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arXiv:2606.21598 · Journal of Physics G Nuclear and Particle Physics

One-body and two-body density matrix elements in a symplectic many-body basis

Jakub Herko, Mark A. Caprio

Published Oct 1, 2026• 0 citations
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Abstract

Abstract The symplectic no-core configuration interaction (SpNCCI) framework is an ab initio many-body method for nuclear structure which makes use of the approximate symplectic symmetry of nuclei by appropriate choice of many-body basis states. In this paper we derive recurrence relations allowing for calculation of one-body and two-body density matrix elements between the SpNCCI basis states. Availability of these matrix elements allows for integration of the SpNCCI framework with other modern many-body methods and for calculation of matrix elements of any one-body or two-body operator.

DOI
10.1088/1361-6471/aeaf04

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•Feb 1, 2020•20 citations
Abstract The symplectic no-core configuration interaction (SpNCCI) framework is an ab initio many-body method for nuclear structure which makes use of the approximate symplectic symmetry of nuclei by appropriate choice of many-body basis states. In this paper we derive recurrence relations allowing for calculation of one-body and two-body density matrix elements between the SpNCCI basis states. Availability of these matrix elements allows for integration of the SpNCCI framework with other modern many-body methods and for calculation of matrix elements of any one-body or two-body operator.
We present the symplectic no-core configuration interaction (SpNCCI) framework, in which the nuclear many-body problem is solved a symmetry-adapted basis that explicitly encodes approximate symmetries associated with nuclear collectivity and deformation. In this framework, calculations are carried out in a basis organized into Sp(3,R) irreducible representations (irreps), each of which can be expressed as an infinite tower of U(3) irreps. In this framework, matrices of realistic relative two-body operators, such as the nuclear Hamiltonian, are computed directly in the Sp(3,R) many-body basis, obviating the need to expand all Sp(3,R) many-body states in, e.g., a U(3)-coupled configuration basis. Instead, many-body matrix elements are obtained via a recurrence relation that expresses a given matrix element in terms of matrix elements between basis states with fewer oscillator quanta. To use this recurrence method for computing matrix elements of relative two-body operators, we must first expand each operator into components of U(3) tensors. To this end, we present a method for decomposing arbitrary operators into U(3) tensor components.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
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.
Matrix elements of two-body operators between many-particle permutational symmetry-adapted functions in hyperspherical coordinates are constructed. The matrix elements are evaluated using the appropriate hyperspherical coefficients of fractional parentage, the Raynal-Revai coefficients, and the hyperspherical recoupling coefficients. We use the power expansion of the two-body operators and obtain an analytic expression for the matrix element of each term. These expressions are studied numerically. The results allow precise evaluations of two-body matrix elements for few-body calculations in nuclear, atomic, and molecular physics.
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.