View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by NORA - Norwegian Open Research Archives Parquet theory in nuclear structure calculations Elise Bergli Thesis submitted to the degree of Philosophiae Doctor Department of Physics University of Oslo February 2010 © Elise Bergli, 2010 Series of dissertations submitted to the Faculty of Mathematics and Natural Sciences, University of Oslo No. 926 ISSN 1501-7710 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. Cover: Inger Sandved Anfinsen. Printed in Norway: AiT e-dit AS. Produced in co-operation with Unipub. The thesis is produced by Unipub merely in connection with the thesis defence. Kindly direct all inquiries regarding the thesis to the copyright holder or the unit which grants the doctorate. Acknowledgements This thesis is primarily, of course, my own work. It has taken quite some time, not all of it pleasant, and I look forward to finally be doing something else. However, looking back, there are several people whom I would like to thank. Without them, I would not have been able to finish. First of all, I thank my adviser Morten Hjorth-Jensen. You have pro- vided more advise and encouragement than most advisers, being accessible and ready to answer questions at almost all times, regardless of time or place. I have really learned a lot from you these years. And I would also like to thank my co-adviser Eivind Osnes for pleasant discussions and having such an immense amount of literature available. I would like to thank all my different office mates during these years: Victoria, Simen, Mateuz, Jon, Konrad, Maxim, Gustav and Øyvind. You have provided lots of interesting discussions and diversions on almost any subject imaginable. Sometimes we have even discussed physics... Most of you have also been able to provide insights and help me with numerous technical problems. Thanks also to Joakim, for all our lunch discussions and the last- minute proof-reading and comments. And then there is as always my family, my most constant source of hap- piness. Thank you for simply being, my life would feel empty without you. Contents Contents i 1 Introduction 1 2 Ab initio methods for nuclear structure calculations 7 2.1Many-bodyformalismandquasiparticles............ 8 2.2Thebarenucleon-nucleon(NN)interaction........... 11 2.3Effectiveinteractionsinfinitemodelspaces........... 13 2.4 Ab Initio approachesforlightandmedium-massnuclei..... 17 2.5Effectiveinteractionsintheshellmodel............. 20 2.6Nucleardensityfunctionaltheory................. 21 3 Many-Body theory and Green’s functions 23 3.1 The one-particle propagator . ........... 23 3.2Thefour-pointGreen’sfunction.................. 26 3.3Selfenergy............................. 30 3.4Theeigenvalueequationmethod................. 34 4 Parquet summation of diagrams 37 4.1 Channels: Equivalent ways of building the interaction operator 38 4.2 Two-time propagators . ..................... 44 4.3Self-consistentParquetequations................. 48 4.4ApproximatesolutiontotheDysonequation.......... 56 4.5Summaryofthedifferentapproximations............ 58 5 A simple model 63 5.1Descriptionofthemodel...................... 63 5.2Resultsforthepairing-onlymodel................ 65 5.3Pair-breakinginteraction...................... 81 5.4 Selected combinations of g and f .................106 i ii CONTENTS 5.5Summary..............................108 6Resultsfor4He 111 6.1 Results for unperturbed propagator structure (UPS) approxi- mation................................112 6.2 Results for fixed-energy Dyson equation (FED) approximation 138 6.3 Summary of the 4Heresults....................154 7 Conclusions and future work 157 Bibliography 163 1 Introduction Most real systems consist of several interacting bodies, and so many-body physics is an essential ingredient in any realistic description of the physical world. Quantum many-body theory concerns the fundamental properties of systems where the quantum nature of the constituents plays an important role. These include both small systems like nucleons, nuclei, atoms and small molecules, and large-scale systems as electrons in matter and nuclear matter in neutron stars. The problems encountered are notoriously complex and hard to solve, and many strategies exists to cope with different systems. The fundamental challenge common to all such problems is that describing the motion of all particles involved amounts to keeping track of far too much information for any straight-forward course of attack. Methods developed to handle this complex situation and transform it into something tractable are therefore mostly quite general and can be applied to a number of different systems, each providing different testing grounds for the general methods. The physical systems chosen as subject for study in this thesis are low- energy nuclear systems, that is, nuclei and nuclear matter on an energy scale where the appropriate building blocks are the nucleons, the protons and neu- trons. The range of possible combinations of nucleons spans from the single proton in a hydrogen nucleus to the couple of hundreds found in the the heav- iest nuclei, and then on to the (essentially) infinite number of particles in a neutron star. Nuclear systems are very challenging quantum many-body problems due both to the complexity of the strong force and to the number of particles involved, most nuclei being in the difficult mesoscopic range of too many particles to handle each separately and too few for statistical treatments. The aim of ab initio methods in nuclear structure calculations is to de- scribe the nucleus and nuclear matter from first principles only, with as little use of phenomenological determination as possible, ultimately building a co- herent framework enabling calculations of low-energy nuclear structure and reaction ratios starting from the field theories of sub-nuclear particle physics and the standard model of Quantum Chromo Dynamic (QCD). If successful, 1 2 CHAPTER 1. INTRODUCTION not only will we gain the satisfaction of being able to study nuclei and nuclear reactions from fundamental principles, but also the possibility of predicting key properties of short-lived nuclei present in stellar interiors and generated during supernova explosions, furthering our understanding of the ratios of dif- ferent nuclei found in our part of the universe. On a more practical level, numerical computations across the range of exotic nuclei opens up for far eas- ier determination of isotopes with specific characteristics that can be used in medicinal diagnostic and therapeutic treatments. Accurate computations could also be an ingredient in fine-tuning nuclear reactor designs to increase safety and efficiency. The complexity of the computations increases very rapidly with increasing particle number. A layered approach is necessary, where only the relevant degrees of freedom for a given system are taken into account. Starting from QCD, the first step is to generate a theory where only nucleons exist, and all effects of the underlying strong force is taken into account by the effective force between the nucleons. This effective force acts between two nucleons in vacuum, and is called the bare interaction. Then we can start to put the nucleons together to form nuclei. Nuclei are quantum systems, and as such, there exists several general approaches which can be applied to these systems. One of the major obstacles is that the interaction between pairs of nucleons changes dramatically in the presence of other nucleons. The complexity of the computations increases and makes approaches to nuclear structure calcu- lations based on the bare two-body interaction directly very impractical for heavier nuclei. A popular simplification is to generate a new effective interac- tion which takes the presence of the surrounding nucleons into account. This thesis concerns one possible framework, the summation of Parquet diagrams within Green’s function theory, to obtain an effective interaction and use it to calculate properties of interest for finite nuclei. The Green’s functions (propagator) method is one of the standard the- oretical descriptions of quantum mechanics, underlying the huge family of Feynman diagram methods employed in most high-energy physics. In the many-body context, two features of many-body systems provides the basis for applying propagator techniques. Firstly, almost all quantum systems of strongly interacting particles (where two particles exchange large amounts of energy each time they interact) can be described as a weakly interacting system of quasi particles (where the quasi particles exchange much smaller amounts of energy when they interact), given the right choice of definition of the quasi particles. Secondly, to find most of the important physical proper- ties of a system, it is not necessary to know the behaviour of each particle exactly, rather it is enough to know the average behaviour of one or two typ- 3 ical particles. The quantities describing this average behaviour are the one- and two-body propagators. Once these are known, most physical properties of the system can be derived. The mathematical transparency, the generality and the close connection between the fundamental definitions and the physical observables makes Green’s function based theories ideal for studies of general properties of many-body theories and the relations between different methods. Within the field of nuclear structure calculations, this framework has