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Journal of Physics A Mathematical and General

Functional integrals for Hartree Fock Bogoliubov many-body matrix elements

G. Puddu

Published Sep 5, 2002• 1 citation
Publisher

Abstract

We discuss exact functional integral expressions for many-body matrix elements of the type ⟨ψ, A |e −βĤ |ψ, A ⟩, between particle number projected Hartree–Fock–Bogoliubov (HFB) wavefunctions with time-reversal symmetry. A proof of positivity is given for a class of Hamiltonians and when the HFB wavefunctions with time-reversal symmetry are particle number projected to an even number of particles. We show explicitly how to reduce the propagator in the functional integral to a Hermitian positive definite propagator for particle pairs. This result generalizes that previously obtained using Bardeen–Cooper–Schrieffer wavefunctions.

DOI
10.1088/0305-4470/35/37/304

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We discuss exact functional integral expressions for many-body matrix elements of the type ⟨ψ, A |e −βĤ |ψ, A ⟩, between particle number projected Hartree–Fock–Bogoliubov (HFB) wavefunctions with time-reversal symmetry. A proof of positivity is given for a class of Hamiltonians and when the HFB wavefunctions with time-reversal symmetry are particle number projected to an even number of particles. We show explicitly how to reduce the propagator in the functional integral to a Hermitian positive definite propagator for particle pairs. This result generalizes that previously obtained using Bardeen–Cooper–Schrieffer wavefunctions.
We obtain exact functional integrals expressions for many-body matrix elements of the type ⟨ψ, J ,π, N , Z |e -βĤ |ψ, J ,π, N , Z ⟩, between states having good angular-momentum, parity and particle number, in the case of particle-number-preserving functional integrals expressions for e -βĤ are used. Compact expressions are obtained without using angular-momentum projectors. In the case of an even number of particles, a new proof of positivity for the pairing plus quadrupole model is given, which is angular-momentum and parity projected to J π = 0 + . It is shown that the propagator in the functional integral is reduced to a Hermitian positive-definite propagator by the use of angular-momentum eigenstates.
Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei 162Dy,148Sm^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}162Dy,148Sm, and 150Sm^{150}\mathrm{Sm}150Sm. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.
Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.
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.
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.