Ground-state Momentum Distribution Research Articles

Momentum distributions and numerical methods for strongly interacting one-dimensional spinor gases

One-dimensional spinor gases with strong $\ensuremath{\delta}$ interaction fermionize and form a spin chain. The spatial degrees of freedom of this atom chain can be described by a mapping to spinless noninteracting fermions and the spin degrees of freedom are described by a spin-chain model with nearest-neighbor interactions. Here, we compute momentum and occupation-number distributions of up to 16 strongly interacting spinor fermions and bosons as a function of their spin imbalance, the strength of an externally applied magnetic field gradient, the length of their spin, and for different excited states of the multiplet. We show that the ground-state momentum distributions resemble those of the corresponding noninteracting systems, apart from flat background distributions, which extend to high momenta. Moreover, we show that the spin order of the spin chain---in particular antiferromagnetic spin order---may be deduced from the momentum and occupation-number distributions of the system. Finally, we present efficient numerical methods for the calculation of the single-particle densities and one-body density matrix elements and of the local exchange coefficients of the spin chain for large systems containing more than 20 strongly interacting particles in arbitrary confining potentials.

High-momentum tail in the Tonks–Girardeau gas under general confining potentials

We prove that the ground state momentum distribution of a one-dimensional system of impenetrable bosons exhibits a k −4 tail for any confining potential. We also derive an expression for easily computing the asymptotic occupation numbers and verify our results with an exact numerical approach.

Momentum distribution of electrons in parabolic quantum dots

We investigate the momentum distribution of interacting electrons confined in a two-dimensional parabolic quantum dot for varying values of the interaction strength and for different number of electrons. We use an optimized Jastrow-type wave function for the ground state in which the Slater determinant part has been calculated using density-functional theory and evaluate the ground-state momentum distribution using the Fermi hypernetted-chain equations. We discuss the effects of interactions on the momentum distribution and reveal the existence of a shell structure in momentum space in analogy with what happens to the spatial density profile. The momentum distribution can thus be used as a further tool to investigate the effects of the interaction in parabolic quantum dots.

On the (anisotropic) uniform metallic ground states of fermions interacting through arbitrary two-body potentials in d dimensions

We demonstrate that the skeleton of the Fermi surface pertaining to a uniform metallic ground state (corresponding to fermions with spin index σ) is determined by the Hartree–Fock contribution to the dynamic self-energy . That is to say, in order for , it is necessary (but for anisotropic ground states in general not sufficient) that the following equation be satisfied: where stands for the underlying non-interacting energy dispersion and ϵF for the exact interacting Fermi energy. The Fermi surface consists of the set of k points which in addition to satisfying the above equation fulfil where1 The set of k points which satisfy the first of the above two equations but fail to satisfy the second constitute the pseudogap region of the putative Fermi surface of the interacting system. We consider the behaviour of the ground-state momentum distribution function for k in the vicinity of and show that, whereas for the uniform metallic ground states of the conventional single-band Hubbard Hamiltonian, described in terms of an on-site interaction, and (here and denote vectors infinitesimally close to , located respectively inside and outside the underlying Fermi sea), for interactions of non-zero range these inequalities can be violated (without thereby contravening the stability condition ). This aspect is borne out by the pertaining to the normal states of for instance liquid 3He (corresponding to a range of applied pressure) as determined by means of quantum Monte Carlo calculations. We further demonstrate that for Fermi-liquid metallic states of fermions interacting through interaction potentials of non-zero range (e.g. the Coulomb potential), the zero-temperature limit of does not need to be equal to ½ for ; this in strict contrast with the uniform Fermi-liquid metallic states of the single-band Hubbard Hamiltonian (if such states at all exist). This aspect should be taken into account while analysing the deduced from the angle-resolved photoemission data concerning real materials. We discuss, in the light of the findings of the present work, the growing experimental evidence with regard to the ‘frustration’ of the kinetic energy of the charge carriers in the normal states of the copper-oxide-based high-temperature superconducting compounds.

Population accumulation in dark states and subrecoil laser cooling.

