Fock Terms Research Articles

Attenuation of the neutron spin–orbit splitting in a relativistic Hartree–Fock calculation in the 48Ca and 208Pb nuclei with the removal of the s1/2 protons

On the basis of a relativistic Hartree–Fock (RHF) calculation, we describe the reduction of the neutron and proton spin–orbit (SO) splittings when passing from the 48Ca and 208Pb nuclei to the nuclei characterized by the πs−21/2 configuration with respect to the starting double magic nuclei (in their ground states). As in earlier relativistic mean field calculations by several authors, we have observed a correlation between the magnitude of the SO splitting in p-doublets and the density of protons in the nuclear core, which is modified by the proton hole created with the removal of the s1/2 protons near the Fermi surface in the double magic nuclei studied. The SO interaction, in our model, due to the strong effect of the Fock terms, does not have a perturbative character. The contribution of the SO potential to the SO splitting is negative and increases in magnitude when going from the magic nuclei considered to the 46Ar and 206Hg nuclei. The specific values of the SO splittings, in our case, arise as a consequence of a fine adjustment of the central and the SO potentials (which are strongly state dependent), and of the kinetic energy also, rather than induced by just the change of the SO potential owing to the modification of the charge density inside the nucleus. From the magic nucleus to the πs−21/2 configuration nucleus, in the p-doublets, it happens that the more bound SO partner with j+ is shifted up more so than its less bound partner with j−, bringing about a minor SO splitting or even an incipient inversion of the relative position of the two SO partners. Pions slightly enhance this different behaviour of the two SO partners and, consequently, favour the quenching of the SO splitting.

Neutron star equation of state in density dependent relativistic Hartree–Fock theory

The equation of state of neutron stars is studied in the newly developed density dependent relativistic Hartree–Fock (DDRHF) theory with the effective interaction PKO1 and applied to describe the properties of neutron stars. The results are compared with the recent observational data of compact stars and those calculated with the relativistic mean field (RMF) effective interactions. The maximum mass of neutron stars calculated with PKO1 is about 2.45 M⊙, which consists with high pulsar mass from PSR B1516+02B recently reported. The influence of Fock terms on the cooling of neutron stars is discussed as well.

Neutron star properties in density-dependent relativistic Hartree-Fock theory

With the equations of state provided by the newly developed density dependent relativistic Hartree-Fock (DDRHF) theory for hadronic matter, the properties of the static and $\beta$-equilibrium neutron stars without hyperons are studied for the first time, and compared to the predictions of the relativistic mean field (RMF) models and recent observational data. The influences of Fock terms on properties of asymmetric nuclear matter at high densities are discussed in details. Because of the significant contributions from the $\sigma$- and $\omega$-exchange terms to the symmetry energy, large proton fractions in neutron stars are predicted by the DDRHF calculations, which strongly affect the cooling process of the star. The critical mass about 1.45 $M_\odot$, close to the limit 1.5 $M_\odot$ determined by the modern soft X-ray data analysis, is obtained by DDRHF with the effective interactions PKO2 and PKO3 for the occurrence of direct Urca process in neutron stars. The maximum masses of neutron stars given by the DDRHF calculations lie between 2.45 M$_\odot$ and 2.49 M$_\odot$, which are in reasonable agreement with high pulsar mass $2.08 \pm 0.19 M_\odot$ from PSR B1516+02B. It is also found that the mass-radius relations of neutron stars determined by DDRHF are consistent with the observational data from thermal radiation measurement in the isolated neutron star RX J1856, QPOs frequency limits in LMXBs 4U 0614+09 and 4U 1636-536, and redshift determined in LMXBs EXO 0748-676.

