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arXiv:cond-mat/9609284

Numerical results for generalized RVB wavefunctions

G. Fano, F. Ortolani, L. Ziosi

Published Sep 30, 1996• 0 citations
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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 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.

DOI
10.48550/arxiv.cond-mat/9609284

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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 2X2X22 X 2 X 22X2X2 and the 4X44 X 44X4 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.
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.
The exact many-body states of a 10-site square cluster with periodic boundary conditions of a S = 1/2 antiferromagnetic Heisenberg lattice model are investigated. A general form of a Resonating Valence Bond wavefunction (RVB), written in fermion representation is examined. It is shown that the optimum variational wavefunction has complex singlet bond distribution amplitudes of s+id symmetry . It has a very large overlap with the groundstate, belongs to the same space group representation and its energy functional has a minimum very close to the ground-state energy. On the other hand, if the amplitudes are restricted to real numbers, the local minimum has slightly higher energy and d-wave symmetry.
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 have calculated the energy of the resonating-valence-bond (RVB) state made of nearest-neighbor singlet pairs in the two-dimensional Hubbard model at half filling and the energy of holes in the RVB state near half filling. The energy of the RVB state was obtained by extrapolating the result in a 6\ifmmode\times\else\texttimes\fi{}6 square lattice with the periodic boundary condition to the infinite lattice. The energy of holes was obtained by examining the first five moments of the density of states of holes in the RVB state. We find that the holes are more stabilized in the RVB state than in the N\'eel state for J/t\ensuremath{\gtrsim}0.1, where J and t are the exchange interaction and the nearest-neighbor hopping integral, respectively. We discuss the possibility of the RVB state in the high-Tc{T}_{c}Tc​ Cu oxides.
A fully variational resonance-valence-bond (RVB) study is performed for two-, three- and four-legged isotropic spin-1/2 Heisenberg ladders under periodic boundary conditions along the rungs and periodic/twisted boundary conditions along the chains. By using the full lattice groups of these systems very accurate results have been obtained for the ground-state bulk limit properties in the case of the dimer RVB basis. The results reported here represent the best one can do within the dimer RVB ansatz and should be useful when comparing with results obtained from an a priori fixed structure of the wavefunction. Further results are presented for some finite-sized systems, extending beyond the dimer RVB approach by including RVB functions containing spin pairings between non-nearest-neighbour sites. These results indicate that the three-legged ladder with periodic boundary conditions along the rungs falls into the same universality class of ladders possessing short-range RVB wavefunctions as the two- and four-legged ladders.
The ground states of the antiferromagnetic Heisenberg model on the triangular and square lattices are discussed with use of the combinatorial method on the basis of Anderson's RVB (resonating-valence-bond) variational wave-function. Some properties of RVB states are studied systematically for finite system-size N up to 20 and are compared to the exact values. For the triangular lattice the estimated energy value E RVB =-2.08 is closer to the exact value than that of the spin-wave theory. The value E RVB =-2.36 is estimated for the square lattice. For the square lattice our RVB state is closer to the exact ground state than the Néel state is in the sense of a wave-function. The dimer problem has a close relation to RVB states and is used to count exactly the number of singlet-pair configurations.