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Progress of Theoretical Physics

Resonating Random Phase Approximation for Excitations in a Superconducting Fermion System with Large Quantum Fluctuations

Seiya Nishiyama, Hideo Fukutome

Published Aug 1, 1991• 6 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.86.371

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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 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.
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.
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.
In the present paper we give the explicit expressions for the ground state of a many boson system in different mean field approximations, such as Hartree-Bose, Bogoliubov, the particle-hole random phase approximation (RPA), and its coupling with the particle-particle (RPA). The ground states obtained satisfy the requirement that the annihilation operators of the ``elementary excitations'' annihilates them. In all cases the ground state wave functions can be understood as a condensate of pairs of bosons.
A derivation of the random phase approximation is given, which is based on the generalized Schwinger representation of the fermion-pairs. The method is made of two stages : First, the truncation of the boson space is done, and secondly comes the separation of boson operators into static and fluctuating parts. The pairing vibration in the super phase and the so-called quadrupole oscillation are treated by the method, which leads to the same results as those of RPA. Then the relations of these bosons to ones appearing in RPA are clarified.