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Parquet theory in nuclear structure calculations

Elise Bergli

Published Jan 1, 2010• 1 citation
OpenAlex

Abstract

The thesis concerns a numerical implementation of the Parquet summation of diagrams within Green's functions theory applied to calculations of nuclear systems. The main motivation has been to investigate whether it is possible to develop this approach to a level comparable in accuracy and reliability to other ab initio nuclear structure methods. \n\nThe Green's functions approach is theoretically well-established in many-body theory, but to our knowledge, no actual application to nuclear systemshas been previously published. It has a number of desirable properties, foremostthe gently scaling with system size compared to direct diagonalization and the closeness to experimentally accessible quantities. The main drawback is the\nnumerical instabilities due to the pole structure of the one-particle propagator, leading to convergence difficulties. This issue is one of the main focal points of the work presented in this thesis, and strategies to improve the convergence properties are described and investigated. \n\nWe have applied the method both to a simple model which can be solved by exact diagonalization and to the more realistic 4He system. The results shows that our implementation is close to the exact solution in the simple model as long as the interaction strengths are small. As the number of particles increases,\nconvergence is increasingly hard to obtain. In the 4He case, we obtain results in the vicinity of the results from comparable approaches. The numerical in-stabilities in the current implementation still prevents the desired accuracy and\nstability necessary to achieve the current benchmark standards.

