Body Density Matrix Research Articles

Analyzing the Nuclear Structure of 13O-13B and 13N-13C Mirror Nuclei

In the context of the shell model, electromagnetic transitions were used to analyze the nuclear structure of a mirror nucleus with the same mass number, A = 13. The shell model investigation was performed by calculating the root mean square (rms) for the proton, neutron, matter and charge radii, occupancies, the excitation energies, as well as electromagnetic moments, using the elements of a body density matrix of the PSDMOD two-body effective interactions carried out in the psdpn-shell model space. The present results adopted the harmonic oscillator's single particle eigen functions and Hartree-Fock approximation. In addition, the effect of core polarization was added using the effective charge and effective g factors to calculate electromagnetic moments. To acquire a fair analysis of the information, the core polarization has to be present. The outcomes were compared with experimental data

Indications for the Substantial Predominance of Short-Range Effects on Inelastic Coulomb Form Factors for Various States in the 26Mg Nucleus

Short-range effects on C2, C3, as well as C4 form factors in the 26Mg nucleus, were examined. The charge density distribution in this nucleus was also tested by means of one and two body fragments of cluster enlargement in cooperation with single-particle wave functions of harmonic potential. The correlation of Jastrow form was employed to inset the influence of short-range into the two body fragment of cluster enlargement. The nucleus of 26Mg was assumed to own a 16O-core with (A-16) nucleons dispersed over the sd-model space. The form factors in 26Mg nucleus ascend from the core-polarization and model space involvements. The form of Tassie model, subject to the charge density, was used to determine the transition density of core polarization. The one body density matrix elements required for determining the transition density of model space for various transitions in 26Mg were found via carrying out shell model computations using the OXBASH program with the universal-sd interaction of Wildenthal. The present calculations were subjected to the oscillator and correlation parameters symbolized by and respectively. These parameters are self-sufficiently generated for every specific nucleus by fitting between the calculated and observed elastic form factors. For determining the charge density, elastic form factors and inelastic Coulomb form factors for dissimilar excited states in 26Mg, one value is needed for and This study shows indications for the substantial predominance of short-range influences on current computations, where considering these influences look to be requisite for carrying out a distinguished adjustment in calculated results which ultimately leads to a remarkable explication of the data throughout all the considered momentum transfers.

Mean field and beyond mean field calculations of hypernuclei for study of electroproduction

We present two methods, the Nucleon-Lambda Tamm Dancoff Approximation (NΛ TDA) and the Equation of Motion Phonon Method (EMPM) suitable for calculating hypernuclear energy spectra and structure. These methods are applicable for hypernuclei of wide range of masses with one Λ particle replacing one nucleon in an even-even nuclear cores. Using an effective Lambda-nucleon (ΛN) potential both methods were applied to calculate the energy spectrum of 12ΛB and also one body density matrix elements (OBDME). The OBDME were applied to calculate the electroproduction cross section of 12ΛB. We obtained better agreement with the experimental data by using EMPM than NΛ TDA. We plan to provide theoretical prediction (by applying the same methods and ΛN potentials) of the cross section in electroproduction of 40ΛK and 48ΛK.

Finite temperature off-diagonal long-range order for interacting bosons

Characterizing the scaling with the total particle number ($N$) of the largest eigenvalue of the one--body density matrix ($\lambda_0$), provides informations on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting $\lambda_0\sim N^{\mathcal{C}_0}$, then $\mathcal{C}_0=1$ corresponds to ODLRO. The intermediate case, $0<\mathcal{C}_0<1$, corresponds for translational invariant systems to the power-law decaying of (non-connected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in $\mathcal{C}_0$ [and in the corresponding quantities $\mathcal{C}_{k \neq 0}$ for excited natural orbitals] exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions. We show that $\mathcal{C}_{k \neq 0}=0$ in the thermodynamic limit. In $1D$ it is $\mathcal{C}_0=0$ for non-vanishing temperature, while in $3D$ $\mathcal{C}_0=1$ ($\mathcal{C}_0=0$) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to $D=2$, studying the $XY$ and the Villain models, and the weakly interacting Bose gas. The universal value of $\mathcal{C}_0$ near the Berezinskii--Kosterlitz--Thouless temperature $T_{BKT}$ is $7/8$. The dependence of $\mathcal{C}_0$ on temperatures between $T=0$ (at which $\mathcal{C}_0=1$) and $T_{BKT}$ is studied in the different models. An estimate for the (non-perturbative) parameter $\xi$ entering the equation of state of the $2D$ Bose gases, is obtained using low temperature expansions and compared with the Monte Carlo result. We finally discuss a double jump behaviour for $\mathcal{C}_0$, and correspondingly of the anomalous dimension $\eta$, right below $T_{BKT}$ in the limit of vanishing interactions.

