J. Arponen

VARIATIONAL PRINCIPLES AND LINKED CLUSTER EXP S EXPANSIONS FOR STATIC AND DYNAMIC MANY BODY PROBLEMS

Abstract The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes. This allows the calculation of both nonlinear dynamical behaviour and of dynamical linear response and Greens functions. Practical approximation schemes are considered. In a SUB n approximation the m-body amplitudes are restricted to the order m ⋖ n and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specificed in terms of tree structures, the normal or extended generalized time ordering g.t.o. trees. The extended exp S method is in the SUB n approximation able to sum, in addition to the normal SUB n diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUB n truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUB 1, in the extended exp S method agrees with the mean field theory, which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended exp S method already in low-order approximations can realistically treat a great variety of diverse many-body problems, even including systems which may undergo ground-state phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB 2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the supercondictive phase transition of the BCS model Hamiltonian and the deformation phase transition of the Lipkin model are properly obtained by the extended exp S method in a low-order approximation.

Electron liquid in collective description. III. Positron annihilation

Abstract We give a new, computationally effective, formulation for the problem of a charged impurity in an interacting electron gas. Our approach is based on the previously introduced formalism in which the collective excitations of the electron system are represented by interacting bosons. This enables one to include in a logical way the most important terms of the electron-electron interaction beyond the random-phase approximation (RPA). The numerical application to positron annihilation shows that the added non-RPA terms remove from the annihilation rate λ and the correlation energy E the divergences, which have troubled the earlier theories. The rate λ and the energy E are both continuous functions of the electron density and approach their correct limiting values for low densities. The numerical values of λ at physical densities correspond closely to the measured positron lifetimes in metals. A very noteworthy characteristic of the present theory is that the sum rule for the displaced charge is fulfilled typically to the accuracy of 1%. Numerical results are also given for the positron kinetic energy, the various components of the positron correlation energy, the pair-correlation function, and the electrostatic potential at various electron densities.

Internal structure of neutron stars

Abstract The equation of state and the structure and composition of neutron star matter are investigated in the density region 3.1 × 10 11 −2 × 10 15 g/cm 3 . Below the density 3.1 × 10 11 g/cm 3 the matter is a solid consisting of neutron-rich nuclei in a degenerate electron gas. At 3.1 × 10 11 g/cm 3 neutrons start to drip out of the nuclei; as the density increases, the lattice spacing continuously decreases while the geometrical size of the nuclei only slightly increases, until at about 15 × 10 13 g/cm 3 the nuclei disappear by coalescing into a homogeneous liquid in an almost continuous phase transition. The maximum proton number per nucleus is 40, which is obtained between the densities 1−2.5 × 10 13 g/cm 3 ; after that the proton number decreases until at the solid-to-liquid phase boundary it is about 20. In the liquid-core region, muons appear at the density 20.5 × 10 13 g/cm 3 . The calculations use the differential Thomas-Fermi theory in connection with a new model for the nuclear matter energy. The model is based on the parametrization of the Brueckner G -matrix in conjunction with a few empirical requirements concerning the saturation of the symmetric nuclear matter. It is shown that very small modifications in the details of the nuclear matter energy may lead to considerable differences in the resulting neutron star structure; for example the density of matter at the solid-to-liquid phase boundary is found to be very sensitive to such details. One of the main reasons for this is that the stability of the liquid against clustering into separate nuclei depends mainly on the second derivatives of the nuclear matter energy with respect to neutron and proton densities; for these second derivatives there exist no very reliable theoretical or phenomenological predictions so far. To check that the Thomas-Fermi theory together with the model for the nuclear matter energy works satisfactorily, a test calculation is made about finite nuclei of different sizes between A = 16 and 400.

Independent-cluster parametrizations of wave functions in model field theories. I. Introduction to their holomorphic representations

Abstract The configuration-interaction method (CIM), normal coupled-cluster method (NCCM), and extended coupled-cluster method (ECCM) form a rather natural hierarchy of formulations of increasing sophistication for describing interacting systems of quantum-mechanical particles or fields. They are denoted generically as independent-cluster (IC) parametrizations in view of the way in which they incorporate the many-body correlations via sets of amplitudes that describe the various correlated clusters within the interacting system as mutually independent entities. They differ primarily by the way in which they incorporate the exact locality and separability properties. Each method is shown to provide, in principle, an exact mapping of the original quantum-mechanical problem into a corresponding classical Hamiltonian mechanics in terms of a set of multiconfigurational canonical field amplitudes. In perturbation-theoretic terms the IC methods incorporate infinite classes of diagrams at each order of approximation. The diagrams differ in their connectivity or linkedness properties. The structure of the ECCM in particular makes it capable of describing such phenomena as phase transitions, spontaneous symmetry breaking, and topological states. We address such fundamentally important questions as the existence and convergence properties of the three IC parametrizations by formulating the holomorphic representation of each one for the class of single-mode bosonic field theories which include the anharmonic oscillators. These highly nontrivial models provide a stringent test for the coupled-cluster methods. We present a particularly detailed analysis of the asymptotic behaviour of the various amplitudes which exactly characterize each IC method. More generally, the holomorphic representation allows us to give a completely algebraic description of all aspects of each scheme. In particular, this includes the topological connectivity properties of the various terms or diagrams in their expansions. We construct a generating functional for the calculation of the expectation values of arbitrary operators for each of the IC parametrizations. The functional is used in each case to formulate the quantum mechanical action principle and to perform the mapping into the corresponding classical phase space.

