Zero-pairing limit of Hartree-Fock-Bogoliubov reference states

Duguet, T.; Bally, B.; Tichai, A.

doi:10.48550/arxiv.2006.02871

Cited by 3 publications

(31 citation statements)

References 33 publications

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“…First of all, here the copies appear with the smaller periodicity M α (instead of 2M φ ), which is to be expected given the factor 2π (instead of π) in the discretized operator of Eq. (114). Also, the mirror images are not perfect copies of each other but present some noticeable asymmetries.…”

Section: By Applying the Discretized Operator Pk0mentioning

confidence: 99%

“…V. As a consequence, however, the first point of the discretization is at π Mγ instead of zero. Therefore, the operator (114) does not reduce to the identity for M γ = 1, and the case of "no projection" has thus to be treated separately.…”

Section: E Numerical Implementationmentioning

confidence: 99%

“…As already mentioned when defining them through Eqns. (114) and (116), respectively, the use of the symmetries (132a-132h) of the rotated matrix elements imposes some conditions on the form of the discretized operators PK0…”

Section: B Symmetries Of the Rotated Matrix Elementsmentioning

confidence: 99%

“…In the zero-pairing limit of the HFB approximation, however, it is possible to obtain spherically symmetric states also for openshell systems. Nevertheless, as their resulting density matrices are not the ones of a single Slater determinant[114,115], we will discard such possibility from the further discussion.…”

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confidence: 99%

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Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

Bally,

Bender

2020

Preprint

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Background: Many quantal many-body methods that aim at the description of self-bound nuclear or mesoscopic electronic systems make use of auxiliary wave functions that break one or several of the symmetries of the Hamiltonian in order to include correlations associated with the geometrical arrangement of the system's constituents. Such reference states have been used already for a long time within self-consistent methods that are either based on effective valence-space Hamiltonians or energy density functionals, and they are presently also gaining popularity in the design of novel ab-initio methods. A fully quantal treatment of a self-bound many-body system, however, requires the restoration of the broken symmetries through the projection of the many-body wave functions of interest onto good quantum numbers.Purpose: The goal of this work is three-fold. First, we want to give a general presentation of the formalism of the projection method starting from the underlying principles of group representation theory. Second, we want to investigate formal and practical aspects of the numerical implementation of particle-number and angularmomentum projection of Bogoliubov quasiparticle vacua, in particular with regard of obtaining accurate results at minimal computational cost. Third, we want to analyze the numerical, computational and physical consequences of intrinsic symmetries of the symmetry-breaking states when projecting them.Methods: Using the algebra of group representation theory, we introduce the projection method for the general symmetry group of a given Hamiltonian. For realistic examples built with either a pseudo-potential-based energy density functional or a valence-space shell-model interaction, we then study the convergence and accuracy of the quadrature rules for the multi-dimensional integrals that have to be evaluated numerically and analyze the consequences of conserved subgroups of the broken symmetry groups. Results:The main results of this work are also threefold. First, we give a concise, but general, presentation of the projection method that applies to the most important potentially broken symmetries whose restoration is relevant for nuclear spectroscopy. Second, we demonstrate how to achieve high accuracy of the discretizations used to evaluate the multi-dimensional integrals appearing in the calculation of particle-number and angular-momentum projected matrix elements while limiting the order of the employed quadrature rules. Third, for the example of a point-group symmetry that is often imposed on calculations that describe collective phenomena emerging in triaxially deformed nuclei, we provide the group-theoretical derivation of relations between the intermediate matrix elements that are integrated, which permits for a further significant reduction of the computational cost of the method. These simplifications are valid whatever the number parity of the quasiparticle states and therefore can be used in the description of even-even, odd-mass, and odd-odd nuclei. Conclusions:The quantum-number...

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Section: By Applying the Discretized Operator Pk0mentioning

confidence: 99%

Section: E Numerical Implementationmentioning

confidence: 99%

Section: B Symmetries Of the Rotated Matrix Elementsmentioning

confidence: 99%

mentioning

confidence: 99%

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Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

Bally,

Bender

2020

Preprint

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“…In practice, the limit is achieved by adding a constraining term to the grand potential Ω such that the operator used for the minimization becomes the Routhian [2] Ω(δ)…”

Section: Zero-pairing Limitmentioning

confidence: 99%

Zero-pairing and zero-temperature limits of finite temperature Hartree-Fock-Bogoliubov theory

Duguet,

Ryssens

2020

Preprint

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Zero-pairing and zero-temperature limits of finite-temperature Hartree-Fock-Bogoliubov theory

Duguet

Ryssens

2020

Phys. Rev. C

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Background: Recently, variational Hartree-Fock-Bogoliubov (HFB) mean-field equations were shown to possess a mathematically well-defined zero-pairing limit, independently of the closed-or open-shell character of the system under consideration. This limit is nontrivial for open-shell systems such that HFB theory does not reduce to the Hartree-Fock (HF) formalism in all cases. Purpose: The present work extends this analysis to finite-temperature HFB (FTHFB) theory by investigating the behavior of this more general formalism in the combined zero-temperature and zero-pairing limits. Methods: The zero-pairing and zero-temperature limits of the FTHFB statistical density operator constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction. Results: While the FTHFB density operator reduces to the projector corresponding to a pure HF Slater determinant for closed-shell nuclei, the FTHFB formalism does not reduce to the HF theory in all cases in the zero-temperature and zero-pairing limits, i.e., for open-shell nuclei. However, the fact that a nucleus can be of open-shell character in these joint limits is necessarily the result of some symmetry restrictions. Whenever it is the case, the nontrivial description obtained for open-shell systems is shown to depend on the order with which both limits are taken, i.e., the two limits do not commute for these systems. When the zero-temperature limit is performed first, the FTHFB density operator is demoted to a projector corresponding to a pure state made out of a linear combination of a finite number of Slater determinants with different (even) numbers of particles. When the zero-pairing limit is performed first, the FTHFB density operator remains a statistical mixture of a finite number of Slater determinants with both even and odd particle numbers. While the entropy (pairing density) is zero in the first (second) case, it does not vanish in the second (first) case in spite of the temperature (pairing) tending towards zero. The difference between both limits can have striking consequences for the (thermal) expectation values of observables. For instance, the particle-number variance does not vanish in either case and has limiting values that differ by a factor of two in both cases. Conclusions: While in the textbook situation associated with closed-shell nuclei Hartree-Fock-Bogoliubov (finite-temperature Hartree-Fock) theory reduces to Hartree-Fock theory in the zero-pairing (zero-temperature) limit, the present analysis demonstrates that a nontrivial and unexpected limit is obtained for this formalism in open-shell systems. This result sheds a new light on certain aspects of this otherwise very well-studied many-body formalism.

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Zero-pairing limit of Hartree-Fock-Bogoliubov reference states

Cited by 3 publications

References 33 publications

Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

Zero-pairing and zero-temperature limits of finite temperature Hartree-Fock-Bogoliubov theory

Zero-pairing and zero-temperature limits of finite-temperature Hartree-Fock-Bogoliubov theory

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