Hyperspherical Harmonics Research Articles

Analytical Computation of Hyper-Ellipsoidal Harmonics

The four-dimensional ellipsoid of an anisotropic hyper-structure corresponds to the four-dimensional sphere of an isotropic hyper-structure. In three dimensions, both theories for spherical and ellipsoidal harmonics have been developed by Laplace and Lamé, respectively. Nevertheless, in four dimensions, only the theory of hyper-spherical harmonics is hitherto known. This void in the literature is expected to be filled up by the present work. In fact, it is well known that the spectral decomposition of the Laplace equation in three-dimensional ellipsoidal geometry leads to the Lamé equation. This Lamé equation governs each one of the spectral functions corresponding to the three ellipsoidal coordinates, which, however, live in non-overlapping intervals. The analysis of the Lamé equation leads to four classes of Lamé functions, giving a total of 2n + 1 functions of degree n. In four dimensions, a much more elaborate procedure leads to similar results for the hyper-ellipsoidal structure. Actually, we demonstrate here that there are eight classes of the spectral hyper-Lamé equation and we provide a complete analysis for each one of them. The number of hyper-Lamé functions of degree n is (n + 1)2; that is, n2 more functions than the three-dimensional case. However, the main difficulty in the four-dimensional analysis concerns the evaluation of the three separation constants appearing during the separation process. One of them can be extracted from the corresponding theory of the hyper-sphero-conal system, but the other two constants are obtained via a much more complicated procedure than the three-dimensional case. In fact, the solution process exhibits specific nonlinearities of polynomial type, itemized for every class and every degree. An example of this procedure is demonstrated in detail in order to make the process clear. Finally, the hyper-ellipsoidal harmonics are given as the product of four identical hyper-Lamé functions, each one defined in its own domain, which are explicitly calculated and tabulated for every degree less than five.

Baryonic thermal screening mass at NLO

We determine the resummed 1-loop correction to a baryonic thermal screening mass. The calculation is carried out in the framework of a dimensionally reduced effective theory, where quarks are heavy fields due to their non-zero Matsubara frequencies. The correction due to interactions is computed at O(g2) in the coupling constant. In order to solve a 3-body Schrödinger equation, we exploit a two-dimensional generalization of the hyperspherical harmonics method. At electroweak scale temperatures, the NLO correction represents a ∼ 4.6% increase of the free-theory value 3πT of the screening mass.

Explicit n-particle harmonic oscillator states in products of SU(1,1) representations

In this paper, we present a simple algorithm for the explicit construction of [Formula: see text]-particle harmonic oscillator states simultaneously belonging to irreducible representations of [Formula: see text] (or [Formula: see text]) and [Formula: see text]. For degenerate representations, the construction can be done using generating functions or hyperspherical harmonics. The cases with [Formula: see text] and [Formula: see text] are investigated at greater length. The analysis is extended to nondegenerate states, although the states are not so easily obtained as in the degenerate case. Finally, we briefly touch on the [Formula: see text] structure of Bell states.

Trions in two-dimensional monolayers within the hyperspherical harmonics method: Application to transition metal dichalcogenides

Trions in two-dimensional monolayers within the hyperspherical harmonics method: Application to transition metal dichalcogenides

Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-dimensional sphere (d≥2). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for diverging sequences of Laplace eigenvalues. Our approach takes advantage of a recent result by Bally, Caramellino and Poly (2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (2015) for Hermite-rank 2 functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish second order Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough. Our argument requires some novel estimates on moments of products of Gegenbauer polynomials that may be of independent interest, which we prove via the link between graph theory and diagram formulas.

Monotone complexity measures of multidimensional quantum systems with central potentials

In this work, we explore the (inequality-type) properties of the monotone complexity-like measures of the internal complexity (disorder) of multidimensional non-relativistic electron systems subject to a central potential. Each measure quantifies the combined balance of two spreading facets of the electron density of the system. We show that the hyperspherical symmetry (i.e., the multidimensional spherical symmetry) of the potential allows Cramér–Rao, Fisher–Shannon, and Lopez-Ruiz, Mancini, Calbet–Rényi complexity measures to be expressed in terms of the space dimensionality and the hyperangular quantum numbers of the electron state. Upper bounds, mutual complexity relationships, and complexity-based uncertainty relations of position–momentum type are also found by means of the electronic hyperangular quantum numbers and, at times, the Heisenberg–Kennard relation. We use a methodology that includes a variational approach with a covariance matrix constraint and some algebraic linearization techniques of hyperspherical harmonics and Gegenbauer orthogonal polynomials.

