Three-body Operators Research Articles

Pfaffian formulation for matrix elements of three-body operators in multiple quasiparticle configurations

We present a Pfaffian formula to calculate matrix elements of three-body operators in symmetry-restoration beyond-mean-field methods, including the case of multiple quasiparticle (qp) configurations. Detailed derivation is provided and the validity of the new formula is checked numerically. The corresponding efficiency is discussed, which turns out to be about five times (one order of magnitude) higher than the conventional method when four (less than four) qp operators are considered.

In-medium similarity renormalization group with three-body operators

Over the past decade the in-medium similarity renormalization group (IMSRG) approach has proven to be a powerful and versatile ab initio many-body method for studying medium-mass nuclei. So far, the IMSRG was limited to the approximation in which only up to two-body operators are incorporated in the renormalization group flow, referred to as the IMSRG(2). In this work, we extend the IMSRG(2) approach to fully include three-body operators yielding the IMSRG(3) approximation. We use a perturbative scaling analysis to estimate the importance of individual terms in this approximation and introduce truncations that aim to approximate the IMSRG(3) at a lower computational cost. The IMSRG(3) is systematically benchmarked for different nuclear Hamiltonians for ${}^{4}\text{He}$ and ${}^{16}\text{O}$ in small model spaces. The IMSRG(3) systematically improves over the IMSRG(2) relative to exact results. Approximate IMSRG(3) truncations constructed based on computational cost are able to reproduce much of the systematic improvement offered by the full IMSRG(3). We also find that the approximate IMSRG(3) truncations behave consistently with expectations from our perturbative analysis, indicating that this strategy may also be used to systematically approximate the IMSRG(3).

Connected three-body terms in single-reference unitary many-body theories: Iterative and perturbative approximations.

This work introduces various approaches to include connected three-body terms in unitary many-body theories, focusing on the driven similarity renormalization group (DSRG). Starting from the least approximate method-the linearized DSRG truncated to one-, two-, and three-body operators [LDSRG(3)]-we develop several approximate LDSRG(3) models with reduced computational cost. Through a perturbative analysis, we motivate a family of iterative LDSRG(3)-n and -n' (n = 1, 2, 3, 4) methods that contain a subset of the LDSRG(3) diagrams. Among these variants, the LDSRG(3)-2 scheme has the same computational complexity of coupled cluster theory with singles, doubles, and triples (CCSDT), but it outperforms CCSDT in the accuracy of the predicted correlation energies. We also propose and implement two perturbative triples corrections based on the linearized DSRG truncated to one- and two-body operators augmented with recursive semi-quadratic commutators [qDSRG(2)]. The resulting qDSRG(2)+(T) approach matches the accuracy of the "gold-standard" coupled cluster theory with singles, doubles, and perturbative triples model on the energetics of twenty-eight closed-shell atoms and small molecules.

A three-dimensional momentum-space calculation of three-body bound state in a relativistic Faddeev scheme

In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body t–matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.

Triaxial nuclear shapes of even–even ruthenium isotopic chain in the O(6) dynamical symmetry of interacting boson model with the three-body quadrupole interaction

Even–even nuclei in the $$ A\sim100 $$ mass region are investigated within the framework of the sd-interacting boson model (sdIBM1) with three-body O(6) symmetric quadrupole operator $$ \left[ {QQQ} \right]^{\left( 0 \right)} $$ for generating states with triaxial deformation. The rigid triaxial asymmetric rotor model of collective rotation is used to extract the asymmetry parameter $$ \gamma $$ from both the energy ratio of the second $$ 2_{2}^{ + } $$ state to the first $$ 2_{1}^{ + } $$ state and the reduced E2 transition probabilities from $$ 2_{1,2}^{ + } $$ states to the ground state. Using the method of the intrinsic coherent states, the classical limit of potential energy surfaces which represents the expectation value of the total Hamiltonian is obtained. The critical points of the phase transition are investigated. The triaxiality is analyzed for even–even ruthenium (Ru) isotopic chain as an example of illustrating the effect of the three-body quadrupole operator. By using computer simulated search program, a fitting procedure is performed to get the best parameters of the IBM Hamiltonian for each nucleus in order to obtain a minimum root mean square deviation between the calculated and the experimental some low-lying energy levels and B(E2) transition rates.

The intermediate Hamiltonian Fock-space coupled-cluster method with approximate evaluation of the three-body effects.

