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Progress of Theoretical Physics

Note on the Improper Bogoliubov Transformation: An Example in the Many-Quark System

M. Iwasaki

Published Nov 1, 1996• 2 citations
Publisher

Abstract

We study the improper Bogoliubov transformation which is an improper rotation in the SO(2N) group in Fermion systems. The mathematical properties of the Bogoliubov transformation are investigated generally, when the dimension N is odd. It is shown that the improper transformation appears naturally and some quasi-particles become gapless. This leads to the property that some physical quantities have singularities in momentum space across the Fermi surface. Moreover it is pointed out that such a case may be realized in quark matter.

DOI
10.1143/ptp.96.1043

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We study the improper Bogoliubov transformation which is an improper rotation in the SO(2N) group in Fermion systems. The mathematical properties of the Bogoliubov transformation are investigated generally, when the dimension N is odd. It is shown that the improper transformation appears naturally and some quasi-particles become gapless. This leads to the property that some physical quantities have singularities in momentum space across the Fermi surface. Moreover it is pointed out that such a case may be realized in quark matter.
Properties and implications of the SO(2N+1) regular representation of a fermion system, which is a representation on the SO(2N+1) group proposed recently by us, are studied. By using the independent variables of the SO(2N+1) group, a representation of fermion annihilation-creation and pair operators is obtained which is an extension of the Dyson representation to cover both paired and unpaired modes. The structures of the SO(2N+1) spinor space and the SO(2N+1) wave function are determined. An integral and a differential Schrödinger equation on the SO(2N+1) spinor space are derived which are applicable to both even and odd fermion systems. The structures of the symmetry projected SO(2N+1) wave function and Schrödinger operator are determined. It is shown that odd fermion systems can be treated with an analogous way as even fermion systems by using a special SO(2N+1) canonical transformation which is very similar to the Bogoliubov transformation but corresponds to an improper rotation in the O(2N) group.
We give a variational foundation for the improper Hartree-Bogoliubov (HB) formalism by which odd fermion systems can be treated in an analogous way as the formalism of even fermion systems. Because of the difference of the normalization for the improper HB amplitudes from that for the proper ones, the improper HB equation cannot be diagonalized with the conventional BCS ansatz. We decompose the improper HB amplitudes into the amplitudes describing paired and unpaired modes of fermions and the variational procedure is applied to them. We obtain a new form of the improper HB equation with respect to the paired mode amplitudes. It includes the effects of unpaired modes but can be diagonalized with the conventional manner.
A new many-body theory for fermions is proposed wihich is based on the SO(2N+1) Lie algebra of the fermion operators consisted of the annihilation-creation operators and the pair operators. A New cannonical transformation, which is the extension of the Bogoliubov transformation to the SO(2N+1) group, is introduced. A new bose representation for the fermion Lie operators is obtained by mapping the fermion Lie operators into the regular representation of the SO(2N+1) group. The annihilation-creation operators and the pair operators of fermions are represented by the closed first order differential operators on the SO(2N+1) group. An exact representation of fermion wavefunctions in a form similar to the wavefunction of the generator coordinate method is obtained making use of the SO(2N+1) canonical transformation. The physical fermion space is shown to be the irreducible spinor representation of the SO(2N+1) group. The dynamics of fermions in the bose representation space is shown to represent rotations of a 2N+1 dimensional rotator.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.
Non-Fermi liquid behavior is found for the first time in a two-dimensional (2D) system with non-singular interactions using Haldane’s bosonization scheme. The bosonized system is solved exactly by a generalized Bogoliubov transformation. The fermion momentum distribution, calculated using a generalized Mattis-Lieb technique, exhibits a non-universal power law in the vicinity of the Fermi surface for intermediate interaction strengths. In recent years, there has been great interest in possible non-Fermi-liquid-like behavior in the ground state of two-dimensional (2D) strongly correlated fermionic systems [1]. It is well-known that in one dimension the ground state of a fermionic system with arbitrarily weak interactions is not a Fermi liquid but a state usually known as a Luttinger liquid [2]. For the 2D case, many studies have suggested that, in the limit of weak interactions, this behavior does not appear [3]. Studies of systems with a small number of particles also suggest Fermi liquid behavior in 2D [4]. Although Khveshchenko et al have found non-Fermi-liquid behavior in 2D, they worked only with a singular long-range current-current interaction [5]. The one-dimensional Luttinger liquid solution can be obtained by a bosonization
We consider many-fermion systems with singular Fermi surfaces, which contain Van Hove points where the gradient of the band function k ↦ e(k) vanishes. In a previous paper, we have treated the case of spatial dimension d ≥ 3. In this paper, we focus on the more singular case d = 2 and establish properties of the fermionic self-energy to all orders in perturbation theory. We show that there is an asymmetry between the spatial and frequency derivatives of the self-energy. The derivative with respect to the Matsubara frequency diverges at the Van Hove points, but, surprisingly, the self-energy is C1 in the spatial momentum to all orders in perturbation theory, provided the Fermi surface is curved away from the Van Hove points. In a prototypical example, the second spatial derivative behaves similarly to the first frequency derivative. We discuss the physical significance of these findings.