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Physical Review B

Many-body wave functions approximated by the superposition of spin-projected nonorthogonal Slater determinants in the resonating Hartree-Fock method

Norikazu Tomita

Published Jan 29, 2004• 38 citations
Publisher

Abstract

A resonating Hartree-Fock (Res-HF) method is revisited and improved by the complete spin projection for the interacting Fermion system. This method approximates a many-body wave function by the superposition of nonorthogonal Slater determinants (S-dets). The nonorthogonality of the S-dets makes it possible to describe the large quantum fluctuations efficiently, since each S-det naturally includes the full-electron-excitation effects from other S-dets. The molecular orbitals in every S-det, as well as the superposition coefficients, are variationally determined. So far, however, the spin contamination, caused by the conventional half-projection, has been a serious obstacle to obtain the accurate wave functions for large-size fermion systems, especially in the intermediate and large correlation regimes. In this paper, we apply the complete spin projection method to the Res-HF calculations. As an example, the improved Res-HF method is applied to the one-dimensional Hubbard model. It will be shown that the complete spin projection improves the Res-HF wave functions significantly. In fact, the correlation energies explained by the improved Res-HF method are better than those by the variational Monte Carlo method in all the correlation regimes. The correlation structures at arbitrary fillings are also well described by the improved Res-HF method.

