Fock Terms Research Articles

Pseudo-Spin Symmetry and the Hints for Unstable and Superheavy Nuclei

The pseudo-spin symmetry (PSS) provides an important angle to understand nuclear microscopic structure and the novel phenomena found in unstable nuclei. The relativistic Hartree–Fock (RHF) theory, that takes the important degrees of freedom associated with the π-meson and ρ-tensor (ρ-T) couplings into account, provides an appropriate description of the PSS restoration in realistic nuclei, particularly for the pseudo-spin (PS) doublets with high angular momenta (l˜). The investigations of the PSS within the RHF theory are recalled in this paper by focusing on the effects of the Fock terms. Aiming at common artificial shell closures appearing in previous relativistic mean-field calculations, the mechanism responsible for the PSS restoration of high-l˜ orbits is stressed, revealing the manifestation of nuclear in-medium effects on the PSS, and thus, providing qualitative guidance on modeling the in-medium balance between nuclear attractions and repulsions. Moreover, the essential role played by the ρ-T coupling, that contributes mainly via the Fock terms, is introduced as combined with the relations between the PSS and various nuclear phenomena, including the shell structure and the evolution, novel halo and bubble-like phenomena, and the superheavy magicity. As the consequences of the nuclear force in complicated nuclear many-body systems, the PSS itself and the mechanism therein can not only deepen our understanding of nuclear microscopic structure and relevant phenomena, but also provide special insight into the nature of the nuclear force, which can further enrich our knowledge of nuclear physics.

Quantum Magnetism in Wannier-Obstructed Mott Insulators

We develop a strong coupling approach towards quantum magnetism in Mott insulators for Wannier-obstructed bands. Despite the lack of Wannier orbitals, electrons can still singly occupy a set of exponentially localized but nonorthogonal orbitals to minimize the repulsive interaction energy. We develop a systematic method to establish an effective spin model from the electron Hamiltonian using a diagrammatic approach. The nonorthogonality of the Mott basis gives rise to multiple new channels of spin-exchange (or permutation) interactions beyond Hartree–Fock and superexchange terms. We apply this approach to a Kagome lattice model of interacting electrons in Wannier-obstructed bands (including both Chern bands and fragile topological bands). Due to the orbital nonorthogonality, as parameterized by the nearest-neighbor orbital overlap g, this model exhibits stable ferromagnetism up to a finite bandwidth W∼Ug, where U is the interaction strength. This provides an explanation for the experimentally observed robust ferromagnetism in Wannier-obstructed bands. The effective spin model constructed through our approach also opens up the possibility for frustrated quantum magnetism around the ferromagnet-antiferromagnet crossover in Wannier-obstructed bands.

Relativistic Hartree-Fock model of nuclear single-particle resonances based on real stabilization method

With the development of radioactive ion beam devices along with associated nuclear experimental detection technologies, the research areas in atomic nuclei have been further expanded, illustrating many new aspects of nuclear excitation as well as the physics of exotic nuclei far from the <i>β</i>-stability line. For weakly bound nuclei, the Fermi surface may lie near the continuum, which facilitates the easy scattering of valence nucleons into the continuum to occupy the resonance state. These continuum effects are of crucial importance in explaining the unusual structure of unstable nuclei. In this work, with the real stabilization method in coordinate space, nuclear structure model for single-particle resonances is developed within the framework of the relativistic Hartree-Fock (RHF) theory. In order to extract potential single-particle resonance structures, we study the evolution of single-particle states with box size in the continuum. To avoid the instability of nuclear binding energy, the pairing correlations are not taken into account in the calculation. As an important motivation, the roles of Fock terms in determining the energy, widths and spin-orbit splitting are discussed for low-lying neutron resonance states of <inline-formula><tex-math id="M5">\begin{document}$^{120}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M5.png"/></alternatives></inline-formula>Sn. By comparing with the relativistic mean field (RMF) model, it is found that the inclusion of exchange terms in the RHF model changes the in-medium balance of nuclear interactions and the equilibrium of nuclear dynamics, which in turn affects the description of the single-particle effective potential. For several neutron resonance states in <inline-formula><tex-math id="M6">\begin{document}$^{120}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M6.png"/></alternatives></inline-formula>Sn with finite resonant width, RHF model predicts lower resonant energy and smaller widths than RMF. For the single-particle states around the continuum threshold, the featured signals of resonance can depend sensitively on the effective interactions. In addition, for the spin-partner states <inline-formula><tex-math id="M7">\begin{document}$\nu {\mathrm{i}}_{13/2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M7.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$\nu {\mathrm{i}}_{11/2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20231632_M8.png"/></alternatives></inline-formula> in resonance states, the effect of Fock terms on their spin-orbit splitting is analyzed. In comparison with the bound states, the wave functions of resonant spin-partner states can differ remarkably from each other, changing the effective potential and single-particle energies correspondingly. Thus, additional components in the single-particle effective potential may also contribute to the spin-orbit splitting of resonance states, aside from the spin-orbit interaction. In order to elucidate the mechanism of Fock term in single-particle resonance physics, in the subsequent study more numerical techniques that have been recently developed will be incorporated into the RHF methodology.

