The Nuclear Energy Density Functional Formalism

Abstract

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 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 mathematically meaningful fashion even if E[g ′ , g] does not derive from a genuine Hamilton operator. In particular, the restoration of symmetries can be entirely formulated without making 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 physical standpoint. The price at which the latter can be enforced as well in the future is eventually alluded to.

Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian

“…in MollerPlesset MBPT [63]. 13 The mixing coefficients {cα} are still computed from MBPT without particle number projection. Note that an alternative particle-number-restored MBPT based on a projective formula has been recently proposed [25] but not yet applied.…”

“…Off-diagonal coupling between low and high (relative) momenta can be tamed-down, at the price of inducing (hopefully weak) higher-body forces, by pre-processing arXiv:1610.04063v1 [nucl-th] 13 Oct 2016 the nuclear Hamiltonian via, e.g., a unitary similarity transformation [3]. Based on the transformed Hamiltonian, dynamical correlations 1 can be dealt with at a polynomial cost via standard many-body techniques typically based on systematic particle-hole-type expansions.…”

“…While traditionally developed within the frame of effective nuclear mean-field (i.e. energy density functional) approaches [11][12][13], this idea has been recently embraced to develop and implement ab initio Gorkov SCGF [14][15][16], multi-reference IMSGR [17,18] and Bogoliubov CC [19] many-body techniques to tackle pairing correlations 2 . This is achieved by allowing the reference state to break U (1) global gauge symmetry associated with particle-number conservation.…”

Skyrme functional from a three-body pseudopotential of second order in gradients: Formalism for central terms

“…(36) with transition (i.e. offdiagonal) density matrices [5], and multiplying the entire EDF kernel with a norm kernel.…”

Ab initio-driven nuclear energy density functional method

“…The violation of the Pauli principle in the EDF kernels relates itself to the traditional way of parametrizing many-body correlations under the form of density dependencies or to a relaxation of specific interrelations between the coefficients of the various terms at play. Solutions to better formulate the multi-reference EDF method and in particular the restoration of symmetries constitute a top priority today [24,25,26].…”