 668+0.001, in very good agreement with the value v= -, conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form I 5 I" with x =1.34+0.02. However, results for the derivative of this quantity show a different functional dependence upon 5, at least for 5 & 0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as 5 approaches 1.

The finite-cell scaling method is applied to the one-dimensional extended Hubbard Hamiltonian. The results obtained for the correlation functions show that, in the half-filled-band case, the hopping mixes the charge-ordered and the spin-ordered states. As with the real-space renormalizationgroup method, we find that the transition between these states is continuous and that the system, in the vicinity of the transition, is in a mixed charge-orderedspin-ordered state.

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