The exponential parametrization of the wave function used in the coupled-cluster approaches has proven very successful in the ab initio description of atomic and molecular systems. This concerns first of all the single-reference version of the method that is designed for states dominated by a single Slater determinant. Usually, the coupled-cluster methods with one- and two-body excitation operators in the exponent form the basic computational schemes. The inclusion of three-body effects in the cluster operator to increase the accuracy of the results is numerically expensive, so their approximate evaluation is rather used in practice. In the case of the single-reference coupled-cluster approach, the problem of approximate evaluation of three-body effects in the cluster operator has been well studied, and computational schemes of both noniterative and iterative nature have been proposed. The situation is different in the case of multireference coupled-cluster methods which are required to describe open shell and quasidegenerate states. The multireference approaches in their standard effective Hamiltonian formulations are more complicated and less frequently used in routine calculations; however, one of them, the so-called Fock-space coupled-cluster method, becomes very effective if reformulated within the intermediate Hamiltonian framework. Both the basic version of the method with one- and two-body clusters and the extended one that includes up to three-body operators in the exponent are implemented. The latter approach provides more accurate results, but its relatively high numerical cost limits its applicability. For this reason, going beyond the basic scheme with one- and two-body clusters through an approximate evaluation of the impact of three-body clusters is of great interest. In the paper, we investigate different ways of approximate inclusion of the three-body effects in the Fock-space coupled-cluster method designated for excitation energy calculations.

Effects of an induced three-body force in the incident channel of ( d,p ) reactions

A widely accepted practice for treating deuteron breakup in $A(d,p)B$ reactions relies on solving a three-body $A+n+p$ Schr\"odinger equation with pairwise $A$-$n$, $A$-$p$ and $n$-$p$ interactions. However, it was shown in [Phys. Rev. C \textbf{89}, 024605 (2014)] that projection of the many-body $A+2$ wave function into the three-body $A+n+p$ channel results in a complicated three-body operator that cannot be reduced to a sum of pairwise potentials. It contains explicit contributions from terms that include interactions between the neutron and proton via excitation of the target $A$. Such terms are normally neglected. We estimate the first order contribution of these induced three-body terms and show that applying the adiabatic approximation to solving the $A+n+p$ model results in a simple modification of the two-body nucleon optical potentials. We illustrate the role of these terms for the case of $^{40}$Ca($d,p$)$^{41}$Ca transfer reactions at incident deuteron energies of 11.8, 20 and 56 MeV, using several parameterisations of nonlocal optical potentials.

Coherence of three-body Förster resonances in Rydberg atoms

We have observed recently the Stark-tuned three-body F\"orster resonances ${\rm 3}\times nP_{3/2} (|M|)\to nS_{1/2} +(n+1)S_{1/2} +nP_{3/2} (|M^{*} |)$ at long-range interactions of a few cold Rb Rydberg atoms [D.B.Tretyakov et al., Phys. Rev. Lett. 119, 173402 (2017)]. The three-body resonance appears at a different dc electric field with respect to the ordinary two-body resonance ${\rm 2}\times nP_{3/2} (|M|)\to nS_{1/2} +(n+1)S_{1/2} $ and corresponds to a transition when the three interacting atoms change their states simultaneously (two atoms go to the $S$ states, and the third atom remains in the $P$ state but changes its moment projection), with the negligible contribution of the two-body resonance to the population transfer. It thus has a Borromean character and represents an effective three-body operator, which can be used to directly control the three-body interactions in quantum simulations and quantum gates implemented with Rydberg atoms. In this paper we theoretically investigate the coherence of such three-body resonances and we show that high-contrast Rabi-like population oscillations are possible for the localized Rydberg atoms in a certain spatial configuration. This paves the way to implementing three-qubit quantum gates and quantum simulations based on three-body Rydberg interactions.

Quenching of nuclear matrix elements for 0νββ decay by chiral two-body currents

We examine the leading effects of two-body weak currents from chiral effective field theory on the matrix elements governing neutrinoless double-beta decay. In the closure approximation these effects are generated by the product of a one-body current with a two-body current, yielding both two- and three-body operators. When the three-body operators are considered without approximation, they quench matrix elements by about 10%, less than suggested by prior work, which neglected portions of the operators. The two-body operators, when treated in the standard way, can produce much larger quenching. In a consistent effective field theory, however, these large effects become divergent and must be renormalized by a contact operator, the coefficient of which we cannot determine at present.

