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arXiv:2104.14820 · Physical Review C

Normal-ordering approximations and translational (non)invariance

T. Djärv, A. Ekström, C. Forssén, G. R. Jansen

Published Aug 23, 2021• 11 citations
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Abstract

Normal ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this paper we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for $^{4}\mathrm{He}$ and $^{16}\mathrm{O}$ by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for $^{4}\mathrm{He}$, it is also shown to be much smaller for $^{16}\mathrm{O}$ although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases.

DOI
10.1103/physrevc.104.024324

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Normal ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this paper we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for 4He^{4}\mathrm{He}4He and 16O^{16}\mathrm{O}16O by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for 4He^{4}\mathrm{He}4He, it is also shown to be much smaller for 16O^{16}\mathrm{O}16O although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases.
The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schrodinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the so-called normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schrodinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle-number conservation is broken (and potentially restored), a particle-number-conserving normal-orderedk-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number-projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. The PNOkB approximation of an arbitrary N-body operator is formulated. Based on this systematic approach, it is demonstrated that naive extensions of the normal-ordered two-body (NO2B) approximation employed so far on the basis of symmetry-conserving reference states lead to particle non-conserving operators. Alternatively, the PNOkB procedure is now available to generate particle-number-conserving approximate operators. The formal analysis is validated numerically. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed. The future formulation of an angular-momentum-conserving normal-ordered k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.
Reference-state-based many-body methods start from Hamiltonians that are normal ordered with respect to the reference state. In low-energy nuclear physics applications, normal-ordered Hamiltonians consisting of two- and three-nucleon forces are usually truncated at the two-body rank with residual three-nucleon operators being discarded. Benchmark computations have shown that this truncation is accurate, but we lack an understanding about why it works. We show that the normal-ordered two-body truncation is exact for zero-range three-body forces when nuclei are computed using the coupled cluster with singles and doubles method. As the nuclear three-nucleon force is short ranged and a three-body contact is a leading term in effective field theories of quantum chromodynamics, our result provides an analytical basis for the popular normal-ordered two-body approximation.
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.
A method is presented for the calculation of the one-body and two-body density matrices and their Fourier transforms in momentum space, that is consistent with the requirement for translational invariance, in the case of a nucleus (a finite self-bound system). We restore translational invariance by using the so-called fixed center-of-mass approximation for constructing an intrinsic nuclear ground state wavefunction by starting from a non-translationally invariant wavefunction and applying a projection prescription. We discuss results for the one-body and two-body momentum distributions of the 4He nucleus calculated with the Slater determinant of the harmonic oscillator orbitals, as the initial non-translationally invariant wavefunction. Effects of such an inclusion of CM correlations are found to be quite important in the momentum distributions.
The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wavefunctions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators a^+ and a for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set {a^+} times other exponential of the set {a}. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of "Tassie-Barker" factors, and point out other model-independent results. The intrinsic wavefunction of the nucleus in its ground state is constructed from a nontranslationally-invariant (nTI) one via existing projection techniques. As an illustration, the one-body and two-body momentum distributions (MDs) for the 4He nucleus are calculated with the Slater determinant of the harmonic-oscillator model as the trial, nTI wavefunction. We find that the TI introduces important effects in the MDs.
If one assumes a translationally invariant motion of the nucleons relative to the c. m. position in single particle mean fields a correlated single particle picture of the nuclear wave function emerges. A single particle product ansatz leads for that Hamiltonian to nonlinear equations for the single particle wave functions. In contrast to a standard not translationally invariant shell model picture those single particle s-, p- etc states are coupled. The strength of the resulting coupling is an open question. The Schroedinger equation for that Hamiltonian can be solved by few- and many -body techniques, which will allow to check the validity or non-validity of a single particle product ansatz. Realistic nuclear wave functions exhibit repulsive 2-body short range correlations. Therefore a translationally invariant single particle picture -- if useful at all -- can only be expected beyond those ranges. Since exact A = 3 and 4 nucleon ground state wave functions and beyond based on modern nuclear forces are available, the translationally invariant shell model picture can be optimized by an adjustment to the exact wave function and its validity or non-validity decided.