One-body Density Matrices Research Articles

Dmscatter: A fast program for WIMP-nucleus scattering

Recent work [1,2], using an effective field theory framework, have shown the number of possible couplings between nucleons and the dark-matter-candidate Weakly Interacting Massive Particles (WIMPs) is larger than previously thought. Inspired by an existing Mathematica script that computes the target response [2], we have developed a fast, modern Fortran code, including optional OpenMP parallelization, along with a user-friendly Python wrapper, to swiftly and efficiently explore many scenarios, with output aligned with practices of current dark matter searches. A library of most of the important target nuclides is included; users may also import their own nuclear structure data, in the form of reduced one-body density matrices. The main output is the differential event rate as a function of recoil energy, needed for modeling detector response rates, but intermediate results such as nuclear form factors can be readily accessed. Program summaryProgram Title: dmscatterCPC Library link to program files:https://doi.org/10.17632/6fm7d4m2py.1Developer's repository link:https://github.com/ogorton/dmscatterLicensing provisions: MITProgramming languages: Fortran, PythonNature of problem: Simulating the event rate of nuclear recoils from collisions with dark matter for a variety of different nuclear targets and different nucleon-dark matter couplings.Solution method: The event rate is an integral over the product of the nuclear and dark matter response functions, weighted by the expected dark matter flux. To compute the nuclear response, reduced one-body nuclear density matrices are coupled to multipole expansion of operators computed analytically in a harmonic oscillator basis. The nuclear structure input, in the form of one-body densities matrices, are supplied by outside code; we provide a library of the most prominent targets.

Calculation magnetic dipole moments, electric quadrupole moments and form factors for some Ti isotopes

This paper presents calculations of the magnetic moments, the quadrupole moments, and the form factors of a number of titanium isotopes as determined using the shell model space. The calculations of the shell model have adopted the model space (MS) with core-polarization effects, which were included for two types of effective g factors and different effective charges. The wave functions are generated using the harmonic oscillator potential (HO) with size parameters b, which for these isotopes are calculated. The one body density matrices (OBDM) for these isotopes were obtained using the NuShellX@MSU code, which used an adopted fpd6 interaction and fp model space. Reasonable results are obtained that show the collapse of the magicity property at neutron number N = 28.

Applications of Time-Dependent Density-Matrix Approach

The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of $^{40}$Ca and $^{48}$Ca and the other the fusion reactions of $^{16}$O + $^{16}$O.

Insights into one-body density matrices using deep learning.

The one-body reduced density matrix (1-RDM) of a many-body system at zero temperature gives direct access to many observables, such as the charge density, kinetic energy and occupation numbers. It would be desirable to express it as a simple functional of the density or of other local observables, but to date satisfactory approximations have not yet been found. Deep learning is the state of the art approach to performing high dimensional regressions and classification tasks, and is becoming widely used in the condensed matter community to develop increasingly accurate density functionals. Autoencoders are deep learning models that perform efficient dimensionality reduction, allowing the distillation of data to the fundamental features needed to represent it. By training autoencoders on a large data-set of 1-RDMs from exactly solvable real-space model systems, and performing principal component analysis, the machine learns to what extent the data can be compressed and hence how it is constrained. We gain insight into these machine learned constraints and employ them to inform approximations to the 1-RDM as a functional of the charge density. We exploit known physical properties of the 1-RDM in the simplest possible cases to perform feature engineering, where we inform the structure of the models from known mathematical relations, allowing us to integrate existing understanding into the machine learning methods. By comparing various deep learning approaches we gain insight into what physical features of the density matrix are most amenable to machine learning, utilising both known and learned characteristics.

Truncation scheme of time-dependent density-matrix approach III

The time-dependent density-matrix theory (TDDM) where the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for reduced density matrices is truncated by approximating a three-body density matrix with one-body and two-body density matrices is applied to the Lipkin model. It is shown that in the large $N$ limit the ground state in TDDM approaches the exact solution. Various truncation schemes for the three-body density matrix are also tested for an extended three-level Lipkin model.

