Momentum Distribution Function Research Articles

Current and quasi-probability phase-space distributions

Quantum mechanical current has many features in common with classical current, but sometimes its behaviour is unusual and leads to many interesting peculiarities as has been noted by many authors. We show that many peculiarities often occur for wave functions that exhibit multiple trajectories or regions in phase space. Such wave functions have been called 'multi-part' wave functions and the trajectories are the quantum currents of each part. We contrast the behaviour of quantum current with classic current, and we pinpoint some of the curiosities. We argue and show that, in general, multi-part wave functions produce highly oscillating currents, which in itself is not peculiar, but what is peculiar is that the current can range outside the range of the momentum wave function in a variety of different situations. However, we also show that for some multi-part wave functions, the quantum mechanical current is not peculiar and is totally consistent with what one would expect for classical currents. We give the conditions for which this occurs, which are very restrictive. We also explore the interpretation of quantum mechanical current as a first conditional moment of a quantum quasi distribution, and contrast it with the conditional moment that is obtained from a classical phase-space distribution. We explore, for both the classical and quantum case, the relationship of the range of the current with the range of the momentum distribution function. We further show that for windowed type distributions something interesting happens, in that it is possible to choose a window which yields a classical type result, namely, such that the total current for a multi-part wave function is a weighted average of the currents corresponding to each part. We give the conditions for when this is the case.

The Luttinger liquid concept for interacting electrons in one dimension

The theoretical description of interacting fermions in one spatial dimension issimplified by the fact that the low-energy spectrum of noninteracting fermions isidentical to the one for a harmonic chain. This allows us to describe interactingfermions as a system of coupled oscillators. The exact solution of theTomonaga–Luttinger model is discussed and results for the momentumdistribution and spectral functions are presented. The general Luttingerliquid phenomenology is briefly discussed and various attempts at theexperimental verification of the theoretical predictions are examined.Extended defects in semiconductors might also be candidate systems.

Radio Halo Formation through Magnetoturbulent Particle Acceleration in Clusters of Galaxies

We develop a magnetoturbulent model for the cosmic-ray (CR) electrons seen in the radio halo clusters of galaxies. Steady-state momentum distribution functions of the CR electrons are calculated for given spectra of the turbulent Alfven waves. The radio spectrum produced by the obtained CR electron distribution is compared to the observed radio spectrum of the Coma radio halo. We find that the observed radio spectrum of the Coma Cluster is well reproduced when the spectral index of the turbulent Alfven waves is ~2.8. The obtained energy spectrum of the turbulent Alfven waves is steeper than that expected from the turbulence theory, suggesting back-reaction of the particle acceleration. The fitting procedure constraints the amplitude of the turbulent Alfven waves as well as the spectral index. Then we estimate the dissipation of the turbulent Alfven waves, which is found to be less than the cooling rate by X-ray radiation. We suggest that the turbulence that is sufficient for particle acceleration is developed in the clusters containing the radio halo. It is most likely that cluster mergers create the turbulence and seed relativistic electrons.

Low-energy states with different symmetries in thet−Jmodel with two holes on a 32-site lattice

We study the low energy states of the t-J model with two holes on a 32-site lattice with periodic boundary conditions. In contrary to common belief, we find that the state with d_{x^2-y^2} symmetry is not always the ground state in the realistic parameter range 0.2\le J/t\le 0.4. There exist low-lying finite-momentum p-states whose energies are lower than the d_{x^2-y^2} state when J/t is small enough. We compare various properties of these low energy states at J/t=0.3 where they are almost degenerate, and find that those properties associated with the holes (such as the hole-hole correlation and the electron momentum distribution function) are very different between the d_{x^2-y^2} and p states, while their spin properties are very similar. Finally, we demonstrate that by adding ``realistic'' terms to the t-J model Hamiltonian, we can easily destroy the d_{x^2-y^2} ground state. This casts doubt on the robustness of the d_{x^2-y^2} state as the ground state in a microscopic model for the high temperature superconductors.