Reciprocal transformation of seniority number restricted wave function.

In this work, an extension of Malmqvist's algorithm is efficiently implemented for the reciprocal transformation of a seniority number restricted multideterminant wave function constructed by nonorthogonal orbitals to an equivalent one constructed by biorthogonal orbitals. Test calculations show that the novel algorithm is capable of applying to transform quite large nonorthogonal orbital based wave function of distributing 22 electrons on 22 orbitals with the maximum seniority number sector and for the evaluation of a three-body operator, at an affordable memory cost of about 800 GB, which is only about one tenth of the size of the corresponding complete active space. Moreover, the algorithm displays improved computational efficiency, due to the compactness of seniority number restricted multi-determinant wave functions.

Flow equations for cold Bose gases

We derive flow equations for cold atomic gases with one macroscopically populated energy level. The generator is chosen such that the ground state decouples from all other states in the system as the renormalization group flow progresses. We propose a self-consistent truncation scheme for the flow equations at the level of three-body operators and show how they can be used to calculate the ground state energy of a general N-body system. Moreover, we provide a general method to estimate the truncation error in the calculated energies. Finally, we test our scheme by benchmarking to the exactly solvable Lieb–Liniger model and find good agreement for weak and moderate interaction strengths.

Observation of the Borromean Three-Body Förster Resonances for Three Interacting Rb Rydberg Atoms.

Three-body Förster resonances at long-range interactions of Rydberg atoms were first predicted and observed in Cs Rydberg atoms by Faoro etal. [Nat. Commun. 6, 8173 (2015)NCAOBW2041-172310.1038/ncomms9173]. In these resonances, one of the atoms carries away an energy excess preventing the two-body resonance, leading thus to a Borromean type of Förster energy transfer. But they were in fact observed as the average signal for the large number of atoms N≫1. In this Letter, we report on the first experimental observation of the three-body Förster resonances 3×nP_{3/2}(|M|)→nS_{1/2}+(n+1)S_{1/2}+nP_{3/2}(|M^{*}|) in a few Rb Rydberg atoms with n=36, 37. We have found here clear evidence that there is no signature of the three-body Förster resonance for exactly two interacting Rydberg atoms, while it is present for N=3-5 atoms. This demonstrates the assumption that three-body resonances can generalize to any Rydberg atom. As such resonance represents an effective three-body operator, it can be used to directly control the three-body interactions in quantum simulations and quantum information processing with Rydberg atoms.

Three-Body Halo States in Effective Field Theory: Renormalization and Three-Body Interactions in the Helium-6 System

In this paper we study the renormalization of Halo effective field theory applied to the Helium-6 halo nucleus seen as an alpha –neutron–neutron three-body state. We include the 0^+ dineutron channel together with both the 3/2^- and 1/2^- neutron–alpha channels into the field theory and study all of the six lowest-order three-body interactions that are present. Furthermore, we discuss three different prescriptions to handle the unphysical poles in the P-wave two-body sector. In the simpler field theory without the 1/2^- channel present we find that the bound-state spectrum of the field theory is renormalized by the inclusion of a single three-body interaction. However, in the field theory with both the 3/2^- and 1/2^- included, the system can not be renormalized by only one three-body operator.

Faddeev-type calculation of(d,n)transfer reactions in three-body nuclear systems

Exact Faddeev-type three-body equations are applied to the study of the proton transfer reactions $(d,n)$ in the system consisting of a nuclear core and two nucleons. The integral equations for the three-body transition operators are solved in the momentum-space framework including the Coulomb interaction via the screening and renormalization method. For a weakly bound final nucleus the calculation of the $(d,n)$ reaction is more demanding in terms of the screening radius as compared to the $(d,p)$ reaction. Well-converged differential cross section results are obtained for $^{7}\text{Be}(d,n)^{8}\text{B}$, $^{12}\text{C}(d,n)^{13}\text{N}$, and $^{16}\text{O}(d,n)^{17}\text{F}$ reactions. A comparison with the corresponding $(d,p)$ reactions is made. The calculations fail to reproduce the shape of the angular distribution for reactions on $^{12}\text{C}$ but provide quite successful description for reactions on $^{16}\text{O}$, especially for the transfer to the $^{17}\text{F}$ excited state $1/{2}^{+}$ when using a nonlocal optical potential.