Quenched Λ spin-orbit splitting by a relativistic Fock diagram in single- Λ hypernuclei

We extend the relativistic Hartree-Fock (RHF) theory to study the structure of single-$\Lambda$ hypernuclei. The density dependence is taken in both meson-nucleon and meson-hyperon coupling strengths, and the induced $\Lambda$-nucleon ($\Lambda N$) effective interactions are determined by fitting $\Lambda$ separation energies to the experimental data for several single-$\Lambda$ hypernuclei. The equilibrium of nuclear dynamics described by the RHF model in normal atomic nuclei, namely, the balance between nuclear attractive and repulsive interactions, is then found to be drastically changed in single-$\Lambda$ hypernuclei, revealing a different role of Fock terms via $\Lambda$ hyperon from the nucleon exchange. Since only one hyperon exists in a single-$\Lambda$ hypernucleus, the overwhelmed $\Lambda N$ and $\Lambda\Lambda$ attractions via the Hartree than the $\Lambda\Lambda$ repulsion from the Fock terms require an alternation of meson-hyperon coupling strengths in RHF to rebalance the effective nuclear force with the strangeness degree of freedom, leading to an improved description of $\Lambda$ Dirac mass and correspondingly a systematically reduced $\sigma$-$\Lambda$ coupling strength $g_{\sigma\Lambda}$ in current models as compared to those relativistic mean-field (RMF) approaches without Fock terms. As a result, the effective $\Lambda$ spin-orbit coupling potential in the ground state of hypernuclei is suppressed, and these RHF models predict correspondingly a quenching effect in $\Lambda$ spin-orbit splitting in comparison with the RMF cases. Furthermore, the $\Lambda$ spin-orbit splitting could decrease efficiently by evolving the hyperon-relevant couplings $g_{\sigma\Lambda}$ and $g_{\omega\Lambda}$ simultaneously, where to reconcile with the empirical value the RHF models address a larger parameter space of meson-hyperon couplings.

Designing a novel organometallic chalcone with an enormous second-harmonic generation response

Density Functional Theory (DFT) and the Hyper-Rayleigh scattering formalism are applied to investigate the frequency-dependent nonlinear optical (NLO) behavior of a novel organometallic chromophore composed of two chalcones bonded by a silver atom. In this investigation, the appropriate LanL2DZ and 6-311++G(d,p) basis sets are applied to describe the metal and the remaining elements, respectively. DFT-based methods which include an accurate description of the exchange Hartree–Fock term are essential to the description of the electrical α, β, and γ properties, which are the optical coefficients discussed. The results indicate that low polarity solvents like benzene improves β and γ values. For instance, from benzene to the water environment, the M06-2X DFT method indicates a decrease from 81.94×103 to 7.12×103 a.u. Moreover, the results in benzene are quite superior to that obtained for urea (0.04×103 a.u.) or 4-nitroaniline (0.73×103 a.u.), which are more standard for NLO materials. The prediction for the second hyperpolarizability (35.37×106 a.u.) is ca. 4.5 times greater than that reported for a single chalcone. The material is suitable for NLO applications.