Continuous Head-related Transfer Function Representation Based on Hyperspherical Harmonics

Continuous Head-related Transfer Function Representation Based on Hyperspherical Harmonics

Few-body model approach to the lowest bound S-state of non-symmetric exotic atoms and ions

Lowest bound S-state energy of Coulombic three-body systems (N Z+ μ − e −) consisting of a positively charged nucleus of charge number Z (N Z+), a negatively charged muon (μ −) and an electron (e −), is investigated in the framework of few-body (i.e., two- and three-body) cluster model approach. For the three-body cluster model, we adopted the hyperspherical harmonics expansion (HHE) method. An approximated two-body model calculation is also performed for all the three-body systems considered here. A Yukawa-type screened Coulomb potential with an arbitrary screening parameter (λ) is chosen for the two-body subsystems of the three-body system. In the resulting Schrödinger equation (SE), the three-body relative wave function is expanded in the complete set of hyperspherical harmonics (HH). The use of the orthonormality of HH in the SE leads to a set of coupled differential equations (CDEs) which are solved numerically for a manageable basis size to get the energy (E). The pattern of convergence in energy relative to increasing basis size is also investigated. Results are compared with some of those found in the literature.

Efficient representation of head-related transfer functions in continuous space–frequency domains

Utilizing spherical harmonic (SH) domain has been established as the default method of obtaining continuity over space in head-related transfer functions (HRTFs). This paper concerns different variants of extending this solution by replacing SHs with four-dimensional (4D) continuous functional models in which frequency is imagined as another physical dimension. Recently developed hyperspherical harmonic (HSH) representation is compared with models defined in spherindrical coordinate system by merging SHs with one-dimensional basis functions. The efficiency of both approaches is evaluated based on the reproduction errors for individual HRTFs from HUTUBS database, including detailed analysis of its dependency on chosen orders of approximation in frequency and space. Employing continuous functional models defined in 4D coordinate systems allows HRTF magnitude spectra to be expressed as a small set of coefficients which can be decoded back into values at any direction and frequency. The best performance was noted for HSHs and SHs merged with reverse Fourier–Bessel series, with the former featuring better compression abilities, achieving slightly higher accuracy for low number of coefficients. The presented models can serve multiple purposes, such as interpolation, compression or parametrization for machine learning applications, and can be applied not only to HRTFs but also to other types of directivity functions, e.g. sound source directivity.

Hyperspherical Cluster Model for Bosons: Application to Sub-threshold Halo States in Helium Drops

To describe long-range behaviour of one particle removed from a few- or a many-body system, a hyperspherical cluster model has been developed. It has been applied to the ground and first excited states of helium drops with five, six, eight and ten atoms interacting via a two-body soft gaussian potential. Convergence of the hyperspherical cluster harmonics expansion is studied for binding energies, root-mean-squared radii and overlaps of the wave functions of two helium drops differing by one atom. It was shown that with increasing model space the functional form of such overlaps at large distances converges to the correct asymptotic behaviour. The asymptotic normalization coefficients that quantify the overlaps’ amplitudes in this region are calculated. It was also shown that in the first excited state one helium atom stays far apart from the rest forming a two-body molecule, or a halo. The probability of finding the halo atom in the classically-forbidden region of space depends on the definition of the latter and on the valence atom binding energy. The total norm of the overlap integrals, the spectroscopic factor, represents the number of partitions of a many-body state into a chosen state of the system with one particle removed. The spectroscopic factors have been calculated and their sum rules are discussed giving a further insight into the structure of helium drops.

Theoretical Study of the d(d, p)^{3}H and d(d, n)^{3}He Processes at Low Energies.

We present a theoretical study of the processes d(d,p)^{3}H and d(d,n)^{3}He at energies of interest for energy production and for big-bang nucleosynthesis. We accurately solve the four body scattering problem using the abinitio hyperspherical harmonics method, starting from nuclear Hamiltonians which include modern two- and three-nucleon interactions, derived in chiral effective field theory. We report results for the astrophysical S factor, the quintet suppression factor, and various single and double polarized observables. A first estimate of the theoretical uncertainty for all these quantities is provided by varying the cutoff parameter used to regularize the chiral interactions at high momentum.

