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AIP conference proceedings

Unified theory of ground state properties of nuclei and nuclear dynamics

VICTOR A. KHODEL, Vasily R. Shaginyan

Published Jan 1, 1992• 0 citations
Publisher

Abstract

A new microscopic approach to the theory of dynamic correlation, ground state and collective excitations of many‐body systems is suggested. These dynamic and ground state properties are known as being interconnected and should be considered on the same base. A useful tool for this goal is a response function formalism. An exact functional equation for effective interaction and linear response function is obtained. The reduction method of the functional equation to a closed set of equations is given. In the simplest case the functional equation may be transformed into nonlinear integrodifferential equation. Few examples of many‐body systems are considered on the base of this equation. The developed method is applied to gas‐liquid phase transition, for evaluation of transport coefficients and collective excitation width.

DOI
10.1063/1.42006

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A new microscopic approach to the theory of dynamic correlation, ground state and collective excitations of many‐body systems is suggested. These dynamic and ground state properties are known as being interconnected and should be considered on the same base. A useful tool for this goal is a response function formalism. An exact functional equation for effective interaction and linear response function is obtained. The reduction method of the functional equation to a closed set of equations is given. In the simplest case the functional equation may be transformed into nonlinear integrodifferential equation. Few examples of many‐body systems are considered on the base of this equation. The developed method is applied to gas‐liquid phase transition, for evaluation of transport coefficients and collective excitation width.
Ground-state energies and frequencies of collective modes for the Lipkin model are evaluated within the framework of a microscopic functional approach using functional relations between the ground-state energy, the matrix of the effective interaction R^ between particles in the medium, and the linear response function of a system. A strict functional equation for R^ is derived. A special version of the well-known local approximation is elaborated to transform this equation into a system of integro-differential equations, which is solved numerically. Most of the numerical calculations are carried out in the region of moderate coupling constants, where the ground state possesses the same symmetry as the noninteracting system. The deviation of the calculated ground-state energies from the exact values is found not to exceed 0.1% in a wide region of the variation of the interaction strength, including the vicinity of the phase-transition point. This is better by approximately one order of magnitude than the results from the established many-body approaches.
A general implementation of the equation-of-motion (EOM) formalism for correlated many-body states is presented and applied to the description of collective excitations in atomic nuclei. While EOM approaches are traditionally formulated on top of independent-particle reference states, the present work extends the method to correlated reference states generated by modern many-body solvers. This formulation enables a consistent treatment of ground-state correlations and excited-state dynamics within a unified framework. Particular emphasis is placed on collective nuclear excitations employing chiral nuclear Hamiltonians in an ab initio context. The approach is motivated by the renewed interest in EOM techniques across several fields, including quantum chemistry and quantum computing, where they provide efficient and systematically improvable descriptions of excitation spectra. The present results demonstrate that the EOM framework offers a flexible and powerful tool for the microscopic description of nuclear spectroscopy beyond the traditional mean-field paradigm.
We derive a closed equation of motion for the current density of an inhomogeneous quantum many-body system under the assumption that the time-dependent wave function can be described as a geometric deformation of the ground-state wave function. By describing the many-body system in terms of a single collective field we provide an alternative to traditional approaches, which emphasize one-particle orbitals. We refer to our approach as continuum mechanics for quantum many-body systems. In the linear response regime, the equation of motion for the displacement field becomes a linear fourth-order integrodifferential equation, whose only inputs are the one-particle density matrix and the pair-correlation function of the ground state. The complexity of this equation remains essentially unchanged as the number of particles increases. We show that our equation of motion is a Hermitian eigenvalue problem, which admits a complete set of orthonormal eigenfunctions under a scalar product that involves the ground-state density. Further, we show that the excitation energies derived from this approach satisfy a sum rule which guarantees the exactness of the integrated spectral strength. Our formulation becomes exact for systems consisting of a single particle and for any many-body system in the high-frequency limit. The theory is illustrated by explicit calculations for simple one- and two-particle systems.
Density functional theory provides the basis for uncounted studies of ground-state properties of many-body systems. However, the connection between the energy density functional and the underlying microscopic interactions of a given theory is not fully understood. In this thesis we use renormalization-group techniques in combination with density functional theory to study many-body systems from microscopic interactions. We apply our formalism to different one-dimensional systems. We start with systems of identical fermions interacting via a short-range repulsive and long-range attractive interaction which serves as a simple one-dimensional toy model for nuclei. After that we study systems of spin-1/2 fermions, where we assume interactions only between fermions with different spins and consider both a non-local interaction as in the previous case and a contact interaction. In particular, the contact interaction plays a prominent role for ultracold Fermi gases. We calculate ground-state properties such as the energy, density, intrinsic density and density correlation functions and compare our results to values obtained from other approaches. Moreover, we show how energies of excited states and the absolute square of the ground-state wavefunction can be extracted from the density-density correlation function. The relation between our formalism and conventional density functional theory as well as many-body perturbation theory is discussed which may help to guide the development of ab initio functionals for quantitative studies of nuclei in the future.
The self consistent version of the density functional theory (DFT) is presented, which allows to calculate the ground state and dynamic properties of finite multi-electron systems such as atoms, molecules and clusters. The exact functional equation for the effective interaction, using which one can construct the action functional, density functional, the response functions, and excitation spectra of the considered systems, are discussed. We have also related the eigenvalues of the single-particle Kohn-Sham equations to the real single-particle spectra. We begin with the standard action functional and show that it is useful in calculating the linear response functions via which all physical characteristics of a many-body system can be expressed. It is shown, that function can be causal (retarded), noncausal and advanced ones. This resolves the well known paradox related to the causality and symmetry properties of the response functions and the effective interaction.