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Progress of Theoretical Physics

A Variational Foundation of the Improper Hartree-Bogoliubov Formalism

Seiya Nishiyama, M. Iwasaki, Hideo Fukutome, M. Yamamura

Published Aug 1, 1980• 2 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.64.558

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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.
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature TTT, the level density and the strength of the pairing force, are investigated analytically and numerically.
Hartree-Bogolyubov (HB) theory is formulated in a basis-independent way, i.e., in terms of linear and antilinear operators acting in the one-particle space. For that purpose, some basic antilinear algebra is presented. The pairing tensor and the pairing potential are shown to represent two antilinear skew-Hermitian operators. The polar factorization of the first of them (the correlation operator ${\stackrel{^}{t}}_{a}$), i.e., ${\stackrel{^}{t}}_{a}={(\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}{\stackrel{^}{\ensuremath{\rho}}}^{2})}^{\frac{1}{2}}{\stackrel{^}{P}}_{a}$, shows that HB theory has only two variational (trial) operators: the density operator $\stackrel{^}{\ensuremath{\rho}}$ and the antilinear pairing operator ${\stackrel{^}{P}}_{a}$ which is defined by the properties ${{\stackrel{^}{P}}_{a}}^{+}={{\stackrel{^}{P}}_{a}}^{\ifmmode\dagger\else\textdagger\fi{}1}=\ensuremath{-}{\stackrel{^}{P}}_{a}$. These two operators commute. The former is the unique and very well-known variational operator of Hartree-Fock (HF) theory, and the latter represents a new variational freedom typical of HB theory. Most calculations, as for instance the Bardeen-Cooper-Schrieffer (BCS) approximation, restrict this freedom by choosing ${\stackrel{^}{P}}_{a}$ to be the time-reversal operator. The basic dynamical (Euler-Lagrange) equations of HB theory are obtained directly by varying linear and antilinear operators. They are expressed in a compact form, using only commutators and anticommutators of the kinematical and the dynamical operators: ${\stackrel{^}{A}}_{a}\ensuremath{\equiv}{[\stackrel{^}{h},{\stackrel{^}{t}}_{a}]}_{+}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}\frac{1}{2}]}_{+}=0, \stackrel{^}{B}\ensuremath{\equiv}{[\stackrel{^}{h},\stackrel{^}{\ensuremath{\rho}}]}_{\ensuremath{-}}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},{\stackrel{^}{t}}_{a}]}_{\ensuremath{-}}=0,$ where ${\ensuremath{\Delta}}_{a}$ is the pairing potential and $\stackrel{^}{h}$ is the one-particle Hamiltonian.Two identities are found between ${\stackrel{^}{A}}_{a}$ and $\stackrel{^}{B}$ which turn out to be very useful for obtaining solutions. Symmetries of the trial operators and of the solutions are discussed in great detail, special attention being paid to real HB solutions and their connection with some antiunitary symmetries. Several simple solutions are analyzed: (1) the case where $\stackrel{^}{\ensuremath{\rho}}$ is restricted to be a projector, i.e., ${\stackrel{^}{t}}_{a}=0$ (HF case); (2) the case where $\stackrel{^}{\ensuremath{\rho}}$ and ${\stackrel{^}{P}}_{a}$ are restricted to commute with a complete set of observables, which determines the eigensubspaces of $\stackrel{^}{\ensuremath{\rho}}$, and which, in particular cases of rotational and translational symmetries, fixes ${\stackrel{^}{P}}_{a}$ to be equal to the time-reversal operator (BCS case); and (3) the case where ${\stackrel{^}{P}}_{a}$ is any given symmetry operator of the Hamiltonian.
Toward a unified algebraic theory for mean-field Hamiltonian describing paired- and unpaired-mode effects, in this paper, we propose a generalized Hartree–Bogoliubov mean-field Hamiltonian in terms of fermion pair and creation-annihilation operators of the [Formula: see text] Lie algebra. We diagonalize the generalized Hartree–Bogoliubov mean-field Hamiltonian and throughout its diagonalization we can first obtain the unpaired mode amplitudes which are given by the self-consistent field parameters appeared in the Hartree–Bogoliubov theory together with the additional self-consistent field parameter in the generalized Hartree–Bogoliubov mean-field Hamiltonian and by the parameter specifying the property of the [Formula: see text] group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such results. We construct the Killing potential in the coset space [Formula: see text] on the Kähler symmetric space which is equivalent to the generalized density matrix. We show another approach to the fermion mean-field Hamiltonian based on such a generalized density matrix. We derive an [Formula: see text] generalized Hartree–Bogoliubov mean-field Hamiltonian operator and a modified Hartree–Bogoliubov eigenvalue equation. We discuss on the mean-field theory related to the algebraic mean-field theory based on the generalized density matrix and the coadjoint orbit leading to the nondegenerate symplectic form.
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.
The Hartree-Fock-Bogoliubov approximation is very useful for treating both long- and short-range correlations in finite quantum fermion systems, but it must be extended in order to describe detailed spectroscopic properties. One problem is the symmetry-breaking character of the HFB approximation. We present a general and systematic way to restore symmetries and to extend the configuration space using pfaffian formulas for the many-body matrix elements. The advantage of those formulas is that the sign of the matrix elements is unambiguously determined. It is also helpful to extend the space of configurations by constraining the HFB solutions in some way. A powerful method for finding these constrained solutions is the gradient method, based on the generalized Thouless transformation. The gradient method also preserves the number parity of the Bogoliubov transformation, which facilitates the application of the theory to systems with odd particle number.