Self-consistent Green's functions with three-body forces

Abstract

The quantum many-body problem is an everlasting challenge for theoretical physics. The aim sought is to analyze, at the quantum level, the observables arising from a group formed by a certain number of interacting particles N. In the specific case of nuclear physics, the study and interpretation of interacting nuclear systems has spanned from the finite nuclei to the infinite nuclear matter case. In order to characterize such systems, the forces with which nucleons interact in their interior need to be sorted out. Once the Hamiltonian is chosen, the many-body problem for the group of interacting fermions, the nucleons, has to be solved.

In the present thesis we focus our study on infinite nuclear matter, analyzing the properties of the two systems which stand at the extremes in terms of isospin asymmetry: symmetric nuclear matter (SNM), with an equal number on neutrons and protons, and pure neutron matter (PNM), with only neutrons. We base our approach in terms of realistic microscopic potentials, hence we need to choose a many-body technique in order to treat the potential. We choose to perform our calculations exploiting the formalism provided by the self-consistent Green's function's (SCGF) theory. This approach is based on a diagrammatic expansion of the single-particle (SP) Green's function, or propagator. This method was devised to treat the correlated, i.e. beyond mean-field, behavior of strongly interacting systems, such as nuclear matter.

Unfortunately, it is a well established fact that whatever microscopic two-nucleon forces (2NFs) are used in the many-body calculation, empirical saturation properties of nuclear matter fail to be reproduced. Saturation densities appear at high values, presenting energies which are inconsistently too attractive, overbinding nuclear matter. This inconsistency bears some similarities to the case of light nuclei where, on the contrary, the 2NF-only based theory underbinds experimental data. The inclusion of three-nucleon forces (3NFs) has been the indispensable factor to cure this deficiency. Hence, microscopic 3NFs have been mostly devised to provide attraction in light finite systems, small densities, and repulsion in infinite systems, high densities. However, in modeling these 3NFs, the inclusion of phenomenological ingredients has often been the way followed. To bypass the need to adjust the potential with ad hoc contributions, an alternative has been provided by chiral effective field theory (EFT).

Featuring the consistent Hamiltonian at the 2N and 3N level provided by EFT, in this thesis we present calculations including both chiral 2NFs and 3NFs. This is dealt within an extended SCGF formalism, specifically formulated to include consistently 3B forces by means of effective interactions.

We observe the striking effect of the inclusion of 3N forces for the total energy of the many-body ground state. Exploiting the extended SCGF approach presented in this thesis, we see how the modifications induced by the 3B force are larger as the density increases. This increasing with density is what provides the saturation mechanism for nuclear matter. In pure neutron matter, 3NFs are the main cause for the stiffening of the equation of state. In view of astrophysical studies for neutron star masses, this stiffening is a major ingredient for the achievement of theoretical results which can better match recent astrophysical observations.

The idea, on which the development of this thesis was based, has been all the way the quest to introduce consistently three-body forces in the formalism of the SCGF theory. While the main motivation of this study has been nuclear systems, the extended SCGF formalism can be easily applied to other many-body systems, of either atomic or molecular nature.

El problema quàntic de molts cossos ha estat sempre un repte de la física teòrica. L'objectiu al qual s'intenta arribar es la descripció, des d'un punt de vista quàntic, dels observables d'un sistema de moltes partícules en interacció. En el camp específic dels sistemes nuclears, dels més petits com els nuclis als més grans com la matèria nuclear infinita dins de les estrelles, el problema nuclear de molts cossos s'ha intentat resoldre amb la construcció de diferents formalismes. L'objectiu ha sigut descriure aproximadament el comportament de molts nucleons, obtenen resultats els més propers als experimentals i empírics.

En la present tesis ens hem centrat sobre l'estudi de la matèria nuclear infinita. En particular sobre els dos sistemes infinits que es situen en els extrems en termes d'asimetria d'isospín: la matèria nuclear simètrica (SNM), el sistema format d'un igual nombre de protons i nucleons, i la matèria purament neutrónica (PNM), el sistema format solament de neutrons.

Per dur a terme aquests càlculs hem expandit, de manera consistent, el formalisme de funcions de Green auto-consistents per incloure forces a tres cossos. Hem treballat amb un potencial basat en la teoria efectiva de baixes energies aplicada a la Cromodinàmica Quàntica (QCD), i.e. la teoria efectiva quiral. En aquesta teoria, les 2NFs i 3NFs apareixen consistentment.

La inclusió de les forces a tres cossos en els càlculs s'ha demostrat ser fonamental per obtenir el mecanisme de saturació en SNM al voltant de valors acceptables d'energia/densitat i per proporcionar una equació d'estat més rígida per al sistema de PNM. Això demostra, un cop més, la necessitat de considerar les forces de molts cossos per fel càlculs teòrics consistents amb els resultats empírics i experimentals.