The \ensuremath{\Lambda} system resulting from the interaction of counterpropagating orthogonally polarized beams of light with a ${\mathit{J}}_{\mathit{g}}$=1\ensuremath{\rightarrow}${\mathit{J}}_{\mathit{e}}$=1 transition is known to exhibit laser cooling without the presence of a damping force, through velocity selective coherent population trapping. This occurs independent of the sign of the laser's detuning from the atomic resonance frequency. We present measurements and calculations that show a detuning-dependent transient peak or dip in the ground-state momentum distribution centered at ${\mathit{p}}_{\mathit{g}}$=0, however, which is reminiscent of laser cooling by a damping force. We explain this in terms of the character of the quantum-mechanical eigenstates of the total ground-state Hamiltonian of the system, using both analytic expressions and numerical calculations.

K + production in proton and deuteron induced reactions at subthreshold energies

The production ofK+ mesons in proton-nucleus and deuteron-nucleus collisions is analyzed with respect to one-step nucleon-nucleon (NN → N ΛK+),Δ- nucleon (ΔN → N Λ K+) and two-step pion-nucleon (πN → K+Λ) production channels on the basis of experimental ground state momentum distributions and free on-shell production processes. Whereas forK+ production in proton-nucleus reactions the secondary channelπN clearly dominates at subthreshold energies, meson and nucleon induced channels are of similar magnitude in deuteron-nucleus reactions. Contrary to nucleus-nucleus collisions theΔ induced reaction channels are found to be of minor importance. The experimental differentiation of the underlying microscopic reaction channels appears possible via differential proton —K+ coincidence measurements as shown in detail by the microscopic simulations including proton rescattering.

Meson production in proton-nucleus and nucleus-nucleus collisions

The production of K+ or η mesons in proton-nucleus reactions is analysed with respect to primary nucleon-nucleon (NN → NΛK+ or NNη) and secondary pion-nucleon (πN → K+Λ or ηN) production channels on the basis of a phase-space model employing Hartree-Fock groundstate momentum distributions and free on-shell production processes. The phase-space model adopted here compares well for meson production with more involved simulations based on VUU transport equations. Whereas for K+ production in proton-nucleus reactions the secondary channel clearly dominates at subthreshold energies, η mesons arise from primary and secondary processes with roughly the same order of magnitude. Detailed VUU calculations show that the relative weight of the various channels for nucleus-nucleus reactions is quite different as compared to the proton-nucleus case since Δ-resonance excitation and inelastic Δ-nucleon collisions are much more frequent.

High energy gamma rays from 14N+natAg at 35 MeV/nucleon.

The absolute cross section and angular distributions for [gamma]-ray production in the reaction [sup 14]N+[sup nat]Ag at 35 MeV/nucleon have been measured. Calculations of the expected [gamma]-ray yield using an enhanced version of the nucleon exchange transport model have been performed. It has been observed that use of a sharp ground state momentum distribution in the [gamma]-ray calculations does not lead to reproduction of the experimental yields. Diffuse momentum distributions have been incorporated into the model, and the calculations with these momentum distributions give good agreement with experimental yields. Similar calculations have been performed for the production of preequilibrium neutrons and protons for the same system.

? Production in proton-nucleus reactions

The production ofη-mesons in proton-nucleus reactions is analysed with respect to primary nucleon-nucleon (NN→NNη) and secondary pion-nucleon (πN→ηN) production processes on the basis of Hartree-Fock groundstate momentum distributions and free on-shell production processes. The folding model adopted compares well for meson production with more involved simulations based on VUU transport equations. Similar to K+ production in proton-nucleus reactions theη-mesons are primarily produced by theπN →ηN channel. However,η-mesons are absorbed in nuclei via excitation of the N* (1535) resonance which leads to strong distortions of the primordial spectra. On the other hand, the experimental mass dependence of the differential cross sections might yield information about the in-medium properties of this resonance.

Subthreshold K +-production in proton-nucleus reactions

The production of positively charged kaons in proton-nucleus reactions is analysed with respect to primary nucleon-nucleon and two-step proton-nucleon→pion plus pion-nucleon production processes on the basis of Hartree-Fock ground state momentum distributions and free on-shell production processes. Clear evidence is found for the dominance of secondary processes πN→ ΛK + below 1 GeV. Estimates for the production of hypernuclei in p+ 12C and p+U reactions up to 1 GeV are given.