Relativistic chiral Hartree-Fock description of nuclear matter with constraints from nucleon structure and confinement

We present a relativistic chiral effective theory for symmetric and asymmetric nuclear matter taken in the Hartree-Fock scheme. The nuclear binding is ensured by a background chiral invariant scalar field associated with the radial fluctuations of the chiral quark condensate. Nuclear matter saturation is obtained once the scalar response of the nucleon-generating three-body repulsive forces is incorporated. For those parameters related to the scalar sector and quark confinement mechanism inside the nucleon we make use of an analysis of lattice results on nucleon mass evolution with quark mass. The other parameters are constrained as much as possible by standard hadron and nuclear phenomenology. Special attention is paid to the treatment of the propagation of the scalar fluctuations. The rearrangement terms associated with in-medium modified mass and coupling constants are explicitly included to satisfy the Hugenholtz-Van Hove theorem. We point out the important role of the tensor piece of the rho exchange Fock term to reproduce the asymmetry energy of nuclear matter. We also discuss the isospin dependence of the Landau nucleon effective mass.

Effect of Singwi–Tosi–Land–Sjölander local field correction on spin relaxation in n-type GaAs quantum wells at low temperature

We study the effect of the Singwi–Tosi–Land–Sjölander local field correction on spin relaxation/dephasing in n-type GaAs quantum wells at low temperature by constructing and numerically solving the kinetic spin Bloch equations. We calculate the local field factor G ( q ) in quantum wells by numerically solving three equations which link the local field factor, the structure factor, and the dielectric function, self-consistently. Such a correction reduces both the electron–electron Coulomb scattering and the Coulomb Hartree–Fock term. We compare the spin relaxation times with and without this correction under different conditions such as temperature, electron density, well width and spin polarization. We find that this correction leads to a decrease/increase of the spin relaxation time in the strong/weak scattering limit. At high spin polarization, it reduces the Hartree–Fock term and consequently tends to decrease the spin relaxation time. The modification of the spin relaxation time by the local field correction is more or less moderate either due to the coexistence of scattering other than the Coulomb scattering at low spin polarization and/or due to the competing effects from the Coulomb scattering and the Coulomb Hartree–Fock term at high polarization.

Increase of spin dephasing times in a 2D electron system with degree of initial spin polarization

We report on time-resolved Faraday/Kerr rotation measurements on a high-mobility 2D electron system. A variable initial spin polarization is created in the sample by a circularly polarized pump pulse, and the spin polarization is tracked by measuring the Faraday/Kerr rotation of a time-delayed probe pulse. By varying the pump intensity, the initial spin polarization is changed from the low-polarization limit to a polarization degree of several percent. The observed spin dephasing time increases from less than 20 ps to more than 200 ps as the initial spin polarization is increased. To exclude sample heating effects, additional measurements with constant pump intensity and variable degree of circular polarization are performed. The results confirm the theoretical prediction by Weng and Wu [Phys. Rev. B 68 (2003) 075312] that the spin dephasing strongly depends on the initial spin polarization degree. The microscopic origin for this is the Hartree–Fock term in the Coulomb interaction, which acts as an effective out-of plane magnetic field.

Chiral symmetry, confinement and nuclear matter properties

We discuss the possible influence of fundamental QCD properties such as spontaneous chiral symmetry breaking and nucleon substructure on nuclear matter properties. We propose a chiral version of the relativistic $\sigma-\omega$ model in which the attractive background scalar field is associated with the chiral invariant field governing the radial fluctuations of the quark condensate. Nuclear matter stability is ensured once the scalar response of the nucleon depending on the quark confinement mechanism is properly incorporated. The needed parameters are estimated from lattice results and a satisfactory description of bulk properties follows, the only really free parameter being the $\omega NN$ coupling constant. Pion loops can be also incorporated to obtain in a consistent way the finite density chiral susceptibilities. A good description of the asymmetry energy is obtained once the full rho meson exchange and Fock terms are included.

Density-dependent relativistic Hartree–Fock approach

A new relativistic Hartree-Fock approach with density-dependent $\sigma$, $\omega$, $\rho$ and $\pi$ meson-nucleon couplings for finite nuclei and nuclear matter is presented. Good description for finite nuclei and nuclear matter is achieved with a number of adjustable parameters comparable to that of the relativistic mean field approach. With the Fock terms, the contribution of the $\pi$-meson is included and the description for the nucleon effective mass and its isospin and energy dependence is improved.

