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arXiv:nucl-th/9706041

Extended BCS Theories

Roger Balian, H. Flocard, M. Vénéroni

Published Jun 17, 1997• 0 citations
arXivHTMLPDF

Abstract

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 $T$, the level density and the strength of the pairing force, are investigated analytically and numerically.

DOI
10.48550/arxiv.nucl-th/9706041

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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.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, we extend the Pfaffian formula for the many-body traces of HFB density operators introduced by L. M. Robledo in Ref. [1] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. We validate our method through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
I will present our recent results on the calculation of symmetry-projected wave functions [1,2] for electronic structure theory. For Hartree-Fock and spin projection, this is a 50-year old problem in quantum chemistry, going all the way back to Lowdin and his “extended HF” theory. For Hartree-Fock-Bogoliubov and number projection, our approach offers new perspectives on Antisymmetrized Geminal Power (AGP) wavefunctions that were the focus of much attention in the 1980s. In our work, all molecular symmetries (electron number, spin S 2 and Sz , point group, and complex conjugation) are deliberately broken and restored in a self-consistent variation-after-projection approach. The resulting method yields a comprehensive black-box treatment of static correlation with one-electron (mean-field) computational cost. The ensuing wave function is of high quality multireference character competitive with exact diagonalization. The method can also be applied to calculate excited states and spectral functions [3]. The curse of the thermodynamic limit and the quest for a low-cost treatment of residual correlations will also be addressed.
: The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian. This work was supported in part by Sonderforschungsbereich 288 of the Deutsche Forschungsgemeinschaft...
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, I extend the Pfaffian formula for the many-body traces of HFB density operators introduced by Robledo [L. M. Robledo, Phys. Rev. C. 79, 021302(R) (2009)] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. The method is validated through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
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.
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.