Two-body Operators Research Articles

Can the eigenstates of a many-body Hamiltonian be represented exactly using a general two-body cluster expansion?

It is proved that a recent conjecture that the exact ground-state wave function of an arbitrary many-fermion system with one- and two-body interactions may be represented by an exponential cluster expansion employing finite two-body operators, starting from any reference function sufficiently close to the exact eigenfunction, is not valid. We show that the space of initial reference functions which lead to the exact ground state is of dimension equal to the number of two-body operators. If the dimension of the multiparticle space is greater than the number of two-body operators, then the space of good reference functions is of measure zero in it.

Tools for analysing configuration interaction wavefunctions

The configuration interaction (CI) approach to quantum chemical calculations is a well-established means of calculating accurately the solution to the Schrödinger equation for many-electron systems. It represents the many-body electron wavefunction as a sum of spin-projected Slater determinants of orthogonal one-body spin-orbitals. The CI wavefunction becomes the exact solution of the Schrödinger equation as the length of the expansion becomes infinite, however, it is a difficult quantity to visualise and analyse for many-electron problems. We describe a method for efficiently calculating the spin-averaged one- and two-body reduced density matrices ρ Ψ( r ̄ ; r ̄ ′) and Γ Ψ( r ̄ 1, r ̄ 2; r ̄ ′ 1, r ̄ ′ 2) of an arbitrary CI wavefunction Ψ. These low-dimensional functions are helpful tools for analysing many-body wavefunctions; we illustrate this for the case of the electron–electron cusp. From ρ and Γ one can calculate the matrix elements of any one- or two-body spin-free operator O . For example, if O is an applied electric field, this field can be included into the CI Hamiltonian and polarisation or gating effects may be studied for finite electron systems.

Electromagnetic and weak transitions in light nuclei

Recent advances in the study of the p-d radiative and $\mu-^3$ He weak capture processes by our group are presented and discussed. The trinucleon bound and scattering states have been obtained from variational calculations by expanding the corresponding wave functions in terms of correlated hyperspherical harmonic functions. The electromagnetic and weak transition currents include one- and two-body operators. The accuracy achieved in these calculations allows for interesting comparisons with experimental data.

Exactness of two-body cluster expansions in many-body quantum theory.

The Horn-Weinstein formula and the variational principle, combined with numerical results for a few many-electron systems, are used to provide support for a conjecture that the exact ground-state wave function for a Hamiltonian system containing up to two-body terms may be represented by an exponential cluster expansion employing a finite two-body operator.

On the Discrete Spectrum of a Pseudo-Relativistic Two-Body Pair Operator

We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.

The baryonic Y-shape confining potential energy and its approximants

We discuss the validity of replacing the complicated three-body confinement operator of the Y string junction type by three kinds of approximation which are numerically much simpler to handle: a one-body operator with the junction point at the centre of mass, a two-body operator corresponding to half the perimeter of the triangle formed by the three particles, and the average of both. Two different approaches for testing the quality of the approximations are proposed: a geometrical treatment based on the comparison of the potential energy strengths for the various inter quark distances, and a dynamical treatment based on the comparison of the corresponding effective string tensions using a hyperspherical approach. Both procedures give very similar results. It is shown how to simulate the genuine string junction operator by the approximations proposed above. Exact three-body calculations are presented in order to compare quantitatively the various approximations and to confirm our analysis.

Theoretical study of3He(μ−,νμ)3Hcapture

The ^3He(mu^-,nu_mu)^3H weak capture is studied using correlated-hyperspherical-harmonics wave functions, obtained from realistic Hamiltonians consisting of the Argonne $v_{14}$ or Argonne $v_{18}$ two-nucleon, and Tucson-Melbourne or Urbana-IX three-nucleon interactions. The nuclear weak charge and current operators have vector and axial-vector components that include one- and two-body contributions. The strength of the leading two-body operator in the axial-vector current is adjusted to reproduce the Gamow-Teller matrix element in tritium $\beta$-decay. The calculated total capture rate is in excellent agreement with the most recent experimental determination $1496\pm 4$ sec$^{-1}$, when the PCAC value is adopted for the induced pseudo-scalar coupling constant $g_{PS}$. The predictions for the capture rate and angular correlation parameters $A_v$, $A_t$, and $A_\Delta$ are found to be only very weakly dependent on the model input Hamiltonian. The variation of these observables with $g_{PS}$ and the theoretical uncertainties deriving from the model-dependent procedure used to constrain the axial current are investigated.

