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Physical Review A

From one-electron to correlated wave functions in extended systems: A valence-bond investigation

Marie-Bernadette Lepetit, Brahim Oujia, Jean‐Paul Malrieu, Daniel Maynau

Published Apr 1, 1989• 18 citations
Publisher

Abstract

One-electron wave functions of half-filled-band closed-shell systems have been decomposed into their orthogonal single-determinant valence-bond (VB) components (antisymmetrized products of orthogonal atomic orbitals). Despite the ordering effect of the Fermi hole among electrons of the same spin, favoring spin-density and charge-density waves, the relative independence between the \ensuremath{\alpha} and \ensuremath{\beta} parts of the wave function results in an overestimation of the weight of multiply ionic structures. Comparison is made with the internally correlated functions for different graphs and Hamiltonians (Hubbard, Pariser-Parr-Pople, and ab initio types). The scaling effect of the correlation does not appear as essentially dependent on the ionicity of the VB forms but exponentially linked to the mean energy of each VB component.

DOI
10.1103/physreva.39.3274

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One-electron wave functions of half-filled-band closed-shell systems have been decomposed into their orthogonal single-determinant valence-bond (VB) components (antisymmetrized products of orthogonal atomic orbitals). Despite the ordering effect of the Fermi hole among electrons of the same spin, favoring spin-density and charge-density waves, the relative independence between the \ensuremath{\alpha} and \ensuremath{\beta} parts of the wave function results in an overestimation of the weight of multiply ionic structures. Comparison is made with the internally correlated functions for different graphs and Hamiltonians (Hubbard, Pariser-Parr-Pople, and ab initio types). The scaling effect of the correlation does not appear as essentially dependent on the ionicity of the VB forms but exponentially linked to the mean energy of each VB component.
Using the Hubbard model and the corresponding t-J model, we study the properties of correlated states with valence-bond structures. Mixed states with such structures and with antiferromagnetic spin ordering can be constructed by means of local unitary transformations of uncorrelated states. The latter turn out to have lower energy than mean-field antiferromagnetic solutions. Spin correlations for various degrees of doping δ = n −1 are in good agreement with the results of exact calculations for finite systems. In contrast to mean-field solutions, allowance for valence-bond correlations leads to a reasonable value of the critical δ , at which long-range antiferromagnetic order disappears. A calculation of the spectral functions that describe photoemission reveals typical behavior in two bands of effective hole (and electron) excitations, and energy transport in bands as the quasimomentum varies from (0,0) to ( π,π ), consistent with calculations in finite systems. We construct a homogeneous correlated state of fluctuating valence bonds (the band-model analog of states of fluctuating valence bonds), and demonstrate that its energy is lower than that of valence-bond alternant structures.
One- and two-particle reduced density matrices have been calculated for a ``resonating'' Heitler-London wavefunction, in a system which is a finite model of a two-dimensional square planar lattice structure (4×4 net of one-electron atoms with Born-von Karman boundary conditions). The Heitler-London wavefunction describes the electronic ground state of a system with a single narrow half-filled band (the Mott insulator). Various measures of electron correlation (pair correlation functions, etc.) are defined and calculated. The intent of these calculations is to provide ``empirical'' guidelines for many-body methods for calculation of Green's functions for one and two particles (reduced density matrices are initial values for the respective Green's functions). Results show that the Heitler-London description is incompatible with several assumptions and widely used models for the Mott insulator. In particular it is found that (a) pair correlations in the Heitler-London ground state remain local even when ``resonance'' of an extremely large number of valence-bond structures occurs; the long range antiferromagnetic ordering obtained in several models of the Mott insulator (the ``two-sublattice'' or ``different bands for different spins'' models) does not occur. (b) The density matrix for the Heitler-London state does not exhibit a BCS type ``pair condensation'', as has been conjectured to describe the Mott insulator in some recent work. Our results strongly suggest that if the Heitler-London description is appropriate, the ``Hubbard Hamiltonian'', which includes only intraatomic pair interactions, does not offer an adequate model of the Mott insulator.
We study the effects of a bond-charge interaction (or correlated hopping) on the properties of the extended [i.e., with both on-site (U) and nearest-neighbor (V) repulsions] Hubbard model in one dimension at half-filling. Energy gaps and correlation functions are calculated by Lanczos diagonalization on finite systems. We find that, irrespective of the sign of the bond-charge interaction, X, the charge-density-wave (CDW) state is more robust than the spin-density-wave (SDW) state. A small bond-charge interaction term is enough to make the differences between the CDW and SDW correlation functions much less dramatic than when X=0. For X=t and fixed V2t (t is the uncorrelated hopping integral), there is an intermediate phase between a charge ordered phase and a phase corresponding to singly occupied sites, the nature of which we clarify: it is characterized by a succession of critical points, each of which corresponds to a different density of doubly occupied sites. We also find an unusual slowly decaying staggered spin-density correlation function, which is suggestive of some degree of ordering. No enhancement of pairing correlations was found for any X in the range examined. \textcopyright{} 1996 The American Physical Society.
A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.
I present a non-orthogonal and overcomplete set of states that spans the S_tot=0 Hilbert subspace for a fermionic system with four possible spin configurations per site, on a half-filled lattice with an even number of sites. This set consists of possible pairing of the spins into three bond types: singlet bonds and positive and negative charge bonds. Although I present the specific case for the half-filled Hubbard model in one-dimension, these sets of states can be trivially extended to higher dimensions. The overlap of these generalized valence bond states form closed loops with properties that are related to operator expectation values in this basis. I verify the correctness of this basis by solving the model via exact diagonalization and comparing with known results. There is evidence of structure in the ground state that may be useful to simulations using variational wave functions.