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Progress of Theoretical Physics

Theory of Resonating Quantum Fluctuations in a Fermion System: Resonating Hartree-Fock Approximation

Hideo Fukutome

Published Sep 1, 1988• 102 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.80.417

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Journal of the Physical Society of Japan•Oct 15, 1991•19 citations
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.
We develop a general theory to describe a superconducting Fermion system with large quantum fluctuations. The theory is a direct extension of the resonating Hartree-Fock (HF) theory to the Hartree-Bogoliubov case including pair correlations. We start with an exact coherent state representation of a superconducting Fermion system on a special orthogonal group. A state with large quantum fluctuations is approximated by a superposition of non-orthogonal Hartree-Bogoliubov (HB) wave functions with different correlation structures. We obtain the variational equations to determine the coefficients of configuration mixing and the orbitals in the resonating HB wave functions.
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.
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.
The electronic structure of the one dimensional Hubbard model is calculated by the resonating Hartree-Fock (HF) method. Solitons in the spin density wave (SDW) order parameter (OP) make translational bands with a lot of high harmonics. Their effective masses are about 0.1 of the electron mass, so that they are potentially good charge or spin carriers. We consider diradical and zwitterionic breathers, which consist of a neutral and charged soliton-antisoliton pair, respectively, making quantum translational and breathing motions, as local quantum fluctuations in the SDW in the half filled case. The stabilization of the ground state by the resonationg breathers amounts to 76.5% ( U =3) and 70.1% ( U =4) of the correlation energy. This is better than the result of the Gutzwiller method. The fluctuations by breathers explain the rapid decrease of the spin correlation function in short distances and give also low energy excited states with the symmetries and energetic ordering 3 B < 1 A < 1 B that are characteristic in polyenes.