Trends in Atomic Parameters for Crystals and Free Ions across the Lanthanide Series: The Case of LaCl3:Ln(3+).

Analyses of the crystal field energy levels of the series LaCl3:Ln(3+) using a semiempirical Hamiltonian shows that only five ions (Pr, Nd, Pm, Dy, Ho) meet the criteria to avoid overfitting of the atomic part. A new parameter (SNES) has been introduced to represent the strength of the normalized electrostatic repulsion for these ions. This parameter varies linearly (R(2)adj = 0.9994, N = 5) with the reciprocal of the radius of the tripositive lanthanide ion, as expected from the form of repulsive Coulomb interaction. The Slater parameters from the crystal field analyses, F(k)(corr) (i.e., corrected for the effects of the two-particle component of the three-body operator associated with the T(2) parameter), exhibit an exponential variation with the number of electrons, n, in 4f(n). This is explained by reference to the radial part of a hydrogen-like wave function. The ratio of F(k)(corr) with the ab initio free ion Slater parameter F(k)(ab initio) varies linearly with n. Fitted parameters F(k)(corr: free ion) from the free ion data for Pr(3+) and Nd(3+) show that the corresponding ab initio values are between 14 and 27% too high. The spin-orbit coupling constant from crystal field analyses (ζ4f) exhibits a quartic variation with atomic number, and the ratio ζ4f/ζ4f(ab initio) follows an exponential growth model with n. The results serve to confirm the hypothesis that smooth trends can be observed across the Ln(3+) series for the fitted parameters despite the fact that the majority of experimental data is lacking.

Numerical construction of the continuous spectrum Eigenfunctions of the three body Schrödinger operator: Three particles on the axis with short-range pair potentials

Based on a new method of the numerical construction of the three-body Schrodinger operator continuous spectrum eigenfunctions an analysis of the solutions of the problem of three identical particles on the axis with quickly decreasing repulsive pair potentials is offered. The initial problem is reduced to solving an inhomogeneous boundary problem for an elliptical partial differential equation in a twodimensional domain as a circle with radiation boundary conditions, with a ray approximation of the solution with diffraction corrections, contributing to a smoothness of a solution sought, being used. The approach offered allows a natural generalization for a case of slowly decreasing potentials of the Coulomb type and higher configuration space dimensions.

On the Bound States for the Three-Body Schrödinger Equation with Decaying Potentials

The three-body Schrodinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length \({\kappa + 1, \kappa = 0, 1, \ldots}\) As a result, the solutions to the three-body Schrodinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrodinger equation in three dimensions with the inverse power potential of the form \({r^{-{\kappa}-1}}\) . As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with \({\kappa = 0}\) , are reduced to the well-known eigenvalues of bound states at threshold.

Renormalizing the double-beta operator for the shell model

We review work, much of it by the author and collaborators, to take omitted levels into account in shell-model calculations of neutrinoless double-beta decay. Previous work on this type of problem deals mainly with corrections to the shell-model Hamiltonian. Here we focus on corrections to the two-body transition operator that governs the rate of neutrinoless decay. Corrections in the form of two-body effective operators appear to be significant but have yet to be fully pinned down. Three-body effective operators may be significant as well.

Essential and discrete spectra of the three-particle Schrödinger operator on a lattice

We study the position of the essential spectrum of a three-body Schrodinger operator H. We evaluate the lower boundary of the essential spectrum of H and prove that the number of eigenvalues located below the lower edge of the essential spectrum in the H model is finite.

On the Interpretation of the level structure of the Ground 3d5 Manifold of Mn III, Fe IV, Co V and Ni VI

The level structure of the ground 3d5 configuration of Mn2+, Fe3+, Co4+ and Ni5+ ions was theoretically interpreted by means of a least-squares fit of the energy parameters to the observed values within the framework of the single-configuration approximation. In the Hamiltonian in addition to real electrostatic, spin-orbit, and spin-spin interactions, electrostatic and spin-orbit interactions correlated by configuration mixing were included. It was shown that the correct positions of almost all the energy levels are determined when the Hamiltonian includes the terms of the lineal (two-body operators) and nonlinear (three-body operators) theory of the configuration interaction. The most correct theoretical description of the experimental spectra was obtained by taking into account relativistic interactions and correlation effects of spin-orbit interactions. Adjustable parameters of the interactions included into the Hamiltonian were found.