Application of the density matrix renormalization group in momentum space

We investigate the application of the density matrix renormalization group (DMRG) to the Hubbard model in momentum space. We treat the one-dimensional models with dispersion relations corresponding to nearest-neighbor hopping and $1/r$ hopping and the two-dimensional model with isotropic nearest-neighbor hopping. By comparing with the exact solutions for both one-dimensional models and with exact diagonalization in two dimensions, we first investigate the convergence of the ground-state energy. We find variational convergence of the energy with the number of states kept for all models and parameter sets. In contrast to the real-space algorithm, the accuracy becomes rapidly worse with increasing interaction and is not significantly better at half filling. We compare the results for different dispersion relations at fixed interaction strength over bandwidth and find that extending the range of the hopping in one dimension has little effect, but that changing the dimensionality from one to two leads to lower accuracy at weak-to-moderate interaction strength. In the one-dimensional models at half filling, we also investigate the behavior of the single-particle gap, the dispersion of spinon excitations, and the momentum distribution function. For the single-particle gap, we find that proper extrapolation in the number of states kept is important. For the spinon dispersion, we find that good agreement with the exact forms can be achieved at weak coupling if the large momentum-dependent finite-size effects are taken into account for nearest-neighbor hopping. For the momentum distribution, we compare with various weak-coupling and strong-coupling approximations and discuss the importance of finite-size effects as well as the accuracy of the DMRG.

Applications of light-cone wavefunctions to QCD phenomena

Abstract The matrix elements of the local operators such as electromagnetic current, energy momentum tensor, angular momentum, and skewed parton distribution functions have exact representations in terms of light-cone Fock state wavefunctions of bound states such as hadrons. We present formulae which express the form factors of the local operators in terms of the light-cone wavefunctions.

Hartree-Fock-Bogoliubov theory of a charged Bose gas at finite temperature

We critically examine the Hartree-Fock-Bogoliubov (HFB) solution of the equations of motion for condensate fluctuations in a weakly coupled plasma of charged bosons at finite temperature. Analytic expressions are derived for the first two infrared-divergent terms in both the momentum distribution of the noncondensate and the anomalous Bose correlation function at low momenta. Incorporation into the theory of the appropriate form of the Hugenholtz-Pines relation for the chemical potential is needed to cancel an unphysical divergence. Exact cancellation of infrared-divergent terms is demonstrated in the HFB shift of the single-particle excitation energy away from the Bogoliubov value at long wavelengths, with the residual terms raising it towards the plasma frequency at low temperature. Numerical illustrations are presented for a number of properties of the boson plasma as functions of temperature and density in the weak-coupling regime: these are the chemical potential, the condensate fraction, the normal and anomalous momentum distribution functions and the corresponding one-body density matrices, and the dispersion relation of single-particle excitations.

Kinetic approach toη′production from aCP-odd phase

The production of $(\ensuremath{\eta},{\ensuremath{\eta}}^{\ensuremath{'}})$ mesons during the decay of a CP-odd phase is studied within an evolution operator approach. We derive a quantum kinetic equation starting from the Witten--Di Vecchia--Veneziano Lagrangian for pseudoscalar mesons containing a ${\mathrm{U}}_{A}(1)$ symmetry breaking term. The nonlinear vacuum mean field for the flavor singlet pseudoscalar meson is treated as a classical, self-interacting background field with fluctuations assumed to be small. The numerical solution provides the time evolution of the momentum distribution function of the produced ${\ensuremath{\eta}}^{\ensuremath{'}}$ mesons after a quench at the deconfinement phase transition. We show that the time evolution of the momentum distribution of the produced mesons depends strongly on the shape of the effective potential at the end of the quench, exhibiting either parametric or tachyonic resonances. Quantum statistical effects are essential and lead to a pronounced Bose enhancement of the low-momentum states.