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The thesis concerns a numerical implementation of the Parquet summation of diagrams within Green's functions theory applied to calculations of nuclear systems. The main motivation has been to investigate whether it is possible to develop this approach to a level comparable in accuracy and reliability to other ab initio nuclear structure methods. \n\nThe Green's functions approach is theoretically well-established in many-body theory, but to our knowledge, no actual application to nuclear systemshas been previously published. It has a number of desirable properties, foremostthe gently scaling with system size compared to direct diagonalization and the closeness to experimentally accessible quantities. The main drawback is the\nnumerical instabilities due to the pole structure of the one-particle propagator, leading to convergence difficulties. This issue is one of the main focal points of the work presented in this thesis, and strategies to improve the convergence properties are described and investigated. \n\nWe have applied the method both to a simple model which can be solved by exact diagonalization and to the more realistic 4He system. The results shows that our implementation is close to the exact solution in the simple model as long as the interaction strengths are small. As the number of particles increases,\nconvergence is increasingly hard to obtain. In the 4He case, we obtain results in the vicinity of the results from comparable approaches. The numerical in-stabilities in the current implementation still prevents the desired accuracy and\nstability necessary to achieve the current benchmark standards.
Abstract The main task of microscopic many-body theory is to provide an understanding and an explanation of properties of macroscopic systems from no other information than the properties of the underlying Hamiltonian, the particle statistics, and the macroscopic geometry of the system. Variational wave functions have been a very successful technique for examining the ground state and dynamic properties of dense quantum fluids. Specifically the “optimized (Fermi)-hypernetted chain” ((F)HNC-EL) hierarchy of approximations reproduces, for simple interactions like electrons and quantum fluids, the most basic features of a self-bound many-body system, namely binding, saturation, and spinodal decomposition. The relationships between Green’s functions based perturbation theory and variational wave functions have been clarified in much detail in the past. In the language of Feynman diagrams, a self-consistent summation of ring- and ladder-diagrams is the minimum requirement for a satisfactory microscopic description of these basic features. Realistic nucleon–nucleon interactions pose further problems: The most difficult aspect of the nuclear many-body problem is caused by the form of the microscopic nucleon–nucleon interaction which depends at least on the spin, isospin, the relative orientation and angular momentum of the interacting particles. Simple variational methods as used for electrons and quantum fluids are inadequate. Correlation functions that have the same structure as the interactions lead to so-called “commutator diagrams” that have, so far, been mostly neglected. We have shown in the past that these corrections can be very important if the two-body interactions are very different in different spin- or isospin channels. To deal with the problem of realistic nuclear interactions we have combined techniques of the Jastrow–Feenberg variational method and the local parquet-diagram theory. In the language of diagrammatic perturbation theory, “commutator diagrams” can be identified with non-parquet diagrams. We examine the physical processes described by these terms and include the relevant diagrams in a way that is suggested by the Jastrow–Feenberg approach. We show that the corrections from non-parquet contributions are, at short distances, larger than all other many-body effects. We examine here neutron matter as a prototype of systems with state-dependent interactions. Calculations are carried out for neutrons interacting via the so-called v8v_8v8​ v 8 version of four popular interactions. We determine the structure and effective interactions and apply the method to the calculation of the energetics, structure and dynamic properties such as the single-particle self-energy and the dynamic response functions as well as BCS pairing in both singlet and triplet states. We find that many-body correlations lead to a strong reduction of the spin-orbit interaction, and, therefore, to a suppression of the 3P2^3\textrm{P}_23P2​ 3 P 2 and 3P2^3\textrm{P}_23P2​ 3 P 2 - 3F2^3\textrm{F}_23F2​ 3 F 2 gaps. We also find pairing in 3P0^3\textrm{P}_03P0​ 3 P 0 states; the strength of the pairing gap depends sensitively on the potential model employed.
The solution of the quantum many-body problem for medium-mass nuclei using realistic nuclear interactions poses a superbe challenge for nuclear structure research. Because an exact solution can only be provided for the lightest nuclei, one has to rely on approximate solutions when proceeding to heavier systems. Over the past years, tremendous progress has been made in the development and application of systematically improvable expansion methods and an accurate description of nuclear observables has become viable up to mass number A≈100A \approx 100A≈100. While closed-shell systems are consistently described via a plethora of different many-body methods, the extension to genuine open-shell systems still remains a major challenge and up to now there is no \textit{ab initio} many-body method which applies equally well to systems with even and odd mass numbers. The goal of this thesis is the development and implementation of innovative perturbative approaches with genuine open-shell capabilities. This requires the extension of well-known single-reference approaches to more general vacua. In this work we choose two complementary routes for the usage of generalized reference states. First, we derive a new \emph{ab initio} approach based on multi-configurational reference states that are conveniently derived from a prior no-core shell model calculation. Perturbative corrections are derived via second-order many-body perturbation theory, thus, merging configuration interaction and many-body perturbation theory. The generality of this ansatz enables for a treatment of medium-mass systems with arbitrary mass number, as well as the extension to low-lying excited states such that ground and excited states are treated on an equal footing. In a complementary approach, we use reference states that break a symmetry of the underlying Hamiltonian. In the simplest case this corresponds to the expansion around a particle-number-broken Hartree-Fock-Bogoliubov vacuum which is obtained from a mean-field calculation. Pairing correlations are already absorbed in the reference state. The mild scaling allows for investigations up to tin isotopic chains. All benchmarks were performed using state-of-the-art chiral two- plus three-nucleon interactions thus allowing for a universal description of nuclear observables in the medium-mass regime.
