On the breaking and restoration of symmetries within the nuclear energy density functional formalism

We review the notion of symmetry breaking and restoration within the frame of nuclear energy density functional methods. We focus on key differences between wave-function- and energy-functional-based methods. In particular, we point to difficulties encountered within the energy functional framework and discuss new potential constraints on the underlying energy density functional that could make the restoration of broken symmetries better formulated within such a formalism. We refer to Ref.~\cite{duguet10a} for details.Comment: 5 pages, presented at the 17th nuclear physics conference "Marie and Pierre Curie", September 2010, Kazimierz Dolny, Polan

Skyrme functional from a three-body pseudo-potential of second-order in gradients. Formalism for central terms

In one way or the other, all modern parametrizations of the nuclear energy density functional (EDF) do not respect the exchange symmetry associated with Pauli's principle. It has been recently shown that this practice jeopardizes multi-reference (MR) EDF calculations by contaminating the energy with spurious self-interactions that, for example, lead to finite steps or even divergences when plotting it as a function of collective coordinates. As of today, the only viable option to bypass these pathologies is to rely on EDF kernels that enforce Pauli's principle from the outset by strictly and exactly deriving from a genuine, i.e. density-independent, Hamilton operator. We wish to develop the most general Skyrme-like EDF parametrization containing linear, bilinear and trilinear terms in the density matrices with up to two gradients, under the key constraint that it derives strictly from an effective Hamilton operator. The most general three-body Skyrme-like pseudo-potential containing up to two gradient operators is constructed to generate the trilinear part. The present study is limited to central terms. Spin-orbit and tensor will be addressed in a forthcoming paper. (See paper for full abstract)Comment: 38 pages revtex, no figur

Breaking and restoring symmetries within the nuclear energy density functional method

We review the notion of symmetry breaking and restoration within the frame of nuclear energy density functional methods. We focus on key differences between wave-function- and energy-functional-based methods. In particular, we point to difficulties to formulate the restoration of symmetries within the energy functional framework. The problems tackled recently in connection with particle-number restoration serve as a baseline to the present discussion. Reaching out to angular-momentum restoration, we identify an exact mathematical property of the energy density $E^{LM}(\vec{R})$ that could be used to constrain energy density functional kernels. Consequently, we suggest possible routes towards a better formulation of symmetry restorations within energy density functional methods.Comment: 16 pages, 3 figures, contribution to the "Focus issue on Open Problems in Nuclear Structure", Journal of Physics

Skyrme effective pseudopotential up to the next-to-next-to-leading order

The explicit form of the next-to-next-to-leading order ((NLO)-L-2) of the Skyrme effective pseudopotential compatible with all required symmetries and especially with gauge invariance is presented in a Cartesian basis. It is shown in particular that for such a pseudopotential there is no spin-orbit contribution and that the D-wave term suggested in the original Skyrme formulation does not satisfy the invariance properties. The six new (NLO)-L-2 terms contribute to both the equation of state and the Landau parameters. These contributions to symmetric nuclear matter are given explicitly and discussed

Ab initio derivation of model energy density functionals

I propose a simple and manageable method that allows for deriving coupling constants of model energy density functionals (EDFs) directly from ab initio calculations performed for finite fermion systems. A proof-of-principle application allows for linking properties of finite nuclei, determined by using the nuclear nonlocal Gogny functional, to the coupling constants of the quasilocal Skyrme functional. The method does not rely on properties of infinite fermion systems but on the ab initio calculations in finite systems. It also allows for quantifying merits of different model EDFs in describing the ab initio results

The nuclear energy density functional formalism

The present document focuses on the theoretical foundations of the nuclear energy density functional (EDF) method. As such, it does not aim at reviewing the status of the field, at covering all possible ramifications of the approach or at presenting recent achievements and applications. The objective is to provide a modern account of the nuclear EDF formalism that is at variance with traditional presentations that rely, at one point or another, on a {\it Hamiltonian-based} picture. The latter is not general enough to encompass what the nuclear EDF method represents as of today. Specifically, the traditional Hamiltonian-based picture does not allow one to grasp the difficulties associated with the fact that currently available parametrizations of the energy kernel $E[g',g]$ at play in the method do not derive from a genuine Hamilton operator, would the latter be effective. The method is formulated from the outset through the most general multi-reference, i.e. beyond mean-field, implementation such that the single-reference, i.e. "mean-field", derives as a particular case. As such, a key point of the presentation provided here is to demonstrate that the multi-reference EDF method can indeed be formulated in a {\it mathematically} meaningful fashion even if $E[g',g]$ does {\it not} derive from a genuine Hamilton operator. In particular, the restoration of symmetries can be entirely formulated without making {\it any} reference to a projected state, i.e. within a genuine EDF framework. However, and as is illustrated in the present document, a mathematically meaningful formulation does not guarantee that the formalism is sound from a {\it physical} standpoint. The price at which the latter can be enforced as well in the future is eventually alluded to.Comment: 64 pages, 8 figures, submitted to Euroschool Lecture Notes in Physics Vol.IV, Christoph Scheidenberger and Marek Pfutzner editor

Infinite matter properties and zero-range limit of non-relativistic finite-range interactions

We discuss some infinite matter properties of two finite-range interactions widely used for nuclear structure calculations, namely Gogny and M3Y interactions. We show that some useful informations can be deduced for the central, tensor and spin-orbit terms from the spin-isospin channels and the partial wave decomposition of the symmetric nuclear matter equation of state. We show in particular that the central part of the Gogny interaction should benefit from the introduction of a third Gaussian and the tensor parameters of both interactions can be deduced from special combinations of partial waves. We also discuss the fact that the spin-orbit of the M3Y interaction is not compatible with local gauge invariance. Finally, we show that the zero-range limit of both families of interactions coincides with the specific form of the zero-range N3LO Skyrme interaction and we emphasize from this analogy the benefits of N3LO.Comment: 30 page

Skyrme pseudo-potential-based EDF parametrisation for spuriousity-free MR-EDF calculations

First exploratory steps towards a pseudo-potential-based Skyrme energy density functional for spuriousity-free multi-reference calculations are presented. A qualitatively acceptable fit can be accomplished by adding simple three- and four-body contact terms to the standard central plus spin-orbit two-body terms. To achieve quantitative predictive power, higher-order terms, e.g.\ velocity-dependent three-body terms, will be required