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Progress of Theoretical Physics

Theory of Excitations in a Fermion System with Large Quantum Fluctuations: Resonating Random Phase Approximation

Hideo Fukutome

Published Feb 1, 1989• 19 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.81.342

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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 have proposed a theory called the resonating Hartree-Bogoliubov (Res HB) theory to describe large quantum fluctuations in a superconducting Fermion system. One needs also to deal with small quantum fluctuations which correspond to those described by the conventional random phase approximation (RPA). We can derive a new approximation called the Res HB RPA from time dependent small fluctuations of the Res HB ground state. The theory can give a unified description of collective excitations arising from large and small quantum fluctuations and their coupling in a superconducting system. For instance it may be applicable to rotational and vibrational excitations in nuclei with resonance of different deformed shapes.
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 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.
Collective excitations are one of the most common and interesting features of many-body systems. Of particular interest are those collective modes which can be interpreted in terms of vibrations. Nuclei show a large variety of such modes, both low-lying and high-lying. In particular the giant dipole resonance is due to the coherent motion of protons against neutrons. Its analogue in metal clusters is the dipole plasmon excitation, due to the oscillation of electrons against the positive ions. The Random Phase Approximation (RPA) is extensively used as a microscopic theory to study the basic properties of these collective excitations. In the derivation of RPA use is made of the Quasi Boson Approximation (QBA). It consists in replacing the expectation values in the correlated ground state with those in the uncorrelated (Hartree-Fock) one. Strictly related to QBA is the feature of RPA of predicting a harmonic spectrum. On the other hand, the existence of anharmonicities in the multiphonon spectra of nuclei and their relevance in various physical processes are well established. Overcoming QBA has represented the starting point of many attempts aiming at improving RPA. A recently developed approach in this line will be presented and discussed. A very nice property of RPA is to satisfy Energy Weighted Sum Rules (EWSR). However, some violations are present in several extensions of RPA. We will show how to cure this shortcoming.
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.