One-body entanglement as a quantum resource in fermionic systems

We show that one-body entanglement, which is a measure of the deviation of a pure fermionic state from a Slater determinant (SD) and is determined by the mixedness of the single-particle density matrix (SPDM), can be considered as a quantum resource. The associated theory has SDs and their convex hull as free states, and number conserving fermion linear optics operations (FLO), which include one-body unitary transformations and measurements of the occupancy of single-particle modes, as the basic free operations. We first provide a bipartitelike formulation of one-body entanglement, based on a Schmidt-like decomposition of a pure $N$-fermion state, from which the SPDM [together with the $(N-1)$-body density matrix] can be derived. It is then proved that under FLO operations, the initial and postmeasurement SPDMs always satisfy a majorization relation, which ensures that these operations cannot increase, on average, the one-body entanglement. It is finally shown that this resource is consistent with a model of fermionic quantum computation which requires correlations beyond antisymmetrization. More general free measurements and the relation with mode entanglement are also discussed.

Symmetry conserving coupled cluster doubles wave function and the self-consistent odd particle number RPA

Mixing single and triple fermions an exact killing operator of the Coupled Cluster Doubles (CCD) wave function with good symmetry was found in \cite{Tohy13}. Using these operators with the equation of motion (EOM) method the so-called self-consistent odd particle number random phase approximation (odd-RPA) was set up. Together with the stationarity condition of the two body density matrix it is shown that the killing conditions allow to reduce the order of correlation functions contained in the matrix elements of the odd-RPA equations to a fully self consistent equation for the single particle occupation numbers. Excellent results for the latter and the ground state energies are obtained in an exactly solvable model from weak to strong couplings.

Investigation of the Quadrupole Moment and Form Factors of Some Ca Isotopes

Nuclear shell model is adopted to calculate the electric quadrupole moments for some Calcium isotopes 20Ca (N = 21, 23, 25, and 27) in the fp shell. The wave function is generated using a two body effective interaction fpd6 and fp space model. The one body density matrix elements (OBDM) are calculated for these isotopes using the NuShellX@MSU code. The effect of the core-polarizations was taken through the theory microscopic by taking the set of the effective charges. The results for the quadrupole moments by using Bohr-Mottelson (B-M) effective charges are the best. The behavior of the form factors of some Calcium isotopes was studied by using Bohr-Mottelson (B-M) effective charges.

Two body density matrix of model nuclear matter

A lowest-cluster-order variational calculation of the half-diagonal two-body density matrix ρ2(r1,r2,r’1) and the corresponding generalized momentum distribution n(p.Q) is performed for three representative models of nuclear matter containing central correlations. Dynamical correlations produce significant deviations from the results for a noninteracting Fermi gas. Calculations axe in progress that include higherorder cluster corrections as well as state-dependent correlations

Study of the Electric Quadrupole Moments for some Scandium Isotopes Using Shell Model Calculations with Different Interactions

The electric quadrupole moments for some scandium isotopes (41, 43, 44, 45, 46, 47Sc) have been calculated using the shell model in the proton-neutron formalism. Excitations out of major shell model space were taken into account through a microscopic theory which is called core polarization effectives. The set of effective charges adopted in the theoretical calculations emerging about the core polarization effect. NushellX@MSU code was used to calculate one body density matrix (OBDM). The simple harmonic oscillator potential has been used to generate the single particle matrix elements. Our theoretical calculations for the quadrupole moments used the two types of effective interactions to obtain the best interaction compared with the experimental data. The theoretical results of the quadrupole moments for some scandium isotopes performed with FPD6 interaction and Bohr-Mottelson effective charge agree with experimental values.

Superfluidity, Bose-Einstein condensation, and structure in one-dimensional Luttinger liquids

We report diffusion Monte Carlo (DMC) and path integral Monte Carlo (PIMC) calculations of the properties of a one-dimensional (1D) Bose quantum fluid. The equation of state, the superfluid fraction ${\ensuremath{\rho}}_{S}/{\ensuremath{\rho}}_{0}$, the one-body density matrix $n(x)$, the pair distribution function $g(x)$, and the static structure factor $S(q)$ are evaluated. The aim is to test Luttinger liquid (LL) predictions for 1D fluids over a wide range of fluid density and LL parameter $K$. The 1D Bose fluid examined is a single chain of $^{4}\mathrm{He}$ atoms confined to a line in the center of a narrow nanopore. The atoms cannot exchange positions in the nanopore, the criterion for 1D. The fluid density is varied from the spinodal density where the 1D liquid is unstable to droplet formation to the density of bulk liquid $^{4}\mathrm{He}$. In this range, $K$ varies from $Kg2$ at low density, where a robust superfluid is predicted, to $Kl0.5$, where fragile 1D superflow and solidlike peaks in $S(q)$ are predicted. For uniform pore walls, the ${\ensuremath{\rho}}_{S}/{\ensuremath{\rho}}_{0}$ scales as predicted by LL theory. The $n(x)$ and $g(x)$ show long range oscillations and decay with $x$ as predicted by LL theory. The amplitude of the oscillations is large at high density (small $K)$ and small at low density (large $K)$. The $K$ values obtained from different properties agree well verifying the internal structure of LL theory. In the presence of disorder, the ${\ensuremath{\rho}}_{S}/{\ensuremath{\rho}}_{0}$ does not scale as predicted by LL theory. A single ${v}_{J}$ parameter in the LL theory that recovers LL scaling was not found. The one body density matrix (OBDM) in disorder is well predicted by LL theory. The ``dynamical'' superfluid fraction, ${\ensuremath{\rho}}_{S}^{D}/{\ensuremath{\rho}}_{0}$, is determined. The physics of the deviation from LL theory in disorder and the ``dynamical'' ${\ensuremath{\rho}}_{S}^{D}/{\ensuremath{\rho}}_{0}$ are discussed.