Even-Odd Effect on the Thermal Equilibrium State of the Pairing Model. II: Mean Field Approximation and Renormalized Distribution

We define a mean field approximation of the modified grand partition function of the pairing model, which has been defined on the modified grand canonical ensemble with even or odd particle number. Using numerical evaluations, it is shown that this approximation reproduces the even-odd effect exhibited in the case of the modified grand partition function at low temperature very well in a qualitative sense. The expectation values of one-body operators can be expressedsimply in terms of those obtainedwith the conventional mean field approximation when a Fermi-type distribution is replaced by a renormalized distribution. Although Wick’s theorm does not hold in this case, the dominant parts of the expectation values of two-body operators are also given using those obtained by the conventional mean fieldapproximation with the replacement mentionedabove.

Weak transitions inA=6and 7 nuclei

The ${}^{6}\mathrm{He}$ $\ensuremath{\beta}$ decay and ${}^{7}\mathrm{Be}$ electron capture processes are studied using variational Monte Carlo wave functions, derived from a realistic Hamiltonian consisting of the Argonne ${v}_{18}$ two-nucleon and Urbana-IX three-nucleon interactions. The model for the nuclear weak axial current includes one- and two-body operators with the strength of the leading two-body term---associated with $\ensuremath{\Delta}$-isobar excitation of the nucleon---adjusted to reproduce the Gamow-Teller matrix element in tritium $\ensuremath{\beta}$ decay. The measured half-life of ${}^{6}\mathrm{He}$ is underpredicted by theory by $\ensuremath{\simeq}8%,$ while that of ${}^{7}\mathrm{Be}$ for decay into the ground and first excited states of ${}^{7}\mathrm{Li}$ is overpredicted by $\ensuremath{\simeq}9%.$ However, the experimentally known branching ratio for these latter processes is in good agreement with the calculated value. Two-body axial current contributions lead to a $\ensuremath{\simeq}1.7%$ (4.4%) increase in the value of the Gamow-Teller matrix element of ${}^{6}\mathrm{He}$ ${(}^{7}\mathrm{Be}),$ obtained with one-body currents only, and slightly worsen (appreciably improve) the agreement between the calculated and measured half-life. Corrections due to retardation effects associated with the finite lepton momentum transfers involved in the decays, as well as contributions of suppressed transitions induced by the weak vector charge and axial current operators, have also been calculated and found to be negligible.

Longitudinal and transverse quasielastic response functions of light nuclei

The ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}$ longitudinal and transverse response functions are determined from an analysis of the world data on quasielastic inclusive electron scattering. The corresponding Euclidean response functions are derived and compared to those calculated with Green's function Monte Carlo methods, using realistic interactions and currents. Large contributions associated with two-body currents are found, particularly in the ${}^{4}\mathrm{He}$ transverse response, in agreement with data. The contributions of the two-body charge and current operators in the ${}^{3}\mathrm{He},$ ${}^{4}\mathrm{He},$ and ${}^{6}\mathrm{Li}$ response functions are also studied via sum-rule techniques. A semiquantitative explanation for the observed systematics in the excess of transverse quasielastic strength, as function of mass number and momentum transfer, is provided. Finally, a number of model studies with simplified interactions, currents, and wave functions are carried out to elucidate the role played, in the full calculation, by tensor interactions and correlations.

Role of two-body operators in electromagnetic transitions

Effective electromagnetic transition operators pertinent to the model space in the shell model are formulated for many- j-shell configurations and in isospin formalism within the framework of the first-order perturbation theory. The correlated operators thus obtained consist of one-body and two-body transition operators, the former of which is considered to be the origin of effective charges, while the latter cannot be incorporated into the idea of effective charges. Electromagnetic transitions in the 1 f 7/2-shell nuclei are studied systematically by using the effective transition operators so generated. It is found that the two-body contributions are very important not only in the higher-multipole transitions but also in the E2 transitions, to the extent that the concept of effective charge breaks down, and they are indispensable in achieving reasonable agreement with experiment.

Neutron charge form factor at largeq2

The neutron charge form factor ${G}_{\mathrm{En}}(q)$ is determined from an analysis of the deuteron quadrupole form factor ${F}_{C2}(q)$ data. Recent calculations, based on a variety of different model interactions and currents, indicate that the contributions associated with the uncertain two-body operators of shorter range are relatively small for ${F}_{C2}(q),$ even at large momentum transfer q. Hence, ${G}_{\mathrm{En}}(q)$ can be extracted from ${F}_{C2}(q)$ at large ${q}^{2}$ without undue systematic uncertainties from theory.