Photoinduced Dirac-cone flattening in BaNiS2

Using a real-time implementation of the self-consistent $GW$ method, we theoretically investigate the photo-induced changes in the electronic structure of the quasi two-dimensional semi-metal BaNiS$_2$. This material features four Dirac cones in the unit cell and our simulation of the time- and momentum-resolved nonequilibrium spectral function reveals a flattening of the Dirac bands after a photo-doping pulse with a 1.5 eV laser. The simulation results are consistent with the recently reported experimental data on photo-doped BaNiS$_2$ and ZrSiSe, another Dirac semi-metal. A detailed analysis of the numerical data allows us to attribute the nonequilibrium modifications of the Dirac bands to (i) an increased effective temperature after the photo-excitation, which affects the screening properties of the system, and (ii) to nontrivial band shifts in the photo-doped state, which are mainly induced by the Fock term.

Thermodynamic and electromagnetic properties of the eta-pairing superconductivity in the Penson–Kolb model

In the paper, we study the thermodynamic and electromagnetic properties of the Penson–Kolb (PK) model, i.e., the tight-binding model for fermionic particles with the pair-hopping interaction J. We focus on the case of repulsive J (i.e., J<0), which can stabilize the eta-pairing superconductivity with Cooper-pair center-of-mass momentum q→=Q→, Q→=(π/a,π/a,…). Numerical calculations are performed for several d-dimensional hypercubic lattices: d=2 (the square lattice, SQ), d=3 (the simple cubic lattice) and d=∞ hypercubic lattice (for arbitrary particle concentration 0<n<2 and temperature T). The ground state J versus n phase diagrams and the crossover to the Bose–Einstein condensation regime are analyzed and the evolution of the superfluid characteristics are examined within the (broken symmetry) Hartree–Fock approximation (HFA). The critical fields, the coherence length, the London penetration depth, and the Ginzburg ratio are determined at T=0 and T>0 as a function of n and pairing strength. The analysis of the effects of the Fock term on the ground state phase boundaries and on selected PK model characteristics is performed as well as the influence of the phase fluctuations on the eta-pairing superconductivity is investigated. Within the Kosterlitz–Thouless scenario, the critical temperatures TKT are estimated for d=2 SQ lattice and compared with the critical temperature Tc obtained from HFA. We also determine the temperature Tm at which minimal gap between two quasiparticle bands vanishes in the eta-phase. Our results for repulsive J are contrasted with those found earlier for the PK model with attractive J (i.e., with J>0).

Tensor-force effects on shell-structure evolution in N=82 isotones and Z=50 isotopes in the relativistic Hartree-Fock theory

The evolutions of the energy difference between the neutron states $1{i}_{13/2}$ and $1{h}_{9/2}$ in the $N=82$ isotones and that between the proton states $1{h}_{11/2}$ and $1{g}_{7/2}$ in the $Z=50$ isotopes are investigated within the framework of the relativistic Hartree-Fock theory, using the density-dependent effective interactions PKA1, $\mathrm{P}\mathrm{K}\mathrm{O}i$ ($i=1$, 2, 3), and a new interaction developed in this study. By identifying the contributions of the tensor force, which is naturally induced via the Fock terms, we find that the tensor force plays crucial roles in the evolution of the shell structure. The strength of the tensor force is also explored. It is found that moderately increasing the coupling strength of pion-nucleon coupling, i.e., ${f}_{\ensuremath{\pi}}$, will significantly improve the description of the shell-structure evolution. In particular, reducing the density dependence of ${f}_{\ensuremath{\pi}}$ is shown to be preferable, in comparison to enlarging ${f}_{\ensuremath{\pi}}$ with a factor. This is consistent with the idea of ``tensor renormalization persistency'' and provides valuable guidance for the development of the nuclear energy density functional in the relativistic framework.

