Introduction to Modern Methods of Quantum Many-Body Theory and Their Applications

The following sections are included:

• CONTENTS

• PREFACE

Density functional theory is a remarkably successful theory of ordinary matter, despite its ad hoc origins. These lectures describe the theory and its applications starting from an elementary level. The practical theory uses the Kohn-Sham equations, well-chosen energy functionals, and efficient numerical methods for solving the Schroedinger equation. The time-dependent version of the theory is also useful for describing excitations. These notes are based on courses given by one of the authors (GFB) at the Graduiertenkolleg in Rostock, Germany in March 2001 and the Summer School on Microscopic Quantum Many-Body Theories in Trieste, Italy in September 2001.

A pedagogical introduction to the macroscopic description of helium liquids is presented. Particular attention is paid to the methodological aspects in the framework of the variational theory. After a short introduction to the physics of helium liquids, it is shown how a properly chosen trial wave function leads to a correct description of the ground state of an infinite and homogeneous boson system. The extension of the method allows also describing the main trends of the spectrum of elementary excitations. Furthermore, the problem of one ³He impurity on a background of ⁴He atoms is also analyzed and used to introduce perturbation theory in the framework of the Correlated Basis Function theory. The formalism is afterwards extended to treat Fermi systems. Finally, the excitation spectrum in a Fermi liquid and general aspects of the dynamic structure function are also discussed.

The purpose of these lectures is to present a pedagogical and illustrative, rather than exhaustive, review on the Coupled Cluster Method and some of its applications. The basic formalism to calculate the ground state energy of a general many-body system is presented, and a brief outline of its extensions to analyze other observables is also given. The main SUB(n) or CCn hierarchy of approximations is discussed, and some applications to systems interacting through a Coulomb potential are shown. The problems posed by the presence of a repulsive hard-core in the potential are briefly considered, and the HCSUB(n) approximation is formulated. To deal with finite systems such as nuclei or helium droplets, a translationally invariant approximation is formulated. This is first considered for a bosonic system, both in configuration and coordinate representation, and it is then generalized to the fermionic case. Results for the ⁴He and ¹⁶O atomic nuclei are presented, including only pair correlations in the wave function, and using semirealistic interactions of V4 form, and the V6 part of several realistic interactions. It is shown that the problems related to the existence of a very strong repulsion at short distances may be overcome by using a mixed description which incorporates scalar Jastrow correlations. This hybrid J-TICCn description, alternative to the HCSUB(n) approximation, is applied to the study of finite helium clusters, including up to three-body correlations in a Configuration-Interaction scheme. Detailed results are presented and compared with stochastic results for clusters of isotope ⁴He, both for ground and excited states. Ground state properties of ³He droplets are also discussed. To complement the pedagogical purpose of the lectures, some exercises are proposed to help the readers deepen their understanding of the Coupled Cluster approach to the study of many-body systems.

We report on three experiments based on a rubidium Bose-Einstein condensate. We examine the atomic micromotion of the condensate within a time dependent time orbiting potential. We have loaded Bose-Einstein condensates into one-dimensional, off-resonant optical lattices and accelerated them by chirping the frequency difference between the two lattice beams. We have studied the photoionization processes of the condensate.

I review some aspects of Bose–Einstein condensation in infinite and confined systems and discuss the differences between the dense and dilute cases. As an example of dense and strongly interacting system I focus on the zero temperature behavior of atomic ⁴He. The condensate fraction, the momentum distribution and the natural orbits of the homogeneous liquid and of the clusters are studied. I then address a dilute gas of hard–spheres, in the standard thermodynamic limit or trapped in an external potential. The Gross–Pitaevski equation is used to describe a large but finite number of alkali atoms in harmonic traps. The limitations of this approach when the gas parameter becomes large is finally discussed.

The equations of motion method for studying excitations and dynamic structure of quantum fluids is reviewed in this series of lectures. The method is based on the least action principle where one minimizes the action integral of the dynamic system. As a result one gets the continuity equations, which connect the density fluctuations and currents to an external driving force. The external force is assumed to infinitesimal and the response of the system to that is linear. The real poles of the linear response function determine the elementary excitation modes and the imaginary part of the self energy defines the continuum limit and gives the finite lifetime of the decaying modes. Our dynamic wave function contains time-dependent one- and two-particle correlation functions, which includes couplings between three modes. Thus one mode can split into two modes if energy and momentum are conserved. We begin with the Feenberg's β-derivative formulation of the optimized ground state and then derive general equations of motion for the dynamic system from the least action principle. We show how the simplest one-body approximation leads to the Feynman theory of excitations. By including the fluctuating two-body correlation function within the uniform limit one recovers the correlated basic function approximation. The fully consistent theory gives a good account of the elementary excitations and we show results on current patterns in the maxon-roton regions and on the precursor of the liquid-solid phase transition. Finally we apply the method to the excitations of the impurity and derive the hydrodynamic effective mass of the ³He impurity in ⁴He and the ³He dynamic structure function.