Study of the Electric Quadrupole Moments for some Scandium Isotopes Using Shell Model Calculations with Different Interactions

The electric quadrupole moments for some scandium isotopes (41, 43, 44, 45, 46, 47Sc) have been calculated using the shell model in the proton-neutron formalism. Excitations out of major shell model space were taken into account through a microscopic theory which is called core polarization effectives. The set of effective charges adopted in the theoretical calculations emerging about the core polarization effect. NushellX@MSU code was used to calculate one body density matrix (OBDM). The simple harmonic oscillator potential has been used to generate the single particle matrix elements. Our theoretical calculations for the quadrupole moments used the two types of effective interactions to obtain the best interaction compared with the experimental data. The theoretical results of the quadrupole moments for some scandium isotopes performed with FPD6 interaction and Bohr-Mottelson effective charge agree with experimental values.

Nuclear Electric Quadrapole Moments (Q) in &amp;lt;sup&amp;gt;58&amp;lt;/sup&amp;gt;Ni

Nuclear Electric quadrapole moments Q in 58Ni for some selected levels have been investigated and calculated through Nuclear shell model and considering of 56Ni as an inert core with two active neutrons in a model space (2p3/2, 1f5/2 and 2p1/2) and the configuration mixing of the original states is also done. F5Pvh interaction has been utilized as a two body interaction to generate model space vectors with harmonic oscillator potential as a single particle wave function. OXBASH code is used to carry this calculations and the program of Core, Valence, Tassie (CVT) written in FORTRAN go language to calculate the Electric quadrapole moments between excited states themselves. All of these calculations have been carried through model space vectors only. One body density matrix elements (OBDM) for ground and Excited states is calculated in order to carry the calculations using single particle Transition matrix elements between excited states theme selves.

Ground state properties of anti-ferromagnetic spinor Bose gases in one dimension

We investigate the ground state properties of anti-ferromagnetic spin-1 Bose gases in one dimensional harmonic potential from the weak repulsion regime to the strong repulsion regime. By diagonalizing the Hamiltonian in the Hilbert space composed of the lowest eigenstates of single particle and spin components, the ground state wavefunction and therefore the density distributions, magnetization distribution, one body density matrix, and momentum distribution for each components are obtained. It is shown that the spinor Bose gases of different magnetization exhibit the same total density profiles in the full interaction regime, which evolve from the single peak structure embodying the properties of Bose gases to the fermionized shell structure of spin-polarized fermions. But each components display different density profiles, and magnetic domains emerge in the strong interaction limit for $M=0.25$. In the strong interaction limit, one body density matrix and the momentum distributions exhibit the same behaviours as those of spin-polarized fermions. The fermionization of momentum distribution takes place, in contrast to the $\delta$-function-like distribution of single component Bose gases in the full interaction region.

Multicomponent density-functional theory for time-dependent systems

We derive the basic formalism of density functional theory for time-dependent electron-nuclear systems. The basic variables of this theory are the electron density in body-fixed frame coordinates and the diagonal of the nuclear $N$-body density matrix. The body-fixed frame transformation is carried out in order to achieve an electron density that reflects the internal symmetry of the system. We discuss the implications of this body-fixed frame transformation and establish a Runge-Gross-type theorem and derive Kohn-Sham equations for the electrons and nuclei. We illustrate the formalism by performing calculations on a one-dimensional diatomic molecule for which the many-body Schr\"odinger equation can be solved numerically. These benchmark results are then compared to the solution of the time-dependent Kohn-Sham equations in the Hartree approximation. Furthermore, we analyze the excitation energies obtained from the linear response formalism in the single pole approximation. We find that there is a clear need for improved functionals that go beyond the simple Hartree approximation.

