Particle number projection with effective forces

Anguiano, M.; Egido, J.L.; Robledo, L. M.

doi:10.1016/s0375-9474(01)01219-2

Cited by 206 publications

(350 citation statements)

References 35 publications

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“…Nowadays, a lot of efforts is applied to move beyond the mean field approximation and account for missing correlations. Special attention is paid to the restoration of symmetries broken in mean field approaches [1][2][3][4][5][6][7][8][9], for example using projection techniques. An alternative way is to develop a theory in which the trial wave functions preserve certain symmetries.…”

Section: Introductionmentioning

confidence: 99%

Low-lying spectroscopy of a few even-even silicon isotopes investigated with the multiparticle-multihole Gogny energy density functional

et al. 2012

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A multiconfiguration microscopic method has been applied with the Gogny effective interaction to the calculation of low-lying positive-parity states in even-even 26−32 Si isotopes. The aim of the study is to compare the results of this approach with those of a standard method of GCM type and to get insight into the predictive power of multiconfiguration methods employed with effective nucleonnucleon force taylored to mean-field calculations. It is found that the multiconfiguration approach leads to an excellent description of the low-lying spectroscopy of 26 Si, 28 Si and 32 Si, but gives a systematic energy shift in 30 Si. A careful analysis of this phenomenon shows that this discrepancy originates from too large proton-neutron matrix elements supplied by the Gogny interaction at the level of the approximate resolution of the multiparticle-multihole configuration mixing method done in the present study. These proton-neutron matrix elements enter in the definition of both singleparticle orbital energies and coupling matrix elements. Finally, a statistical analysis of highly excited configurations in 28 Si is performed, revealing exponential convergence in agreement with previous work in the context of the shell model approach. This latter result provides strong arguments towards an implicit treatment of highly excited configurations.

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Section: Introductionmentioning

confidence: 99%

Low-lying spectroscopy of a few even-even silicon isotopes investigated with the multiparticle-multihole Gogny energy density functional

et al. 2012

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“…Since then, BCS or the more general Hartree-Fock-Bogoliubov (HFB) theory, combined with effective or phenomenological nuclear forces, has been the standard tool to describe the low-energy properties of heavy nuclei. Improvements over BCS or HFB came through the restoration of broken symmetries, especially particle-number projection, which is still a problem not satisfactorily solved with density-dependent forces [3]. From a different perspective, Richardson found an exact solution of the constant-pairing problem with nondegenerate single-particle energies as early as 1963 [4].…”

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confidence: 99%

Exactly solvable pairing Hamiltonian for heavy nuclei

et al. 2011

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We present a new exactly solvable Hamiltonian with a separable pairing interaction and nondegenerate single-particle energies. It is derived from the hyperbolic family of Richardson-Gaudin models and possesses two free parameters, one related to an interaction cutoff and the other to the pairing strength. These two parameters can be adjusted to give an excellent reproduction of Gogny self-consistent mean-field calculations in the Hartree-Fock basis. Pairing is one of the most important ingredients of the effective nuclear interaction in atomic nuclei, as was recognized early on by Bohr et al. [1] in an attempt to explain the large gaps observed in even-even nuclei. They suggested that the newly proposed Bardeen-Cooper-Schriefer (BCS) [2] theory of superconductivity could be a useful tool in nuclear structure, although care should be taken with the violation of particle number in finite nuclei. Since then, BCS or the more general Hartree-Fock-Bogoliubov (HFB) theory, combined with effective or phenomenological nuclear forces, has been the standard tool to describe the low-energy properties of heavy nuclei. Improvements over BCS or HFB came through the restoration of broken symmetries, especially particle-number projection, which is still a problem not satisfactorily solved with density-dependent forces [3]. From a different perspective, Richardson found an exact solution of the constant-pairing problem with nondegenerate single-particle energies as early as 1963 [4]. Though highly schematic, the constant-pairing force has been used for decades in nuclear structure with several approximations [BCS, random-phase approximation (RPA), projected BCS (PBCS), etc.], but rarely resorting to the exact solution. Almost forgotten, the exact Richardson solution was recovered within the framework of ultrasmall superconducting grains [5], in which not only number projection but also pairing fluctuations were essential to describe the disappearance of superconductivity as a function of the grain size.By combining the Richardson exact solution with the integrable model proposed by Gaudin [6] for quantum spin systems, it was possible to derive three families of integrable models called Richardson-Gaudin (RG) models [7]. The rational family, extensively used since then, contains the Richardson model as a particular exactly solvable Hamiltonian, as well as many other exactly solvable Hamiltonians of relevance in quantum optics, cold-atom physics, quantum dots, etc. [8]. However, the other families did not find a physical realization until very recently when it was shown that the hyperbolic family could model a p-wave pairing Hamiltonian in a two-dimensional lattice [9], such that it was possible to study, with the exact solution, an exotic phase diagram having a nontrivial topological phase and a third-order quantum phase transition [10]. In this Rapid Communication, we will show that the hyperbolic family gives rise to a separable pairing Hamiltonian with two free parameters that can be adjusted to reproduce the properties of hea...

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“…Despite this success, the necessity of increasing the accuracy of theoretical predictions in some cases has motivated several groups to formulate beyond-mean-field models, which explicitly include more correlations in their formal scheme. Several directions have been explored, for instance: projection techniques in the framework of the generator coordinate method [2][3][4][5]; second randomphase approximation (SRPA) calculations with Skyrme and Gogny forces [6,7], as well as with an interaction derived from a realistic force [8]; particle-vibration coupling (PVC) techniques with the Skyrme interaction [9,10] and with a relativistic Lagrangian [11]; multiparticlemultihole configuration mixing (mpmhCM) methods with both Skyrme [12] and Gogny [13] forces.…”

Section: Introductionmentioning

confidence: 99%

Renormalizability of the nuclear many-body problem with the Skyrme interaction beyond mean field

et al. 2017

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Phenomenological effective interactions like Skyrme forces are currently used in mean-field calculations in nuclear physics. Mean-field models have strong analogies with the first order of the perturbative many-body problem and the currently used effective interactions are adjusted at the mean-field level.In this work, we analyze the renormalizability of the nuclear many-body problem in the case where the effective Skyrme interaction is employed in its standard form and the perturbative problem is solved up to second order. We focus on symmetric nuclear matter and its equation of state, which can be calculated analytically at this order. It is shown that only by applying specific density dependence and constraints to the interaction parameters could renormalizability be guaranteed in principle. This indicates that the standard Skyrme interaction does not in general lead to a renormalizable theory. For achieving renormalizability, other terms should be added to the interaction and employed perturbatively only at first order.

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Particle number projection with effective forces

Cited by 206 publications

References 35 publications

Low-lying spectroscopy of a few even-even silicon isotopes investigated with the multiparticle-multihole Gogny energy density functional

Low-lying spectroscopy of a few even-even silicon isotopes investigated with the multiparticle-multihole Gogny energy density functional

Exactly solvable pairing Hamiltonian for heavy nuclei

Renormalizability of the nuclear many-body problem with the Skyrme interaction beyond mean field

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