Effect of Fock terms on nuclear symmetry energy based on Lorentz-covariant decomposition of nucleon self-energies

Using the Lorentz-covariant decomposition of nucleon self-energies with relativistic mean-field approximation, we study the effect of Fock terms on the density dependence of nuclear symmetry energy, E sym. It is found that the Fock contribution suppresses the potential part of E sym at higher densities, and the constraint from the heavy-ion collision data is in favor of the present result including exchange terms with the cutoff parameters given by the CD-Bonn potential. In addition, not only the isovector-vector (ρ) meson but also the isoscalar (σ, ω) and π mesons give influence on the potential part of E sym through the exchange diagram.

Self-consistent random-phase approximation based on the relativistic Hartree-Fock theory: Role of ρ -tensor coupling

The framework of the random-phase approximation (RPA) based on the relativistic Hartree-Fock (RHF) theory is extended to achieve a self-consistent calculation with the $\ensuremath{\rho}$-meson tensor coupling. The model self-consistency is verified by the check of the isobaric analog state, and it is found that the $\ensuremath{\rho}$-tensor and $\ensuremath{\rho}$-vector-tensor couplings play significant roles in maintaining the self-consistency. Using the RHF Lagrangian PKA1, the properties of the Gamow-Teller resonances (GTR) are investigated, in which the roles played by the particle-hole residual interaction of various meson-nucleon couplings are clarified in details. Furthermore, the effects of the tensor force, which is introduced naturally via the Fock terms, are analyzed by comparing the calculations with full Lagrangians and the ones artificially dropping the tensor force components. It is found that for the RHF Lagrangians $\mathrm{PKO}i$ $(i=1,2,3)$ and PKA1, the tensor forces play the role mainly via the RHF mean field rather than via the RPA residual interaction in determining the GTR. Moreover, the tensor-force effects are not as strong as those indicated by the Skyrme Hartree-Fock calculations.

Decomposition of nuclear symmetry energy based on Lorentz-covariant nucleon self-energies in relativistic Hartree-Fock approximation

Using the Lorentz-covariant decomposition of nucleon self-energies based on the Hugenholtz–Van Hove theorem, we study the effect of Fock terms on the nuclear symmetry energy and its slope parameter within relativistic Hartree-Fock approximation. It is found that the exchange contribution suppresses the nuclear symmetry energy and prevents the slope parameter from increasing monotonically at high densities. Furthermore, not only the isovector-vector (ρ) meson but also the isoscalar mesons (σ,ω) and pion give significant influence on the potential term of nuclear symmetry energy through the exchange diagrams.

Pseudospin symmetry restoration and the in-medium balance between nuclear attractive and repulsive interactions

The mechanism that restores the pseudo-spin symmetry (PSS) are investigated under the relativistic Hartree-Fock (RHF) approach, by focusing on the in-medium balance between nuclear attractive and repulsive interactions. It is illustrated that the modelings of both the equilibrium of nuclear dynamics and the in-medium effects can be essentially changed by the $\rho$-tensor coupling that play the role almost fully via the Fock terms, from which the model discrepancy on the PSS restoration is verified. Specifically, the largely different density-dependent behaviors of the isoscalar coupling strengths $g_\sigma$ and $g_\omega$, deduced from the parametrization of the RHF Lagrangian PKA1, play an essential role in restoring the PSS of the high-$l'$ pseudo-spin doublets around the Fermi levels. Qualitatively, a guidance is provided for the modelings of both the equilibrium of nuclear dynamics and the in-medium effects via the PSS restoration.