In the interest of polishing the parquet approach to microscopic calculations of ground-state properties of zero-temperature quantum fluids, we have developed a perturbative scheme for the systematic improvement of the four-point function which retains the correct behavior of the parquet diagrams in both the small-r and small-k limits. Numerical implementation of this scheme indicates significantly improved agreement with binding energies calculated with Green's-function Monte Carlo techniques (i.e., differences in liquid 4He^{4}\mathrm{He}4He are less than 1.5%). The charged Bose gas and the hard-sphere system are also considered. Remaining difficulties in the calculations of the liquid structure function are noted and discussed.
This dissertation consists of two independent parts: density functional theory (Part I), and nuclear quantum effects (Part II). Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory in condensed matter systems. The computational time of KSDFT increases rapidly with respect to the number of electrons in the system, which hinders its practical application to systems of large size. The central quantities in KSDFT are the electron density and the electron energy, which can be fully characterized by the diagonal elements and the nearest off-diagonal elements of the single particle density matrix. However, methods that are currently available require the calculation of the full density matrix. This procedure becomes highly inefficient for systems of large size. Part I of this dissertation develops a new method for solving KSDFT, which directly targets at the calculation of the diagonal and the nearest offdiagonal elements of the single particle density matrix. The new method is developed under the framework of Fermi operator expansion. The new method achieves the optimal expansion cost in the operator level. The electron density and the electron energy is then evaluated from a serires of Green’s functions by a new fast algorithm developed in this dissertation. This dissertation also develops a novel method for discretizing the Hamiltonian of the system that achieves high accuracy with a very small number of basis functions. Combining all these components together, we obtain a unified, accurate, efficient method to solve KSDFT for insulating and metallic systems. Nuclear quantum effects play an important role in a large variety of hydrogen bonded systems such as water and ice due to the small mass of protons (the nuclei of hydrogen atoms). The equilibrium proton dynamics is reflected in the quantum momentum distribution and is the focus of intense research. The standard open path integral formalism for computing the quantum momentum distribution requires the
The equation of state of symmetric nuclear matter is addressed starting both from a realistic interaction derived from nucleon-nucleon scattering processes and from a low-momentum effective potential. The approach is based on finite temperature Green’s functions. The internal energy per particle is estimated from the summation of diagrams and through the Galitskii-Koltun’s sum rule. At present, there is not a unique and reliable theory of nuclear interactions. From the study of nucleon-nucleon scattering processes different models for the basic interaction between nucleons have been developed, but difficulties arise when they are used in calculations which aim to reproduce the properties of bound nuclei. These models all reproduce the experimental N-N phase shifts up to the threshold for pion production with high accuracy, but differ in the short-wavelength region, where less experimental constraints are available. In deriving the properties of dense nuclear matter, if we start from a bare N-N potential we have to take into account the complex correlations induced
The quantum theory of many-particle systems is deeply connected to the mathematical formalism and techniques of quantum field theory, which are crucial in order describe the emergent phenomena that arise in condensed matter physics. The main goal of this thesis is to provide a unified description of quantum field theory methods in the context of many-particle physics in order to extend and develop these techniques to better understand the different quantum properties of matter. This both furthers the conceptual understanding of quantum many-body correlation and also allows for the development of novel theoretical techniques that combine the different formalisms employed in condensed matter theory, quantum chemistry and nuclear physics. To achieve this aim, the thesis is composed of two parts. Part I deals with many-body fermion-fermion interactions and is physically motivated by the non-relativistic electronic structure problem. Here, we introduce the Keldysh contour formulation of quantum field theory to provide a unified presentation of quantum systems both in and out-ofequilibrium. Then, we extend the conventional quantum theory of many-particle systems, based on hermitian quantum mechanics, to describe systems governed by non-hermitan Hamiltonians. This allows us to present a natural unification of two major pillars in the ab initio quantum theory of many-particle systems: coupled-cluster theory and the Green’s function formalism. This unification is further demonstrated through the emergence of the coupled-cluster amplitude equations from the Green’s function formalism. In particular, this analysis has direct bearing on the development of novel theoretical approaches to the electronic structure and nuclear many-body problems. Our findings are also relevant to the study of general non-hermitian as well as open quantum systems. Part II focuses on the quantum field theory of electron-nuclear interactions in infinite solids. Here, we use the Keldysh contour representation presented in Part I to understand excited state dynamics that include electron-electron and electron-lattice interactions in materials. We also derive the analytic continuation of the Green’s function for systems that have continuous eigenspectra. This analysis demonstrates the relationship between the Green’s function formalism and scattering theory, making the role of the continuum and causality conditions explicitly clear in the context of the pole condition of the Dyson equation. Using the analytic continuation of the propagator, we provide an in-depth analysis of electron-phonon and exciton-phonon scattering channels, with a particular focus on phonon-driven exciton dissociation scattering. Our theoretical analysis then firmly demonstrates the connection between the extension of the Bethe-Salpeter equation approach to include electron-phonon interactions and effective field theories based on composite exciton-phonon coupling. We subsequently combine the techniques presented in Part I, based on the Dyson supermatrix formulation, with the analysis presented in Part II to calculate, from first principles, the effects of including electron-phonon interactions on the excited states of heterogeneous semiconductors. Finally, we provide a summary of the fundamental contributions presented in this thesis and the potential further research directions to be pursued.