Time dependent spatial entanglement in atom-field interaction

By using stochastic ensembles of walkers in physical and in one-body Hilbert spaces the recently proposed time-dependent quantum Monte Carlo (TDQMC) method offers the unique capability to calculate one-body density matrices at fully correlated level, without referencing the many-body quantum state. Here TDQMC is applied to study entanglement of simple systems such as Moshinsky atom (oscillator potentials) and atoms with Coulomb potentials in one spatial dimension. Our findings indicate that the dynamic entanglement of atoms exposed to powerful ultrashort laser pulse can be easily manipulated by introducing an appropriate phase modulation where the negative chirp enhances the entanglement while the positive one suppresses it. These findings can be used to explore the correlation properties of different constituents of complex quantum systems subjected to appropriately shaped laser radiation.

Ab initio translationally invariant nonlocal one-body densities from no-core shell-model theory

[Background:] It is well known that effective nuclear interactions are in general nonlocal. Thus if nuclear densities obtained from {\it ab initio} no-core-shell-model (NCSM) calculations are to be used in reaction calculations, translationally invariant nonlocal densities must be available. [Purpose:] Though it is standard to extract translationally invariant one-body local densities from NCSM calculations to calculate local nuclear observables like radii and transition amplitudes, the corresponding nonlocal one-body densities have not been considered so far. A major reason for this is that the procedure for removing the center-of-mass component from NCSM wavefunctions up to now has only been developed for local densities. [Results:] A formulation for removing center-of-mass contributions from nonlocal one-body densities obtained from NCSM and symmetry-adapted NCSM (SA-NCSM) calculations is derived, and applied to the ground state densities of $^4$He, $^6$Li, $^{12}$C, and $^{16}$O. The nonlocality is studied as a function of angular momentum components in momentum as well as coordinate space [Conclusions:] We find that the nonlocality for the ground state densities of the nuclei under consideration increases as a function of the angular momentum. The relative magnitude of those contributions decreases with increasing angular momentum. In general, the nonlocal structure of the one-body density matrices we studied is given by the shell structure of the nucleus, and can not be described with simple functional forms.

Evolution from few- to many-body physics in one-dimensional Fermi systems: One- and two-body density matrices and particle-partition entanglement

We study the evolution from few- to many-body physics of fermionic systems in one spatial dimension with attractive pairwise interactions. We determine the detailed form of the momentum distribution, the structure of the one-body density matrix, and the pairing properties encoded in the two-body density matrix. From the low- and high-momentum scaling behavior of the single-particle momentum distribution we estimate the speed of sound and Tan's contact, respectively. Both quantities are found to be in agreement with previous calculations. Based on our calculations of the one-body density matrices, we also present results for the particle-partition entanglement entropy, for which we find a logarithmic dependence on the total particle number.

Some universal properties of density matrices for finite nuclei (bound systems)

The properties of one-body and two-body density matrices of a nucleus as a nonrelativistic system are studied. Unlike the usual procedure applied in the theory of infinite systems, for a nucleus of A nucleons these quantities are determined as the expectation values of A-body multiplicative operators that depend on the relative coordinates and momenta (Jacobi variables) and affect the intrinsic wave functions. The translational invariance of these operators is thus ensured. An algebraic technique based on the Cartesian representation is developed for handling such operators. Within this technique, the coordinate and momentum operators are linear combinations of production and annihilation operators \(\hat \vec a^ + \) and \(\hat \vec a\) with commutation relations for bosons (oscillators). Each of the multiplicative operators reduces to the normally ordered product of two exponents, one of which depends on set \(\left\{ {\hat \vec a^ + } \right\}\) only and is located to the left of the second, which depends on \(\left\{ {\hat \vec a} \right\}\) only. This procedure offers a new view of the origin of the so-called Tassie–Barker factors and other model-independent results. In addition, the proposed method yields a fair description of the available experimental data and can easily be extended to investigate the role of short-range nucleon-nucleon correlations within these translationally invariant calculations of nucleon density and momentum distributions in light nuclei.