Extended Coupled Cluster Method: Quantum Many-Body Theory Made Classical

We focus attention in this paper on how the general quantum many-body problem can be cast in the form of a variational principle for a specified action functional. After some preliminary discussion in Section 2 concerning the algebra of the many-body operators and the development of a convenient shorthand notation to describe it, we show in Section 3 how each of the configuration-interaction (CI)1 method, the normal coupled cluster method (CCM),2–6 and an extended version of the CCM,7,8 can be derived by specific parametrisations of the ground-state bra and ket wavefunctions in the action functional. In each case we make contact and comparison with time-independent perturbation theory, and we discuss the various “tree-diagram” structures that emerge in each case.

The Lipkin pseudospin model in the variational exp S formalism

Abstract The variational extended exp S formulation of the coupled-cluster method is numerically studied and applied to the many-particle model of Lipkin, Meshkov and Glick.In contrast to the standard exp S scheme the new method is shown to give good results even in the deformed phase without explicit introduction of a new single-particle basis. The ground-state energies are found to satisfy the variational property of being upper limits to the exact energies. Also the excitation energies are in satisfactory overall agreement with the true ones for arbitrary interaction strengths. Special attention is paid to the phase-transition region of the interaction strength parameter.

Theory of electron gas as a system of interacting collective excitations I. Boson formalism

Abstract The interacting fermion system is studied by performing a transformation to boson operators, which correspond to approximate collective excitations of the system. Rules are given for constructing the bosonic operators corresponding to any spin-independent observables of the fermion system. It is shown that the resulting bosonic Hamiltonian of the fermion system contains terms that are at most of sixth order in the boson creation and annihilation operators. The equivalence of the fermion and boson descriptions is shown by perturbation-diagrammatic methods. The noninteracting boson system corresponds to the conventional random phase approximation. This formulation of the interacting fermion system offers a systematic way to incorporate the remaining residual interactions between the bosons by perturbation theory.

On an Effective Gauge Field Description of a Positron Impurity in Polarizable Media

The local polarization around a positron impurity is described by a unitary operator, which defines a dynamical gauge field in interaction with the particle. We study the relation of this gauge field to the elementary collective excitations of the medium. We make contact with the generalized coherent bosonization scheme recently introduced in the extended coupled cluster theory, which suggests a definite parametrization of the polarization unitary operator in terms of a double similarity transformation. We derive the exact equations for the wavefunction and for the CCM amplitudes and show that they satisfy the conservation laws.

A holomorphic representation approach to the regularization of model field theories in coupled cluster form

SummaryWe explain in detail the so-called Bargmann or holomorphic representation, and apply it to the general class of single-mode bosonic field theories. Since these model field theories have no attribute of separability and are, in some sense, maximally nonlocal, they are an especially severe test of the capability of coupled cluster methods to parametrize them satisfactorily. They include the cases of anharmonic oscillators of order 2K (K=2, 3,...), for which ordinary perturbation theory is known to diverge, and we therefore make a special study of such systems. We demonstrate for the first time for any quantum-mechanical problem with infinite Hilbert space that both the normal and extended coupled cluster methods (NCCM and ECCM) have phase spaces which rigorously exist. We analyze completely the asymptotic properties of the complete sets of the NCCM and ECCM amplitudes, either of which fully characterizes the system. It is thereby shown how the holomorphic representation can be used to regularize completely all otherwise formally divergent series that appear. We demonstrate in detail how the entire NCCM and ECCM programmes can be carried through for these systems, including the diagonalization of the classically mapped Hamilitonians in the respective classical NCCM and ECCM phase spaces.

Independent-cluster methods as mappings of quantum theory into classical mechanics

SummaryThe algebraic structures of theconfiguration interaction, normal coupled cluster, andextended coupled cluster methods are reviewed and developed. These methods are pointed out to perform a mapping of the quantum mechanical problem into a classical phase space, where in each case the classical canonical coordinates have characteristically different cluster and locality properties. Special focus is given to the extended coupled cluster method (ECCM), which alone is based on an entirely additively separable coordinate system. The general principles are formulated for systems with both bosonic and fermionic degrees of freedom, allowing both commutative and anticommutative (Grassmann) cluster amplitudes. The properties of the classical images are briefly discussed. It is proposed that phase spaces may exist which are fixed points of quantization.