Charge radii of light and heavy mesons

We calculate the electromagnetic (EM) form factors of the pseudoscalar mesons in the light-front framework. Specifically, these form factors are directly extracted from the relevant matrix elements, instead of choosing the Breit frame. The results show that the charge radius of the meson is related to both the first and second longitudinal momentum square derivatives of the momentum distribution function. The static properties of the EM form factors and the heavy quark symmetry of the charge radii are checked analytically in the heavy quark limit. In addition, we use the Gaussian-type wavefunction to obtain the numerical results.

Electronic momentum distribution in the one-dimensional extended Hubbard model: determinantal Monte Carlo study

The effect of electron–electron (e–e) interaction on trans-polyacetylene ( t-PA) properties is investigated within the framework of an extended Hubbard model in one dimension. For numerical calculation, we use the determinantal version of quantum Monte Carlo approach, which provides a breakthrough to simulate statistical fluctuations in the systems with many degrees of freedom, in order to obtain mean values for observables of physical interest. This allows one to analyze the discrete system of fermions without encountering the numerical instabilities that generally occur from the original problem involving anticommuting fermion operators. We calculate the electronic momentum distribution function n( k) for on-site interaction U and nearest-neighbor interaction E, where U<4 t, U≃4 t and U>4 t, with E∈[ U/2, U], ( t is the hopping matrix elements).

Asymptotics of QCD factorization in exclusive hadronic decays of B mesons

Using the renormalon calculus, we study the asymptotic behavior of the perturbative expansion of the hard-scattering kernels entering the QCD factorization formula for the nonleptonic weak decays B ̄ 0 →D (∗)+M − , where M is a light meson. In the “large- β 0 limit”, the kernels are infrared finite and free of endpoint singularities to all orders of perturbation theory. The leading infrared renormalon singularity corresponding to a power correction of order Λ QCD/ m b vanishes if the light meson has a symmetric light-cone distribution amplitude. We calculate the Borel transforms and the corresponding momentum distribution functions of the hard-scattering kernels, and resum the series of O( β 0 n−1 α s n ) corrections to explore the numerical significance of higher-order perturbative and power corrections. We also derive explicit expressions for the O(β 0 α s 2) contributions to the kernels, and for the renormalon singularities corresponding to power corrections of order ( Λ QCD/ m b ) 2. Finally, we study the limit m c →0 relevant to charmless hadronic decays such as B→ ππ.

Fermi surface topology of Bi2201 studied by temperature-dependent angle-resolved photoemission spectroscopy

We report temperature-dependent angle-resolved photoemission spectroscopy (ARPES) on superstructure-free Bi 1.80Pb 0.38Sr 2.01CuO 6− δ (overdoped, T c<4 K) to study the topology of Fermi surfaces (FS). Temperature dependence of the momentum distribution function n( k) obtained from ARPES has revealed a dispersive band which crosses E F along the ( π,0)–( π, π) cut, indicating the hole-like FS is centered at ( π, π).

Coulomb Luttinger liquid

The effect of long-ranged Coulomb interaction on the low energy properties (momentum distribution function, density of states, electron spectral function, and $4k_F$ correlation function) of one-dimensional electron systems is determined theoretically. Over a wide, physically relevant energy range the behavior is found to be well described by the conventional, short-ranged interacting, Luttinger liquid results, \textit{with a scale-dependent effective exponent}. An accurate empirical formula for the effective exponent is presented.

Momentum dependence of the mean field and equation of state of nuclear matter

The momentum and density dependence of the mean field and other properties of nuclear matter are studied with finite-range effective interactions having different functional forms by evaluating the single-particle momentum distribution function self-consistently. In these calculations relativistic effects and supraluminous behaviour of nuclear matter are also taken into account. The equation of states obtained from these effective interactions are used to study liquid–gas phase transition in nuclear matter.