Weakly interacting massive particle-nucleus elastic scattering response

The community's reliance on simplified descriptions of WIMP-nucleus interactions reflects the absence of analysis tools that integrate general theories of dark matter with standard treatments of nuclear response functions. To bridge this gap, we have constructed a public-domain Mathematica package for WIMP analyses based on our effective theory formulation. Script inputs are 1) the coefficients of the effective theory, through which one can characterize the low-energy consequences of arbitrary ultraviolet theories of WIMP interactions; and 2) one-body density matrices for commonly used targets, the most compact description of the relevant nuclear physics. The generality of the effective theory expansion guarantees that the script will remain relevant as new ultraviolet theories are explored; the use of density matrices to factor the nuclear physics from the particle physics will allow nuclear structure theorists to update the script as new calculations become available, independent of specific particle-physics contexts. The Mathematica package outputs the resulting response functions (and associated form factors) and also the differential event rate, once a galactic WIMP velocity profile is specified, and thus in its present form provides a complete framework for experimental analysis. The Mathematica script requires no a priori knowledge of the details of the non-relativistic effective field theory or nuclear physics, though the core concepts are reviewed here and in arXiv:1203.3542.

Mean-field models for disordered crystals

In this article, we set up a functional setting for mean-field electronic structure models of Hartree–Fock or Kohn–Sham types for disordered quantum systems. In the first part, we establish important properties of stochastic fermionic one-body density matrices, assuming that they are stationary under the ergodic action of a translation group. In particular, we prove Hoffmann-Ostenhof and Lieb–Thirring inequalities for ergodic density matrices, and deduce some weak compactness properties of the set of such matrices. We also discuss the representability problem for the associated one-particle densities. In the second part, we investigate the problem of solving Poissonʼs equation for a given stationary charge distribution, using the Yukawa potential to appropriately define the Coulomb self-interaction in the limit when the Yukawa parameter goes to zero. Finally, in the last part of the article, we use these tools to study a specific mean-field model (reduced Hartree–Fock, rHF) for a disordered crystal where the nuclei are classical particles whose positions and charges are random. We prove the existence of a minimizer of the energy per unit volume and the uniqueness of the ground state density. For (short-range) Yukawa interactions, we prove in addition that the rHF ground state density matrix satisfies a self-consistent equation, and that our model is the thermodynamic limit of the supercell model.

Formulation of functional theory for pairing with particle number restoration

The restoration of particle number within Energy Density Functional theory is analyzed. It is shown that the standard method based on configuration mixing leads to a functional of both the projected and non-projected densities. As an alternative that might be advantageous for mass models, nuclear dynamics and thermodynamics, we propose to formulate the functional in terms directly of the one-body and two-body density matrices of the state with good particle number. Our approach does not contain the pathologies recently observed when restoring the particle number in an Energy Density Functional framework based on transition density matrices and can eventually be applied with functionals having arbitrary density dependencies.

Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction

We investigate small equal-mass two-component Fermi gases under external spherically symmetric confinement in which atoms with opposite spins interact through a short-range two-body model potential. We employ a non-perturbative microscopic framework, the stochastic variational approach, and determine the system properties as functions of the interspecies s-wave scattering length a, the orbital angular momentum L of the system, and the numbers N1 and N2 of spin-up and spin-down atoms (with N1-N2 =0 or 1 and N < 7, where N=N1+N2). At unitarity, we determine the energies of the five- and six-particle systems for various ranges r0 of the underlying two-body model potential and extrapolate to the zero-range limit. These energies serve as benchmark results that can be used to validate and assess other numerical approaches. We also present structural properties such as the pair distribution function and the radial density. Furthermore, we analyze the one-body and two-body density matrices. A measure for the molecular condensate fraction is proposed and applied. Our calculations show explicitly that the natural orbitals and the momentum distributions of atomic Fermi gases approach those characteristic for a molecular Bose gas if the s-wave scattering length a, a>0, is sufficiently small.

Ultracold atoms at unitarity within quantum Monte Carlo methods

Variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the properties of the zero-temperature fermionic gas at unitarity are reported. The ratio of the energy of the interacting to the non-interacting gas for a system of 128 particles is calculated to be 0.4517(3) in VMC and 0.4339(1) in the more accurate DMC method. The spherically-averaged pair-correlation functions, momentum densities, and one-body density matrices are very similar in VMC and DMC, but the two-body density matrices and condensate fractions show some differences. Our best estimate of the condensate fraction of 0.51 is a little smaller than values from other quantum Monte Carlo calculations.

Fermionization of a strongly interacting Bose-Fermi mixture in a one-dimensional harmonic trap

We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient $C$ is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.