BOSE–EINSTEIN CONDENSATION IN BULK AND CONFINED SOLID HELIUM

We address the question if the ground state of solid 4 He has the number of lattice sites equal to the number of atoms (commensurate state) or if it is different (incommensurate state). We point out that energy computation from simulation as performed by now cannot be used to decide this question and that the presently best variational wave function, a shadow wave function, gives an incommensurate state. We have extended the calculation of the one–body density matrix ρ1 to the exact Shadow Path Integral Ground State method. Calculation of ρ1 at ρ = 0.031 Å-3 shows that Vacancy–Interstitial pair processes are present also in the exact computation but the simulated system size is too small to infer the presence of off–diagonal long range order. Variational simulations of 4 He confined in a narrow cylindrical pore are also discussed.

Ground-state properties of one-dimensional ultracold Bose gases in a hard-wall trap

We investigate the ground state of the system of $N$ bosons enclosed in a hard-wall trap interacting via a repulsive or attractive $\ensuremath{\delta}$-function potential. Based on the Bethe ansatz method, the explicit ground state wave function is derived and the corresponding Bethe ansatz equations are solved numerically for the full physical regime from the Tonks limit to the strongly attractive limit. It is shown that the solution takes a different form in different regime. We also evaluate the one body density matrix and second-order correlation function of the ground state for finite systems. In the Tonks limit the density profiles display the Fermi-like behavior, while in the strongly attractive limit the Bosons form a bound state of $N$ atoms corresponding to the $N$-string solution. The density profiles show the continuous crossover behavior in the entire regime. Further, the correlation function indicates that the Bose atoms bunch closer as the interaction constant decreases.

Bose-Einstein Condensation of Incommensurate SolidHe4

It is pointed out that the simulation computation of energy performed so far cannot be used to decide if the ground state of solid 4He has the number of lattice sites equal to the number of atoms (commensurate state) or if it is different (incommensurate state). The best variational wave function, a shadow wave function, gives an incommensurate state, but the equilibrium concentration of vacancies remains to be determined. We have computed the one-body density matrix in solid 4He for the incommensurate state by means of an exact ground state projector method in which incommensurability occurs spontaneously. We find a vacancy induced Bose-Einstein condensation of about 0.23 atoms per vacancy at a pressure of 54 bar. This means that bulk solid 4He is supersolid at low enough temperature if the exact ground state is incommensurate.

Bose-Einstein-condensate superfluid–Mott-insulator transition in an optical lattice

We present in this paper an analytical model for a cold bosonic gas on an optical lattice (with densities of the order of 1 particle per site) targeting the critical regime of the Bose - Einstein Condensate superfluid - Mott insulator transition. We focus on the computation of the one - body density matrix and its Fourier transform, the momentum distribution which is directly obtainable from `time of flight'' measurements. The expected number of particles with zero momentum may be identified with the condensate population, if it is close to the total number of particles. Our main result is an analytic expression for this observable, interpolating between the known results valid for the two regimes separately: the standard Bogoliubov approximation valid in the superfluid regime and the strong coupling perturbation theory valid in the Mott regime.

Layer by layer solidification ofHe4in narrow porous media

We give a variational Monte Carlo description of $^{4}\mathrm{He}$ filling under pressure a porous material modeled by a smooth cylindrical nanopore. Our trial wave function is a shadow wave function which allows different degrees of correlation between $^{4}\mathrm{He}$ atoms depending on the distance from the pore wall but still preserving the full Bose symmetry. The radial density profile shows a strong layering of the $^{4}\mathrm{He}$ atoms which are located in concentric annuli. This system has a very rich phase diagram with at least four different phases. The layer in contact with the pore is always solid. For our narrow pore radius $(R=13\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$, as the density is increased, solidification takes place layer by layer, starting from the pore wall, as is confirmed by the static structure factors. The pore radius is too small to allow a bulklike solid to nucleate in the liquid region at the center of the pore, and in order to have a complete crystalline order in all the layers a pressure greater than $200\phantom{\rule{0.3em}{0ex}}\text{bar}$ is needed. Computing the one body density matrix we are able to estimate the condensate fraction, which is still nonzero even if all the layers are in the solid phase.

Bose-Einstein condensation in solidHe4

We have computed the one--body density matrix rho_1 in solid 4He at T=0 K using the Shadow Wave Function (SWF) variational technique. The accuracy of the SWF has been tested with an exact projector method. We find that off-diagonal long range order is present in rho_1 for a perfect hcp and bcc solid 4He for a range of densities above the melting one, at least up to 54 bars. This is the first microscopic indication that Bose Einstein Condensation (BEC) is present in perfect solid 4He. At melting the condensate fraction in the hcp solid is 5*10^{-6} and it decreases by increasing the density. The key process giving rise to BEC is the formation of vacancy--interstitial pairs. We also present values for Leggett's upper bound on the superfluid fraction deduced from the exact local density.