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Physical review. B, Condensed matter

Numerical results for generalized resonating-valence-bond wave functions: Application to the Hubbard model

G. Fano, F. Ortolani, L. Ziosi

Published Dec 15, 1996• 4 citations
Publisher

Abstract

Numerical results are presented for a generalized resonance-valence-bond state which includes both ionic and covalent contributions in each bond; non-nearest-neighbor bonds are also considered. Variational calculations have been performed and the space-group symmetry has been taken into account. The results for the Hubbard Hamiltonian are compared with the exact ones for the 2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2 and the 4\ifmmode\times\else\texttimes\fi{}4 lattices. The agreement is quite satisfactory for large values of the interaction $U$ (for $U=40$ the overlap with the exact ground-state wave function is 0.999). Several correlation functions are also compared.

DOI
10.1103/physrevb.54.17557

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Numerical results are presented for a generalized resonance-valence-bond state which includes both ionic and covalent contributions in each bond; non-nearest-neighbor bonds are also considered. Variational calculations have been performed and the space-group symmetry has been taken into account. The results for the Hubbard Hamiltonian are compared with the exact ones for the 2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2 and the 4\ifmmode\times\else\texttimes\fi{}4 lattices. The agreement is quite satisfactory for large values of the interaction UUU (for U=40U=40U=40 the overlap with the exact ground-state wave function is 0.999). Several correlation functions are also compared.
Numerical results are presented for a generalized resonance valence bond state which includes both ionic and covalent contributions in each bond ; non nearest-neighbor bonds are also considered. Variational calculations have been performed and the space group symmetry has been taken into account. The results for the Hubbard Hamiltonian are compared with the exact ones for the $ 2 X 2 X 2 $ and the $ 4 X 4 $ lattices. The agreement is quite satisfactory for large values of the interaction U ( for U=40 the overlap with the exact ground state wave function is 0.999 ). Several correlation functions are also compared.
With the purpose of assessing the validity of commonly used approximations, best variational solutions are calculated for a four-site (tetrahedral-cluster) Hubbard model with periodic boundary conditions and compared with exact results. In particular, the optimum number-conserving Bardeen-Cooper-Schrieffer-type variational solution is obtained for two, four, and six electrons, and the optimal resonanting-valence-bond state for four electrons. For an attractive interaction U, the exact ground state of the system is obtained in all four cases. If the interaction U is repulsive, the exact ground state is obtained only when the cluster contains two and six electrons. For four electrons the BCS approximation is only satisfactory for relatively small U. For any positive U the resonating-valence-bond state on the other hand, gives very good (albeit not exact) energies for the ground state.
We relate properties of nearest-neighbor resonating valence-bond (NNRVB) wave functions for SU(g)SU(g)SU(g) spin systems on two-dimensional bipartite lattices to those of fully packed interacting classical dimer models on the same lattice. The interaction energy can be expressed as a sum of nnn-body potentials Vn{V}_{n}Vn​, which are recursively determined from the NNRVB wave function on finite subgraphs of the original lattice. The magnitude of the nnn-body interaction Vn{V}_{n}Vn​ (n>1n>1n>1) is of order $\mathcal{O}({g}^{\ensuremath{-}(n\ensuremath{-}1)})$ for small ${g}^{\ensuremath{-}1}$. The leading term is a two-body nearest-neighbor interaction V2(g){V}_{2}(g)V2​(g) favoring two parallel dimers on elementary plaquettes. For SU(2)SU(2)SU(2) spins, using our calculated value of V2(g=2){V}_{2}(g=2)V2​(g=2), we find that the long-distance behavior of the bond-energy correlation function is dominated by an oscillatory term that decays as $1/|\stackrel{\ensuremath{\rightarrow}}{r}{|}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}1.22$. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wave function, which give $\ensuremath{\alpha}\ensuremath{\approx}1.20$.
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.
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.