Diagonalization of the elliptic Macdonald-Ruijsenaars difference system of typeC2

We study a pair of commuting difference operators arising from the elliptic solution of the dynamical Yang-Baxter equation of type C2. The operators act on the space of meromorphic functions on the weight space of (4,). We show that these operators can be identified with the system by van Diejen and by Komori-Hikami with special parameters. It turns out that our case can be related to the difference Lamé operator (two-body Ruijsenaars operator) and thereby we diagonalize the system on the finite-dimensional space spanned by the level-one characters of the C2(1)-affine Lie algebra.

Matrix Elements of One- and Two-Body Operators in the Unitary Group Approach (II) — Application**The project supported by the Key Track-Following Service Foundation of State Education Commission of China and Science Foundation of Education Commission of Liaoning Province of China

Simple analytical expressions for one- and two-body matrix elements in the unitary group approach to the configuration interaction problems of many-electron systems are obtained based on the previous results for general irreps.

Matrix Elements of One- and Two-Body Operators in the Unitary Group Approach (I) — Formalism**The project supported by the Key Track-Following Service Foundation of State Education Commission of China and Science Foundation of Education Commission of Liaoning Province of China

The tensor algebraic method is used to derive general one- and two-body operator matrix elements within the representations, which are useful in the unitary group approach to the configuration interaction problems of quantum many-body systems.

Brueckner based generalized coupled cluster theory: Implicit inclusion of higher excitation effects

A generalization of the single reference Coupled Cluster parameterization for the ground state wave function is proposed that includes substitution operators that annihilate the reference determinant, but which act nontrivially on the correlated part of the wave function. It is shown that an inclusion of such two-body operators can mimic the effect of conventional connected triple and higher excitation operators. Results obtained with Brueckner based Generalized Coupled Cluster Doubles theory (BGCCD-version x) are found to be comparable in accuracy to CCSD(T) and CCSDT for a number of difficult test cases. In the current version of the BGCCD approach we obtain correlated ionization potentials and electron affinities as a by-product of a ground state calculation. This multistate nature of the BGCCD-X approach can give rise to problems with intruder states similar as in Fock Space Coupled Cluster theory.

Nucleon magnetic moments in an extended chiral constituent quark model

We present results for the nucleon magnetic moments in the context of an extended chiral constituent quark model based on the mechanism of the Goldstone boson exchange, as suggested by the spontaneous breaking of chiral symmetry in QCD. The electromagnetic charge-current operator is consistently deduced from the model Hamiltonian, which includes all force components for the pseudoscalar, vector and scalar meson exchanges. Thus, the continuity equation is satisfied for each piece of the interaction, avoiding the introduction of any further parameter. A good agreement with experimental values is found. The role of isoscalar two-body operators, not constrained by the continuity equation, is also investigated.

Implementing unitary operators in quantum computation

We present a general method which expresses a unitary operator by the product of operators allowed by the Hamiltonian of spin-1/2 systems. In this method, the generator of an operator is found first, and then the generator is expanded by the base operators of the product operator formalism. Finally, the base operators disallowed by the Hamiltonian, including more than two-body interaction operators, are replaced by allowed ones by the axes transformation and coupling order reduction technique. This method directly provides pulse sequences for the nuclear magnetic resonance quantum computer, and can be generally applied to other systems.

Multiple Collective Vibrations and Nonlinear Dynamics in Mean-Field Approaches

We study the connection between the Fourier transforms of one- and two-body observations and the excitation of multiple vibrations. We show that one-body observables give information on the excitation of a single vibration while two-body operators are needed to measure the excitation of double vibrations. Therefore, while mean-field approaches correctly predict the excitation of a single vibration, the study of multiple vibrations requires the estimation of many-body observables which goes beyond the mean-field approximation. These findings are illustrated in an algebraical model.

Weak capture of protons by protons

The cross section for the proton weak capture reaction ${}^{1}\mathrm{H}{(p,e}^{+}{\ensuremath{\nu}}_{e}{)}^{2}\mathrm{H}$ is calculated with wave functions obtained from a number of modern, realistic high-precision interactions. To minimize the uncertainty in the axial two-body current operator, its matrix element has been adjusted to reproduce the measured Gamow-Teller matrix element of tritium $\ensuremath{\beta}$ decay in model calculations using trinucleon wave functions from these interactions. A thorough analysis of the ambiguities that this procedure introduces in evaluating the two-body current contribution to the $\mathrm{pp}$ capture is given. Its inherent model dependence is in fact found to be very weak. The overlap integral ${\ensuremath{\Lambda}}^{2}(E=0)$ for the $\mathrm{pp}$ capture is predicted to be in the range 7.05--7.06, including the axial two-body current contribution, for all interactions considered.