Models for relativistic coulomb sum rules: Expansions in moments of the nuclear momentum density

Relativistic Coulomb sum rules for quasielastic electron scattering from nuclei are developed using a class of relativistic models for the nuclear ground-state momentum distribution. Approximate sum rules at constant 3- or 4-momentum transfer are expressed as expansions in moments of the momentum distribution. New sum-rule functions are derived which, even for very large values of energy and momentum where relativistic effects become dominant, approach simple asymptotic values; in doing so they approximately retain the flavor of the nonrelativistic Coulomb sum rule which approaches Z. Specific ways of achieving an optimum separation of effects relating to the electromagnetic response of a single nucleon and of a many-body system of structureless particles are discussed, including a study of sensitivities to alternative parameterizations of G En. Comparisons of results using different momentum distributions for the case of 16O are presented.

A new approach to impulsive neutron scattering

The authors give a simple derivation of the impulse approximation for scattering of neutrons with large momentum and energy transfers from a single particle in a potential well. It does not rely on any definition of 'interaction time'. It clearly shows how the apparently free-particle nature of the final state allows the deduction of the ground-state momentum distribution from the scattered intensity and gives an unambiguous criterion for the validity of the impulse approximation.

Structure of elementary excitations and temperature dependence of the momentum distribution in liquid 4He.

We study the momentum-space structure of the elementary excitations in liquid $^{4}\mathrm{He}$ by calculating the change \ensuremath{\delta}${n}_{\mathrm{k}\ensuremath{\rightarrow}}$(p\ensuremath{\rightarrow}) in the momentum distribution of atoms on creating an excitation of momentum k\ensuremath{\rightarrow}. Jastrow and Jastrow plus triplet wave functions are used for the ground state, and the excitations are created with Feynman and Feynman-Cohen excitation operators. We find that the excitations in the long-wavelength limit are harmonic vibrations with equal amount of change in kinetic and potential terms. At large k, however, they become ``single-particle''-like; and most of the energy comes from removing one particle from the ground-state momentum distribution and putting it at states with p\ensuremath{\rightarrow}\ensuremath{\sim}k\ensuremath{\rightarrow}. The \ensuremath{\delta}${n}_{\mathrm{k}\ensuremath{\rightarrow}}$(p\ensuremath{\rightarrow}) is used to calculate the momentum distribution n(T,p) of the liquid at low temperatures (1 K). The temperature dependence of the fraction of atoms in the p=0 condensate and the 1/${p}^{2}$ and 1/p singularities of n(T,p) are discussed.

Basis for quasi-two-body scaling

Recent measurements of high energy inclusive cross sections in $^{6}\mathrm{Li}$, close to the elastic limit, allow for experimental verification of the "coherent recoil" postulate of "quasi-two-body scaling." Theoretical arguments for the dominance of coherent recoil up to internal nucleon momenta $k\ensuremath{\simeq}1.2 \frac{\mathrm{GeV}}{c}$ are presented, along with a new "derivation" of the relationship between quasi-two-body scaling and an "effective" ground state momentum distribution. A possible relationship between inclusive and elastic cross sections is explored.[NUCLEAR REACTIONS Quasi-two-body scaling, "effective" momentum distributions.]

EXCHANGE CORRECTIONS AND THE CONVERGENCE OF THE PERTURBATION EXPANSION FOR THE MOMENTUM DISTRIBUTION FUNCTION FOR AN INTERACTING ELECTRON GAS AT METALLIC DENSITIES

The ground-state momentum distribution of an interacting electron gas at metallic densities is investigated by means of many-particle perturbation theory. Approximation methods and computational techniques are developed for the evaluation of the leading exchange corrections to the Random Phase Approximation. It is concluded that the straightforward perturbation expansion, which includes only single direct and exchange self-energy insertions, does not converge at the lower metallic densities and that it is necessary to include "higher-order" effects by renormalizing the electron propagator. When this is done, a large gap at the Fermi surface remains in the momentum distribution even for rs ~ 6.