Restoration of pseudo-spin symmetry in N = 32 and N = 34 isotonesdescribed by relativistic Hartree-Fock theory * * Supported by National Natural Science Foundation of China (11675065, 11711540016)

The restoration of pseudo-spin symmetry (PSS) along the and N = isotonic chains and the physics behind are studied by applying the relativistic Hartree-Fock theory with the effective Lagrangian PKA1. Taking the proton pseudo-spin partners as candidates, the systematic restoration of PSS along both isotonic chains is found from sulphur (S) to nickel (Ni), while an obvious PSS violation from silicon (Si) to sulphur is discovered near the drip lines. The effects of the tensor force components are investigated, introduced naturally by the Fock terms, which can only partially interpret the systematics from calcium to nickel, whereas they fail for the overall trends. Further analysis following the Schrödinger-like equation of the lower component of Dirac spinor shows that contributions from the Hartree terms dominate the overall systematics of the PSS restoration. Such effects can be self-consistently interpreted by the evolution of the proton central density profiles along both isotonic chains. In particular, the PSS violation is found to tightly relate to the dramatic changes from the bubble-like density profiles in silicon to the central-bumped ones in sulphur.

Hypernuclear stars from relativistic Hartree-Fock density functional theory

The hypernuclear matter is studied within the relativistic Hartree-Fock theory employing several parametrizations of the hypernuclear density functional with density-dependent couplings. The equations of state and compositions of hypernuclear matter are determined for each parametrization and compact stars are constructed by solving their structure equations in spherical symmetry. We quantify the softening effect of Fock terms on the equation of state, as well as discuss the impact of tensor interactions, which are absent in the Hartree theories. Starting from models of density functionals which are fixed in the nuclear sector to the nuclear phenomenology, we vary the couplings in the hyperonic sector around the central values which are fitted to the hyperon potentials in nuclear matter. We use the SU(6) spin-flavor and SU(3) flavor symmetric quark models to relate the hyperonic couplings to the nucleonic ones. We find, consistent with previous Hartree studies, that for the SU(6) model the maximal masses of compact stars are below the two-solar mass limit. In the SU(3) model we find sufficiently massive compact stars with cores composed predominantly of $\Lambda$ and $\Xi$ hyperons and a low fraction of leptons (mostly electrons). The parameter space of the SU(3) model is identified where simultaneously hypernuclear compact stars obey the astrophysical limits on pulsar masses and the empirical hypernuclear potentials in nuclear matter are reproduced.

Microscopically based energy density functionals for nuclei using the density matrix expansion. II. Full optimization and validation

We seek to obtain a usable form of the nuclear energy density functional that is rooted in the modern theory of nuclear forces. We thus consider a functional obtained from the density matrix expansion of local nuclear potentials from chiral effective field theory. We propose a parametrization of this functional carefully calibrated and validated on selected ground-state properties that is suitable for large-scale calculations of nuclear properties. The first component of this functional is a non-local functional of the density and corresponds to the direct part (Hartree term) of the expectation value of local chiral potentials on a Slater determinant. A second component is a local functional of the density and is obtained by applying the density matrix expansion to the exchange part (Fock term) of the expectation value of the local chiral potential. We apply the UNEDF2 optimization protocol to determine the coupling constants of this energy functional. We obtain a set of microscopically-constrained functionals for local chiral potentials from leading-order up to next-to-next-to-leading order with and without three-body forces and contributions from $\Delta$ excitations. These functionals are validated on the calculation of nuclear and neutron matter, nuclear mass tables, single-particle shell structure in closed-shell nuclei and the fission barrier of $^{240}$Pu. Quantitatively, they perform noticeable better than the more phenomenological Skyrme functionals. The inclusion of higher-order terms in the chiral perturbation expansion seems to produce a systematic improvement in predicting nuclear binding energies. This result is especially promising since all the fits have been performed at the single reference level of the energy density functional approach, where important collective correlations such as center-of-mass correction have not been taken into account yet.

Nuclear fourth-order symmetry energy and its effects on neutron star properties in the relativistic Hartree-Fock theory