Dynamic evolution of relativistic electrons in the radiation belts

Geomagnetic storms are responsible for the large increase of trapped electron fluxes in the magnetosphere resulting in the formation of radiation belts. Such belts can also be artificially produced by injection of electron beams. The dynamic evolution of the electron fluxes is very important for satellite protection purpose. Up to now most of the existing studies have been restricted to the stationary description of natural radiation belts. Recently, great effort has been made for their dynamic description [Bourdane et al., 1996]. We have developed a numerical code, which is able to follow the time behavior of the electron population under influence of collisions with tenuous atmosphere and resonant scattering by plasma waves. The code solves the phase‐averaged Fokker‐Planck equation of the electron momentum distribution function on magnetic shells within the Earth's inner magnetosphere. The physical arguments incorporated in the code include phase‐averaged, approximate collision operators and quasi‐linear diffusion due to resonant interaction with whistler plasma waves. The magnetic field is supposed to be tilted and decentered. Numerical experiments to simulate the decay and particles' loss in an artificial Van Allen belt are presented, and comparisons with available published data are discussed to validate the code [Abel and Thorne, 1998].

Electromagnetic properties of heavy mesons in heavy quark limit

The electromagnetic (EM) form fators of the pseudoscalar and vector mesons are calculated in the light-front framework. We extract these form factors from the relevant matrix elements directly, instead of choosing the Breit frame. The results show that the charge radius of the meson are related to both the first and second longitudinal momentum square derivative of the momentum distribution function. In addition, the static properties of the EM form factors and the heavy quark symmetry of the charge radii are checked analytically when we take the heavy quark limit.

Exact solutions for the periodic Anderson model in two dimensions: A non-Fermi-liquid state in the normal phase

Presenting exact solutions for the two-dimensional periodic Anderson model with finite and nonzero on-site interactions $U&gt;0,$ we describe a rigorous non-Fermi-liquid phase in normal phase and two dimensions. This new state emerges in multiband interacting Fermi systems above half filling, being generated by a flat-band effect. The momentum distribution function ${n}_{\stackrel{\ensuremath{\rightarrow}}{k}}$ together with its derivatives of any order is continuous. The state possesses a well-defined Fermi energy ${(e}_{F}),$ but the Fermi momentum concept is not definable, so the Fermi surface in $\stackrel{\ensuremath{\rightarrow}}{k}$ space is missing. The state emerges in the vicinity of a Mott insulating phase when lattice distortions are present and is highly degenerated and paramagnetic. A gap is present at high U in the density of low-lying excitations. During low-lying excitations, quasiparticles are not created above the Fermi level, only the number of particles at ${e}_{F}$ increases.

New non-Fermi-liquid-type behaviour by a two-band system in normal phase

We are reporting a new non-Fermi-liquid-type normal phase that has a well defined Fermi energy, but without showing any non-regularity in the momentum distribution function nk in the whole momentum space, the sharp Fermi momentum concept being undefinable. The system contains a natural built-in gap that is visible in the physical properties of the system at nonzero temperatures. The presence of a flat band in multi-band interacting Fermi systems with more than half filling is the key feature leading to such a ground state, which is not restricted to one spatial dimension and emerges in the proximity of an insulating phase.

Luttinger-model approach to interacting one-dimensional fermions in a harmonic trap

A model of interacting one--dimensional fermions confined to a harmonic trap is proposed. The model is treated analytically to all orders of the coupling constant by a method analogous to that used for the Luttinger model. As a first application, the particle density is evaluated and the behavior of Friedel oscillations under the influence of interactions is studied. It is found that attractive interactions tend to suppress the Friedel oscillations while strong repulsive interactions enhance the Friedel oscillations significantly. The momentum distribution function and the relation of the model interaction to realistic pair interactions are also discussed.

Molecular dynamics study of the collision-induced rotational alignment of N2+ drifting in helium

The full velocity-angular momentum distribution function for gas-phase N2+ drifting in helium is calculated using a molecular dynamics method, and utilized to examine collision-induced rotational alignment in detail. These results are also compared with experimental measurements, most especially those of Anthony et al. [J. Chem. Phys. 112, 10269 (2000)] and those appearing in the preceding article [Anthony et al., J. Chem. Phys. 114, 6654 (2001)]. Both the calculations and experiments show a number of interesting features including, drift velocities which depend upon rotational state, and quadrupolar alignment parameters which change from negative at high velocities to positive at low velocities.