Pseudo-spin symmetry in density-dependent relativistic Hartree–Fock theory

The pseudo-spin symmetry (PSS) is investigated in the density-dependent relativistic Hartree-Fock theory by taking {the} doubly magic nucleus $^{132}$Sn as a representative. It is found that the Fock terms bring significant contributions to the pseudo-spin orbital potentials (PSOP) and make it comparable to the pseudo-centrifugal barrier (PCB). However, these Fock terms in the PSOP are counteracted by other exchange terms due to the non-locality of the exchange potentials. The pseudo-spin orbital splitting indicates that the PSS is preserved well for the partner states $\lrb{\nu 3s_{1/2}, \nu2d_{3/2}}$ of $^{132}$Sn in the relativistic Hartree-Fock theory.

A Computational Study of Nickel Ferrite

Magnetic properties and electronic structure of nickel ferrite are studied using hybrid density functional theory. In the calculation, the exchange correlation is a mixture of Fock exchange, local spin density approximation (LSDA) and generalized gradient approximation (GGA). The weighting factor ( w) of the Fock term and a scaling factor of the 3d orbitals of Fe 3+ ( α) are introduced as fitting parameters. All nearest neighbor exchange constants for inter and intra A and B sites of the spinel structure of nickel ferrite are calculated from the energies of different magnetic structures. Local magnetic moments of the ions on the A and B sites are calculated by the Mülliken population analysis. Moreover, the density of states is calculated to show the insulating nature of nickel ferrite. The band gap and intra-site Coulomb repulsion are extracted from the calculated density of states. The charge density map is calculated to visualize the electrons in nickel ferrite.

Hartree–Fock versus quantum Monte Carlo study of persistent current in a one-dimensional ring with single scatterer

We calculate the persistent current of interacting spinless electrons in a one-dimensional ring containing a single δ barrier. We use the self-consistent Hartree–Fock method and the quantum Monte Carlo method which gives fully correlated solutions. Our Hartree–Fock method treats the non-local Fock term in a local approximation and also exactly (if the ring is not too large). Treating the Fock term exactly we attempt to support our previous Hartree–Fock result obtained in the local approximation, in particular the persistent current behaving like I ∝ L - 1 - α , where L is the ring length and α > 0 is the power depending only on the electron–electron interaction. Finally, we use the Hartree–Fock solutions as an input for our quantum Monte Carlo calculation. The Monte Carlo results exhibit only small quantitative differences from the Hartree–Fock results.

Calculation of exchange constants in manganese ferrite (MnFe 2O 4)

The exchange constants and electronic structure of manganese ferrite (MnFe 2O 4) were calculated using Becke's density functional. The total exchange energy consists of Hartree–Fock (HF) and Becke's density functional terms. We introduced one parameter w as the weight of HF's contribution. We also introduced a parameter α to scale the radial part of the 3d wave functions of Fe 3+ ions. By varying w and α the calculated exchange constants were quantitatively fitted to the experimental values of a spinel ferrite for the first time. Direct (d–d) and indirect (d–p–d) hopping are controlled by the parameters w and α.

The relation between the Gross–Pitaevskii and Bogoliubov descriptions of a dilute Bosegas

I formulate a ‘pseudo-paradox’ in the theory of a dilute Bose gas with repulsive interactions:the standard expression for the ground state energy within the Gross–Pitaevskii (GP)approximation is lower than that in the Bogoliubov approximation, and hence, bythe standard variational argument, the former should prima facie be a betterapproximation than the latter to the true ground state—a conclusion which is of courseopposite to the established wisdom concerning this problem. It is shown that thepseudo-paradox is (unsurprisingly) resolved by a correct transcription of the two-bodyscattering theory to the many-body case; however, contrary to what appearsto be a widespread belief, the resolution has nothing to do with any spuriousultraviolet divergences which result from the replacement of the true interatomicpotential by a delta-function pseudopotential. Rather, it relates to an infrareddivergence which has the consequence that (a) the most obvious form of the GP‘approximation’ actually does not correspond to any well-defined ansatz for the many-bodywavefunction, and (b) that the ‘best shot’ at such a wavefunction always producesan energy which exceeds, or at best equals, that calculated in the Bogoliubovapproximation. In fact, the necessity of the latter may be seen as a consequence of theneed to reduce the Fock term in the energy, which is absent in the two-particleproblem but dominant in the many-body case; it does this by increasing the densitycorrelations, at distances less than or approximately equal to the correlation length , above the value extrapolated from the two-body case. As a by-product I devise analternative formulation of the Bogoliubov approximation which does not require theexplicit replacement of the true interatomic potential by a delta-function pseudopotential.

