Symmetry-projected variational calculations with the numerical suite TAURUS I. Variation after particle-number projection

Bally, Benjamin; Sánchez-Fernández, Adrián; Rodríguez, Tomás R.

doi:10.48550/arxiv.2010.14169

Cited by 3 publications

(3 citation statements)

References 86 publications

(195 reference statements)

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“…To illustrate our discussion on the behavior of the discretized projection operators, we will study the numerical convergence of the angular-momentum projection of several Slater determinants constructed in the sd-shell valence space using the numerical suite TAURUS [135,136]. The use of Slater determinants for such analysis has the advantage that no particle-number projection is required, thus simplifying the calculations and removing a source of numerical inaccuracies (in particular as different [N, Z] components will in general have different angular-momentum decompositions).…”

Section: Convergence Of the Projected Componentsmentioning

confidence: 99%

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

Bally

Bender

2021

Phys. Rev. C

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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 threefold. 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 angular-momentum 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 pseudopotential-based energy density functional or a valence-space shell-model interaction, we then study the convergence and accuracy of the quadrature rules for the multidimensional 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 multidimensional 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 a further significant reduction of the computational cost of the method. These simplifications are valid regardless of 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-numb...

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Section: Convergence Of the Projected Componentsmentioning

confidence: 99%

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

Bally

Bender

2021

Phys. Rev. C

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“…The first code is restricted to spherical symmetry and is based on the actual diagonalization of the HFB matrix [32]. The second code, named TAURUS vap , solves HFB or variation after particle-number projection (VAPNP) equations for symmetry-unrestricted (real) Bogoliubov quasiparticle states [33], thus allowing for spatially deformed solutions. Employing a gradient method, the code can actually solve the variational equations under a large variety of constraints and was recently used to perform first practical calculations [34].…”

Section: A Numerical Set Upmentioning

confidence: 99%

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

2020

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Background:The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. In this context, pairing correlations may be driven towards zero either on purpose via HFB calculations constrained on, e.g., the particle-number variance or simply because internucleon interactions cannot sustain pairing correlations in the first place in, e.g., closed-shell systems. While taking this limit constitutes a text-book problem when the system is of closed-shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system. Purpose: The present work demonstrates that the zero-pairing limit of an HFB state is well-defined independently of the average particle number A it is constrained to. Still, the nature of the limit state is shown to depend on the regime, i.e., on whether the nucleus characterizes as a closed-shell or an open-shell system when taking the limit. Finally, the consequences of the zero-pairing limit on Bogoliubov many-body perturbation theory (BMBPT) calculations performed on top of the HFB reference state are illustrated. Methods: The zero-pairing limit of a HFB state constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction derived within the frame of chiral effective field theory. Results: The zero-pairing limit of the HFB state is mathematically well-defined, independently of the closedor open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number A of interest. First, the textbook situation is recovered for closed-shell systems, i.e., the limit state is reached for a finite value of the pairing strength (the well-known BardeenCooperSchrieffer (BCS) collapse) and takes the form of a single Slater determinant displaying (i) zero pairing energy, (ii) nondegenerate elementary excitations, and (iii) zero particle-number variance. Contrarily, a nonstandard situation is obtained for open-shell systems, i.e., the limit state is only reached for a zero value of the pairing interaction (no BCS collapse) and takes the form of a specific finite linear combination of Slater determinants displaying (a) a nonzero pairing energy, (b) degenerate elementary excitations, and (c) a nonzero particle-number variance for which an analytical formula is derived. This nonzero particle-number variance acts as a lower bound that depends in a specific way on the number of valence nucleons and on the degeneracy of the valence shell. All these findings are confirmed and illustrated numerically. Last but not least, BMBPT calculations of closed-shell (open-shell) nuclei are shown to be well-defined (ill-defined) in the zero-pairing limit. Conclusions: While HFB theory has been in...

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“…To illustrate our discussion on the behavior of the discretized projection operators, we will study the numerical convergence of the angular-momentum projection of several Slater determinants constructed in the sd-shell valence space using the numerical suite TAURUS [136,137]. The use of Slater determinants for such analysis has the advantage that no particle-number projection is required, thus simplifying the calculations and removing a source of numerical inaccuracies (in particular as different [N, Z] components will in general have different angular-momentum decompositions).…”

Section: By Applying the Discretized Operator Pk0mentioning

confidence: 99%

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

Bally,

Bender

2020

Preprint

Self Cite

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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...

show abstract

Symmetry-projected variational calculations with the numerical suite TAURUS I. Variation after particle-number projection

Cited by 3 publications

References 86 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 limit of Hartree-Fock-Bogoliubov reference states

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

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