Medium-Mass Nuclei from Chiral Nucleon-Nucleon Interactions

We compute the binding energies, radii, and densities for selected medium-mass nuclei within coupled-cluster theory and employ a bare chiral nucleon-nucleon interaction at next-to-next-to-next-to-leading order. We find rather well-converged results in model spaces consisting of 15 oscillator shells, and the doubly magic nuclei 40Ca, 48Ca, and the exotic 48Ni are underbound by about 1 MeV per nucleon within the coupled-cluster singles-doubles approximation. The binding-energy difference between the mirror nuclei 48Ca and 48Ni is close to theoretical mass table evaluations. Our computation of the one-body density matrices and the corresponding natural orbitals and occupation numbers provides a first step to a microscopic foundation of the nuclear shell model.

An exact microscopic multiphonon approach to collective modes in nuclei

We propose a method for solving the nuclear eigenvalue problem using a Hamiltonian of general form in a space spanned by multiphonon states built out of particle-hole TammDancoff phonons. The method consists in deriving for a given n-phonon subspace a set of equations of motion of simple structure, where all quantities are expressed in terms of energies and one-body density matrices determined in the (n − 1)-phonon subspace. Their solution, therefore, yields states built out of particle-hole operators applied on top of the (n−1)phonons basis states. Because of this peculiar structure, the states so generated form an overcomplete set. The redundancy, however, is removed completely and efficiently by a procedure based on the Choleski decomposition. The phonon composition of the basis states allows to remove naturally and maximally the spurious admixtures induced by the center of mass motion. The iteration of the equations from n = 0 up to an arbitrary number N of phonons generates an orthogonal basis spanning a space which is the direct sum of 0+1+..+N phonon subspaces. This basis decomposes the full Hamiltonian into N diagonal blocks plus off-diagonal terms coupling the n-phonons with the (n ± 1)and (n ± 2)-phonon subspaces. These off-diagonal terms are easily computed by recursive relations. The Hamiltonian can, therefore, be easily diagonalized by resorting to importance sampling techniques [1,2], which allow to truncate the multiphonon space while monitoring the accuracy of the solutions. The method can be easily upgraded so as to generate a particle-hole or quasiparticle RPA multiphonon basis. It is suitable for describing with high accuracy low-lying multiphonon spectra, whose experimental evidence is rapidly growing, as well as high-energy resonances like the recently discovered double dipole giant resonance. It can account with great accuracy for the anharmonicities associated to all collective modes. Last, but not least, it yields a highly correlated ground state which contains explicitly up to N particle N hole configurations and, therefore, applies to the cases where the quasiboson approximation breaks down. For illustrative purposes, we apply the method to 16O, whose low-lying states are dominated by complex configurations containing up to 4 particle 4 hole states. Such a calculation probes the extreme accuracy of the method, on the one hand, and, on the other hand, provides a complete and exhaustive information on the microscopic structure of low and high energy states in 16O.

The one-body and two-body density matrices of finite nuclei with an appropriate treatment of the center-of-mass motion⋆

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.

Relation between Density-Matrix Theory and Pairing Theory

The time-dependent density-matrix theory (TDDM) gives a correlated ground state as a stationary solution of the time-dependent equations for one-body and two-body density matrices. The small amplitude limit of TDDM (STDDM) is a version of extended RPA theories which include the effects of ground state correlations. It is shown that the solutions of the Hartree-Fock Bogoliubov theory and the quasi-particle RPA satisfy the TDDM and STDDM equations, respectively, when only pairing-type correlations are taken into account in TDDM and STDDM.

Jastrow-type calculations of one-nucleon removal reactions on opens−dshell nuclei

Single-particle overlap functions and spectroscopic factors are calculated on the basis of Jastrow-type one-body density matrices of open-shell nuclei constructed by using a factor cluster expansion. The calculations use the relationship between the overlap functions corresponding to bound states of the $(A-1)$-particle system and the one-body density matrix for the ground state of the $A$-particle system. In this work we extend our previous analyses of reactions on closed-shell nuclei by using the resulting overlap functions for the description of the cross sections of $(p,d)$ reactions on the open $s$-$d$ shell nuclei $^{24}$Mg, $^{28}$Si and $^{32}$S and of $^{32}$S$(e,e^{\prime}p)$ reaction. The relative role of both shell structure and short-range correlations incorporated in the correlation approach on the spectroscopic factors and the reaction cross sections is pointed out.