Geometrical properties of ΩNN three-body states by realistic NN and first principles Lattice QCD ΩN potentials

The Faddeev equations in coordinate space are solved to study the ΩNN and ΩΩN three-body systems using the latest ΩNS25 and ΩΩ S01 interactions developed by the HAL QCD Collaboration. We recalculate the binding energy of the ΩNN system by examining three NN potentials, i.e., modern realistic AV18 potential, Yukawa-type Malfliet-Tjon (MT) interaction, and Gogny-Pires-Tourreil (GPT) soft and local potential. We take into account the contribution of the Coulomb potential. Our numerical calculations for Ωd(T=0) in maximum spin 5/2+ confirm ground state binding energy of 20.953, 19.368, and 20.439 MeV and a matter radius of 1.097, 1.373, and 1.309 fm using MT, GPT, and AV18 NN potentials, respectively. In the case of Ωd(0)5/2+ system, our numerical analysis shows that considering higher partial waves than s wave in NN interactions leads to an increase of about 0.2 MeV using GPT and about 0.1 MeV reduction with AV18 potentials. We study the convergence of three-body binding energies in a cluster model using the hyperspherical harmonics method and investigate the geometrical properties of Ωd(0)5/2+ ground states.

Study of resonances in exotic neutron-rich 14Be by the use of isospectral potential

In this work, Hyperspherical Harmonic Expansion (HHE) formalism aided by supersymmetric quantum mechanics is used to study bound and resonant states of 14Be in the three-body (12Be+ n + n) cluster model, and the analysis of the resonant states close to the binding threshold utilizes supersymmetric quantum mechanics (SSQM). GPT nucleon-nucleon potential together with SBB core-nucleon potential is chosen for the solution of three-body Schrödinger equation to get the lowest bound state energy and wave function. In the next stage, energy and wave function of the bound state is used to derive an isospectral potential that exhibit deep well following a strong barrier facilitating confinement (or trapping) of particle inside it at some positive energy (E>0). The trapping probability when plotted against energy shows a prominent peak at the energy of resonance. WKB approximation is used to determine the width of resonance width. Calculated resonance energy and width of resonance are compared with those measured experimentally for the promising neutron halo candidate, the 14Be.

Onset of the bound states in strange Λ, Ξ-hypernuclei and hyperspherical three-body model calculation for the corresponding double hyperon hypernuclei

In this work, we have explored ground state properties of Be hypernucleus in the light of recent observation of the experiment J-PARC E07 and the onset of stability for Be hypernucleus in the three-body framework using Hyperspherical Harmonics Expansion (HHE) method. An effective core-Y (Y= Λ, Ξ) Woods-Saxon potential with adjustable depth parameter is used to reproduce the observed ground state energy of the two-body subsystem by solving the two-body Schrödinger equation. The core-Y potential so obtained is clubbed with the YY potential model of the Nijmegen group to obtain the observables of the double-hyperon hypernuclei. Computed YY separation energy and YY bond energy are compared with the data available in the literature.

Three-body Faddeev calculations for and hypernuclei* *M.R.H. was Supported by the National Science Foundation under Grant No. NSF-PHY-2000029 with Central State University

We study the ground-state properties of the double hyperon for and nuclei in a three-body model . We solve two coupled Faddeev equations corresponding to the three-body configurations and in configuration space with the hyperspherical harmonics expansion method by employing the most recent hyperon-hyperon interactions obtained from lattice QCD simulations. Our numerical analysis for , using three lattice interaction models, leads to a ground state binding energy in the MeV domain and the separations and in the domains of fm and fm, respectively. The binding energy of the double- hypernucleus leads to MeV and consequently to smaller separations fm and fm. In addition to geometrical properties, we study the structure of ground-state wave functions and show that the main contributions are from the wave channels. Our results are consistent with the existing theoretical and experimental data.