We describe the development of perturbation theory in a basis of correlated wave functions. The formulation of the basic theory is generic in the sense that it does not depend on a specific choice of correlation operator. We will, however, elaborate on the intricate relationships between cluster expansions for Jastrow–Feenberg wave functions on the one side, and perturbative expansions within the basis of Jastrow–Feenberg generated states.

The first part of this lecture series sets the general scene of perturbation theroy in a non–orthogonal basis, as laid out by Feshbach, Feenberg, Clark, and others. We then turn to the calculation of effective interactions, which will require the same technical tools that were developed for the Fermi-Hypernetted-Chain theory of the ground state. We then look at the interpretation of these effective interactions from various points of view and draw the connections to phenomenological theories like Landau's Fermi liquid theory, or the BCS theory of superconductivity. We also demonstrate that Fermi–sea averages of the CBF effective interaction vanish for optimized correlation functions.

Finally, we turn to cases where CBF theory to infinite order is necessary. Employing ideas of the coupled cluster theory, we develop a method for generating CBF perturbative corrections to infinite order by integral equations. Specializing to ring diagrams, we demonstrate that the "chain" diagrams of the FHNC theory are approximations for the ring diagrams of (CBF-)perturbation theory. The generalization of the time–dependent Hartree–Fock theory to correlated states will allow the formulation of a linear response theory for strongly interacting systems or, alternatively, provide the tools for mapping strong, bare interactions onto weak, effective interactions that can be used in phenomenological theories.

Liquid ³He is a model system which allows the investigation of Fermi liquids in a large variety of experimental conditions. In order to understand the effect of the interactions, its properties can be studied as a function of the particle density and temperature. Pure bulk liquid has been extensively investigated, and it constitutes nowadays the canonical example of a three-dimensional Fermi liquid. In addition, liquid ³He films adsorbed on a solid substrate offer the possibility to study two-dimensional Fermi fluids where the interactions can be varied in a much wider range than in the bulk. The nuclear magnetism of these systems reveals the subtle properties associated to the large zero point motion of the light ³He atoms and to their interactions. We discuss in this lecture NMR measurements performed on these Fermi Liquids and the consequences of the results on the value of the Landau parameters which will be required as an input by future microscopic theories.

Studies of the Schrödinger equation solutions where the wave function is expanded in terms of a basis of hyperspherical harmonic functions started a long time ago and have continued until present times. This approach has been applied to different types of microscopic many-particle systems and its degree of success is strongly determined by the number of particles and by their mutual interactions. For three- and four-particle systems and soft interactions the required number of basis functions is not particularly large. For bigger systems and/or strong interactions this number becomes so great that modifications of the original method need to be devised in order to solve the corresponding numerical problem with present-day computing systems. A few of these modifications, such as using the potential basis, the adiabatic and the correlated expansions are discussed here. Results of the calculations are presented for the bound states of the helium atom and of nuclei with A = 3 and 4, interacting through realistic two-nucleon potentials. More recently a lot of effort has been devoted to the extension, and modification, of the hyperspherical harmonic approach to scattering problems involving charged particles. As a consequence, the properties of few-nucleon reactions of astrophysical interest can be studied and the results will be briefly discussed.

The most recent developments in the nuclear many–body problem are briefly reviewed, trying to focus on the major advances reached in the last few years and on the still open problems. Particular attention is devoted to quantum simulations for nuclear and neutron matter, which have recently received much attention, and, given the rapid increase in computer power, pose themeselves amongst the best candidates for a new generation of ab initio calculations in nuclear physics. The newly developed Auxiliary Field Diffusion Monte Carlo method, or other stochastic methods using auxiliary field ideas, appear to embody all the main features to perform simulations of large nuclear systems, solving, under this respect, the long standing spin problem. The use of the Auxiliary Field Diffusion Monte Carlo method to compute properties of many particle systems interacting via spin-isospin dependent nuclear potentials is described. By combining diffusion Monte Carlo for the spatial degrees of freedom and auxiliary field Monte Carlo to separate the spin-isospin operators, ground state energies and other properties can be calculated for medium size nucleon systems. Recent applications of the method to obtain the equation of state and the compressibilty of neutron matter are presented and discussed. These calculations use realistic interactions such as the Argonne and two–nucleon potentials plus the Urbana IX three-nucleon potential. Other properties of astrophysical interest such as the spin susceptibility of neutron matter and the symmetry energy are also discussed. A new homework problem is proposed to test the modern many–body techniques with a nuclear potential which includes tensor, spin–orbit and three–body interactions.

The following sections are included:

• INDEX