Adopting the density dependent relativistic mean-field (RMF) and relativistic Hartree-Fock (RHF) approaches, the properties of the nuclear fourth-order symmetry energy $S_4$ are studied within the covariant density functional (CDF) theory. It is found that the fourth-order symmetry energies are suppressed in RHF at both saturation and supranuclear densities, where the extra contribution from the Fock terms is demonstrated, specifically via the isoscalar meson-nucleon coupling channels. The reservation of $S_4$ and higher-order symmetry energies in the nuclear equation of state then affects essentially the prediction of neutron star properties, which is illustrated in the quantities such as the proton fraction, the core-crust transition density as well as the fraction of crustal moment of inertia. Since the Fock terms enhance the density dependence of the thermodynamical potential, the RHF calculations predict systematically smaller values of density, proton fraction and pressure at the core-crust transition boundary of neutron stars than density dependent RMF ones. In addition, a linear anti-correlation between the core-crust transition density $\rho_t$ and the density slope of symmetry energy $L$ is found which is then utilized to constrain the core-crust transition density as $\rho_t\thicksim[0.069, 0.098]~\rm{fm}^{-3}$ with the recent empirical information on $L$. The study clarifies the important role of the fourth-order symmetry energy in determining the properties of nuclear matter at extreme isospin or density conditions.

Influence of Fock exchange in combined many-body perturbation and dynamical mean field theory

In electronic systems with long-range Coulomb interaction, the nonlocal Fock exchange term has a band-widening effect. While this effect is included in combined many-body perturbation theory and dynamical mean field theory schemes, it is not taken into account in standard extended DMFT (EDMFT) calculations. Here, we include this instantaneous term in both approaches and investigate its effect on the phase diagram and dynamically screened interaction. We show that the largest deviations between previously presented EDMFT and $GW$+EDMFT results originate from the nonlocal Fock term, and that the quantitative differences are especially large in the strong-coupling limit. Furthermore, we show that the charge-ordering phase diagram obtained in $GW$+EDMFT methods for moderate interaction values is very similar to the one predicted by dual boson methods that include the fermion-boson or four-point vertex.

Correlation-Induced Band Competition in SrTiO3/LaAlO3

ABSTRACTThe oxide interface SrTiO3/LaAlO3 supports a 2D electron liquid displaying superconductivity and magnetism, while allowing for a continuous control of the electron density using a gate. Our recent measurements have shown a similar surprising nonmonotonic behavior as function of the gate voltage (carrier density) of three quantities: the superconducting critical temperature and field, the inverse Hall coefficient, and the frequency of quantum oscillations. While the total density has to be monotonic as function of gate, the last result indicates that one of the involved bands has a nonmontonic occupancy as function of the chemical potential. We show how electronic interactions can lead to such an effect, by creating a competition between the involved bands and making their sturcture non-rigid, and thus account for all these effects. Adding Fock terms to our previous Hartree treatment makes this scenario even more generic.

Impact of Fock terms on the isospin properties of nuclear matter

Several topics on the isospin properties of nuclear matter studied within the density-dependent relativistic Hartree-Fock theory are summarized. In detail, the effects of the Fock terms on the nuclear symmetry energy are listed, including the extra enhancement from the Fock terms of the isoscalar meson-nucleon coupling channels, the extra hyperon-induced suppression effect originating from the Fock channel, self-consistent tensor effects embedded automatically in the Fock diagrams, the enhanced density-dependent isospin-triplet potential part of the symmetry energy at high densities, a reduced kinetic symmetry energy at supranuclear density and so on. The results demonstrate the importance of the Fork diagram, especially from the isoscalar mesonnucleon coupling channels, on the isospin properties of the in-medium nuclear force.

Symmetry energy of nuclear matter in relativistic Hartree-Fock theory

Several topics on the symmetry energy of nuclear matter in relativistic Hartree-Fock theory are reviewed. General formulism of the newly developed density-dependent relativistic Hartree-Fock theory for hadronic matter is presented briefly. The effects of Fock terms on the nuclear symmetry energy are reviewed in detail from three aspects: the extra enhancement from the Fock terms of the isoscalar meson-nucleon coupling channels, the extra hyperon-induced suppression effect originating from the Fock channel, self-consistent tensor effects embedded automatically in the Fock diagrams. The naturally involved tensor force components in the Fock diagrams soften the density-dependent behavior of the symmetry energy and consequently lead neutron stars to be more compact. Finally, the error estimates of theoretical models are performed by doing a variation procedure, and possible ways to improve the relativistic Hartree-Fock models are discussed.