Mesonic fluctuations in a nonlocal Nambu–Jona-Lasinio model

The effects of meson fluctuations are studied in a nonlocal generalization of the Nambu–Jona-Lasinio model, by including terms of next-to-leading order (NLO) in 1/ N c . In the model with only scalar and pseudoscalar interactions NLO contributions to the quark condensate are found to be very small. This is a result of cancellation between virtual mesons and Fock terms, which occurs for the parameter sets of most interest. In the quark self-energy, similar cancellations arise in the tadpole diagrams, although not in other NLO pieces which contribute at the ∼25% level. The effects on pion properties are also found to be small. NLO contributions from real ππ intermediate states increase the sigma-meson mass by ∼30%. In an extended model with vector and axial interactions, there are indications that NLO effects could be larger.

Asymmetric nuclear matter in a Hartree-Fock approach to nonlinear quantum hadrodynamics

The Equation of State (EOS) for asymmetric nuclear matter is discussed starting from a phenomenological hadronic field theory of Serot-Walecka type including exchange terms. In a model with self interactions of the scalar sigma-meson we show that the Fock terms naturally lead to isospin effects in the nuclear EOS. These effects are quite large and dominate over the contribution due to isovector mesons. We obtain a potential symmetry term of "stiff" type, i.e. increasing with baryon density and an interesting behaviour of neutron/proton effective masses of relevance for transport properties of asymmetric dense matter.

Fock exchange terms in nonlinear quantum hadrodynamics

We propose a method to introduce Fock term contributions in relativistic models of fermions coupled to mesons, including self-interactions for the mesonic fields. We show that effects on equilibrium properties and on the dynamical response of the fermionic system can be consistently accounted for. Some implications on equilibrium properties of asymmetric nuclear matter are discussed.

Dirac-Hartree-Bogoliubov approximation for finite nuclei

We develop a complete Dirac-Hartree-Fock-Bogoliubov approximation to the ground state wave function and energy of finite nuclei. We apply it to spin-zero proton-proton and neutron-neutron pairing within the Dirac-Hartree-Bogoliubov approximation (we neglect the Fock term), using a zero-range approximation to the relativistic pairing tensor. We study the effects of the pairing on the properties of the even-even nuclei of the isotopic chains of Ca, Ni and Sn (spherical) and Kr and Sr (deformed), as well as the $N$=28 isotonic chain, and compare our results with experimental data and with other recent calculations.

Non-Perturbative Renormalization Group Equations in Quantum Hadrodynamics

Quantum hadrodynamics are studied with non-perturbative renormalization group equations. It is shown that the contributions of one-loop diagrams are important for the effective potential, since effective couplings of lower-order interactions in the effective potential are only slightly affected by quantum effects at low energy. The relation between the local-potential approximation (LPA) and the traditional Hartree approximation is discussed. The LPA includes the contribution of the Fock term to the nucleon self-energy, the contribution in the random phase approximation to the meson self-energy, and vertex corrections in part, as well as the Hartree contribution to the effective potential.

Fock terms in the quark-meson coupling model

The mean field description of nuclear matter in the quark-meson coupling model is improved by the inclusion of exchange contributions (Fock terms). The inclusion of Fock terms allows us to explore the momentum dependence of meson-nucleon vertices and the role of pionic degrees of freedom in matter. It is found that the Fock terms maintain the previous predictions of the model for the in-medium properties of the nucleon and for the nuclear incompressibility. The Fock terms significantly increase the absolute values of the single-particle, four-component scalar and vector potentials, a feature that is relevant for the spin-orbit splitting in finite nuclei.

Contribution of Fock Term to Properties of Exotic Nuclei

Density dependent relativistic Hartree-Fock theory has been extended to describe properties of exotic nuclei. The contribution of Fock exchange terms is emphasized in this letter. It turns out that the influence of Fock terms on properties of neutron rich nuclei is very different from those for nuclei near stability line.