DOI
10.1103/physrevb.69.045110

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A resonating Hartree-Fock (Res-HF) method is revisited and improved by the complete spin projection for the interacting Fermion system. This method approximates a many-body wave function by the superposition of nonorthogonal Slater determinants (S-dets). The nonorthogonality of the S-dets makes it possible to describe the large quantum fluctuations efficiently, since each S-det naturally includes the full-electron-excitation effects from other S-dets. The molecular orbitals in every S-det, as well as the superposition coefficients, are variationally determined. So far, however, the spin contamination, caused by the conventional half-projection, has been a serious obstacle to obtain the accurate wave functions for large-size fermion systems, especially in the intermediate and large correlation regimes. In this paper, we apply the complete spin projection method to the Res-HF calculations. As an example, the improved Res-HF method is applied to the one-dimensional Hubbard model. It will be shown that the complete spin projection improves the Res-HF wave functions significantly. In fact, the correlation energies explained by the improved Res-HF method are better than those by the variational Monte Carlo method in all the correlation regimes. The correlation structures at arbitrary fillings are also well described by the improved Res-HF method.
A tractable direct optimization algorithm is developed to optimize orbitals in the Slater determinants (S-dets) in a resonating Hartree-Fock (Res HF) wave function. We reduce the variation space using the orbitals to put the first order energy variation in the steepest descent direction. The orbitals in the next iteration are determined so as to minimize the energy functional including up to the second order variation. This algorithm is applied to the one dimensional Hubbard model of half-filling. The optimized S-dets much deviate from the trial S-dets prepared from the HF calculations. The Res HF ground state generated with a few S-dets explains from 99.9 to 95.0% of the ground state correlation energy in all the correlation regimes. We have spin correlation functions with the correct short and long range behaviors and the lowest triplet and singlet spin excitations with correct dispersions, suggesting that the optimization of orbitals incorporates long range spin fluctuations and their mode-mode couplings.
We develop a new general theory for large quantum fluctuations in a Fermion many-body system that cannot be described by fluctuations around the Hartree-Fock ground state but arises from resonance of different correlation structures. We start with an exact coherent state representation of a Fermion system on a unitary group. We show that the Hamiltonian in the coherent state representation has a close connection with the Hartree-Fock energy functional. From this, we can derive a new approximation called the resonating Hartree-Fock approximation in which a state is approximated by a superposition of non-orthogonal Slater determinants with different correlation structures. We derive the variation equations to determine a resonating Hartree-Fock wavefunction. We show that the resonance between degenerate broken symmetry Slater determinants may partially recover the symmetry. We discuss how to choose trial Slater determinants in a resonating Hartree-Fock wavefunction. We suggest that resonance of Slater determinats representing localized defects, such as solitons, polarons and breathers, produced in the long range order of the HF ground state may be the most important content of large quantum fluctuations in condensed matter systems.
We develop a theory of excitations in a Fermion system with large quantum fluctuations that can be described by resonance of non-orthogonal Slater determinants with different correlation structures, namely, by the resonating Hartree-Fock (HF) theory developed previously. We obtain a new approximation for excitations from time dependent small fluctuations of the resonating HF ground state in a similar manner to the time dependent HF derivation of the random phase approximation (RPA). It, therefore, is called the resonating RPA. It gives two kinds of excitations. One is the collective excitations due to secular vibrations of the orbitals in the resonating Slater determinants. These excitations are called vibrons. In contrast to the usual RPA giving only one class of vibrons arising from a Slater determinant, the resonating RPA gives many different classes of vibrons arising from different Slater determinants involved in the ground state. Another kind of excitations arises from secular vibrations of the mixing coefficients of the resonating Slater determinants. These excitations of quite a new kind are shown to correspond to the excited states in the Res HF theory due to the resonance of the Slater determinants and are called resonons. The resonating RPA gives a unified description of vibrons and resonons and their interactions. We discuss new concepts emerging from and suggested by the resonating HF-RPA theory such as resonating energy bands, destruction of long range orders by resonance, solitonic and polaronic resonons and excitons, spin waves with moving solitons and polarons and so on.
The electronic structure of the strongly correlated one dimensional Hubbard model is calculated by the resonating Hartree-Fock (Res HF) method. In the strong correlation regime with the Coulomb repulsion U >∼6, SDW solitons localize almost on the central two sites and only one kind of neutral soliton pair has a much smaller energy than the other pairs. As quantum fluctuations in a periodic chain with 12 sites, we consider up to three diradical breathers consisting of the low energy neutral soliton pair. The Res HF ground state can explain 77.8% ( U =6) and 80.1% ( U =8) of the exact correlation energy. This is much better than the Gutzwiller method to use the SDW as the reference state that explains about 50% of the correlation energy. The breather fluctuations explain the rapid decreases in the short distance of the spin and staggered spin correlation functions. Other correlation functions and the excited states due to the quantum fluctuations are studied.
Mean-field methods such as Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) constitute the building blocks upon which more elaborate many-body theories are based. The HF and HFB wave functions are built out of independent quasiparticles resulting from a unitary linear canonical transformation of the elementary fermion operators. Here, we discuss the possibility of allowing the HF transformation to become nonunitary. The properties of such HF vacua are discussed, as well as the evaluation of matrix elements among such states. We use a simple ansatz to demonstrate that a nonunitary transformation brings additional flexibility that can be exploited in variational approximations to many-fermion wave functions. The action of projection operators on nonunitary-based HF states is also discussed and applied, in a variation-after-projection approach, to the one-dimensional Hubbard model with periodic boundary conditions.
An exact eigenstate of the Lipkin-model Hamiltonian has a parity-symmetry. The resonating Hartree-Fock (Res-HF) ground-state wave function however breaks the parity-symmetry. To restore the parity-symmetry, we develop a parity-projected (P-P) Res-HF approximation to the Lipkin model. The P-P Res-HF wave function is superposed by two P-P Slater determinants. Self-consistent calculation is carried out so as to optimize the P-P Res-HF energy functional. A Res-HF ground state with definite parity brings us the ground-state energy much nearer to the exact one than that the Res-HF and explains most of the ground-state correlation energy. We also show behaviour of P-P Res-HF orbital energies and mixing coefficients calculated by the P-P Res-HF approximation and compare them with those calculated by the parity-unprojected Res-HF approximation. From this comparison, we find very interesting relations between them which have never been seen in the parity-unprojected Res-HF calculation.