Efficient computation of N-point correlation functions in D dimensions

We present efficient algorithms for computing the N-point correlation functions (NPCFs) of random fields in arbitrary D-dimensional homogeneous and isotropic spaces. Such statistics appear throughout the physical sciences and provide a natural tool to describe stochastic processes. Typically, algorithms for computing the NPCF components have [Formula: see text] complexity (for a dataset containing n particles); their application is thus computationally infeasible unless N is small. By projecting the statistic onto a suitably defined angular basis, we show that the estimators can be written in a separable form, with complexity [Formula: see text] or [Formula: see text] if evaluated using a Fast Fourier Transform on a grid of size [Formula: see text]. Our decomposition is built upon the D-dimensional hyperspherical harmonics; these form a complete basis on the [Formula: see text] sphere and are intrinsically related to angular momentum operators. Concatenation of [Formula: see text] such harmonics gives states of definite combined angular momentum, forming a natural separable basis for the NPCF. As N and D grow, the number of basis components quickly becomes large, providing a practical limitation to this (and all other) approaches: However, the dimensionality is greatly reduced in the presence of symmetries; for example, isotropic correlation functions require only states of zero combined angular momentum. We provide a Julia package implementing our estimators and show how they can be applied to a variety of scenarios within cosmology and fluid dynamics. The efficiency of such estimators will allow higher-order correlators to become a standard tool in the analysis of random fields.

Nuclei with Up to $$\varvec{A=6}$$ Nucleons with Artificial Neural Network Wave Functions

The ground-breaking works of Weinberg have opened the way to calculations of atomic nuclei that are based on systematically improvable Hamiltonians. Solving the associated many-body Schr\"odinger equation involves non-trivial difficulties, due to the non-perturbative nature and strong spin-isospin dependence of nuclear interactions. Artificial neural networks have proven to be able to compactly represent the wave functions of nuclei with up to $A=4$ nucleons. In this work, we extend this approach to $^6$Li and $^6$He nuclei, using as input a leading-order pionless effective field theory Hamiltonian. We successfully benchmark their binding energies, point-nucleon densities, and radii with the highly accurate hyperspherical harmonics method.

Analysis of Gegenbauer kernel filtration on the hypersphere

In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated.

Ott–Antonsen ansatz for the D-dimensional Kuramoto model: A constructive approach

Kuramoto’s original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics of the order parameter with a single differential equation. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this article, we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. The method is based on our previous work for the 3D Kuramoto model where the ansatz was constructed using the spherical harmonics decomposition of the distribution function. In the case of motion in a D-dimensional sphere, the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz.

Comparative study of He6 β -decay based on different similarity-renormalization-group evolved chiral interactions

We report on a study of the Gamow-Teller matrix element contributing to $^{6}\mathrm{He}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$ decay with similarity renormalization group (SRG) versions of momentum- and configuration-space two-nucleon interactions. These interactions are derived from two different formulations of chiral effective field theory ($\ensuremath{\chi}\mathrm{EFT})$---without and with the explicit inclusion of $\mathrm{\ensuremath{\Delta}}$ isobars. We consider evolution parameters ${\mathrm{\ensuremath{\Lambda}}}_{\mathrm{SRG}}$ in the range between 1.2 and 2.0 ${\mathrm{fm}}^{\ensuremath{-}1}$ and, for the $\mathrm{\ensuremath{\Delta}}$-less case, also the unevolved (bare) interaction. The axial current contains one- and two-body terms, consistently derived at tree level (no loops) in the two distinct $\ensuremath{\chi}\mathrm{EFT}$ formulations we have adopted here. The $^{6}\mathrm{He}$ and $^{6}\mathrm{Li}$ ground-state wave functions are obtained from hyperspherical-harmonics (HH) solutions of the nuclear many-body problem. In $A=6$ systems, the HH method is limited at present to treat only two-body interactions and non-SRG evolved currents. Our results exhibit a significant dependence on ${\mathrm{\ensuremath{\Lambda}}}_{\text{SRG}}$ of the contributions associated with two-body currents, suggesting that a consistent SRG-evolution of these is needed in order to obtain reliable estimates. We also show that the contributions from one-pion-exchange currents depend strongly on the model (chiral) interactions and on the momentum- or configuration-space cutoffs used to regularize them. These results might prove helpful in clarifying the origin of the sign difference recently found in no-core-shell-model and quantum Monte Carlo calculations of the $^{6}\mathrm{He}$ Gamow-Teller matrix element.