Hartree-Fock Mean Field Research Articles

An ordered-disordered separated graphene nanoribbon: high thermoelectric performance

The key requirement for an enhanced thermoelectric (TE) performance is the presence of asymmetry in transmission function. Focussing on this issue, we propose a unique idea to enhance TE performance in a graphene nanoribbon (GNR) that has not been explored so far to the best of our concern. In the present work, one part of the GNR is considered as a disordered region while the rest of the system is clean. Such an ordered-disordered separated structure yields more asymmetric transmission function over the conventional uniform disordered one. Finally, we include the effect of electron–electron (e–e) interaction to check whether it brings any non-trivial signature on TE performance. The e–e interaction is taken in the form of an on-site Hubbard model and we compute our results within a Hartree–Fock mean field approach. The results obtained in the present work exhibit quite remarkable TE performance along with some non-trivial features.

A fully microscopic model of total level density in spherical nuclei

A fully microscopic model for the description of nuclear level density (NLD) in spherical nuclei is proposed. The model is derived by combining the partition function of the exact pairing solution plus the independent-particle model at finite temperature (EP+IPM) with that obtained by using the collective vibrational states calculated from the self-consistent Hartree-Fock mean field with MSk3 interaction plus the exact pairing and random-phases approximation (SC-HFEPRPA). Two important factors are taken into account in a fully microscopic way, namely the spin cut-off and vibrational enhancement factors are, respectively, calculated using the statistical thermodynamics and partition function of the SC-HFEPRPA without any fitting parameters. The numerical test for two spherical 60Ni and 90Zr nuclei shows that the collective vibrational enhancement is mostly dominated by the quadrupole and octupole excitations. This is the first microscopic model confirming such an effect, which was phenomenologically predicted long time ago and widely employed in several NLD models. In addition, the influence of collective vibrational enhancement on nuclear thermodynamic quantities such as excitation energy, specific heat capacity and entropy is also studied by using the proposed model.

Contribution of the ρ meson and quark substructure to the nuclear spin-orbit potential

The microscopic origin of the spin-orbit (SO) potential in terms of sub-baryonic degrees of freedom is explored and discussed for application to nuclei and hyper-nuclei. We thus develop a chiral relativistic approach where the coupling to the scalar- and vector-meson fields are controlled by the quark substructure. This approach suggests that the isoscalar and isovector density dependence of the SO potential can be used to test the microscopic ingredients which are implemented in the relativistic framework: the quark substructure of the nucleon in its ground-state and its coupling to the rich meson sector where the $\rho$ meson plays a crucial role. This is also in line with the Vector Dominance Model (VDM) phenomenology and the known magnetic properties of the nucleons. We explore predictions based on Hartree and Hartree-Fock mean field, as well as various scenarios for the $\rho$-nucleon coupling, ranked as weak, medium and strong, which impacts the isoscalar and isovector density dependence of the SO potential. We show that a medium to strong $\rho$ coupling is essential to reproduce Skyrme phenomenology in $N=Z$ nuclei as well as its isovector dependence. Assuming an SU(6) valence quark model our approach is extended to hyperons and furnishes a microscopic understanding of the quenching of the $N\Lambda$ spin-orbit potential in hyper-nuclei. It is also applied to other hyperons, such as $\Sigma$, $\Xi$ and $\Omega$.

Mean-field interaction-induced dimensional crossover from two to three dimensions in a weakly interacting Bose gas

We study an interaction-induced dimensional crossover from two to three dimensions in a weakly interacting $^{87}\mathrm{Rb}$ Bose-Einstein condensate (BEC). A highly oblate harmonic optical dipole trap was used where we independently controlled the chemical potential and the temperature of the gas. The dimensional crossover was studied from the thermal state occupancy of atoms. In particular, the growth rate of the thermal component was measured, which contradicted the expected behavior in a fixed dimension. Crossover chemical potential and temperature were obtained at the crossover point. We further validate our observation by numerical calculations based on mean-field Hartree-Fock theory and local density approximation, which not only provided a qualitative picture of the underlying mechanism for the dimensional crossover, but also matched well with the experimentally measured crossover parameter.

Modulated phases of graphene quantum Hall polariton fluids

This work is placed in the context of solid-state systems in the regime of ultra-strong light-matter coupling. To date, the highest light-matter coupling strengths have been measured in experiments with polaritons in semiconductor systems under the conditions of the Integer Quantum Hall effect. Polaritons are excitations resulting from strong coupling of light with a dipole-carrying matter excitation. In Pellegrino et al. (2016), we studied the impact of electron-electron interaction on polaritons in cavities in the case of graphene under the conditions of the Integer Quantum Hall effect. By using a mean-field (Hartree-Fock) approach we have shown the possibility of formation of spatially modulated light-matter phases characterized by a wavelength that is dependent on the value of the applied static magnetic field and the concentration of carriers, which is tunable by varying the gate voltage.

The effective single particle potential and the tadpole

The Energy Density Functional (EDF) theory is extremely successful within the effective force approach, noticeably the Skyrme or Gogny forces, in reproducing the nuclear binding energies and other nuclear properties along the whole mass table. The EDF is in this case represented formally as the Hartree-Fock (HF) mean field of an effective force, and the associated single particle states can be considered the eigenstates of the corresponding effective mean field. In general, the phenomenological single particle spectrum is not accurately reproduced. To overcome this difficulty one can improve the functional in order to incorporate in the mean field additional correlations in an effective way. In an alternative viable scheme one can introduce explicitly many-body correlations which affect the single particle motion at both dynamic and static levels. In particular, the particle-vibration coupling scheme modifies the positions of the individual single particle energies. In this case one also introduce the fragmentation of the single particle states, a feature that cannot be described at the mean field level. At the same time the so-introduced correlations modify the static single particle potential, and the single particle states that diagonalize the corresponding density matrix are not the orbitals of a mean field with occupation numbers 1 (occupied orbitals) and 0 (unoccupied orbitals) anymore. In this paper we show that both static and dynamic effects on the single particle motion can be introduced within a previously developed scheme, based on the conserving approximations, and that the static part can be identified with the so-called tadpole term introduced by Khodel in the self-consistent theory of finite Fermi systems. The treatment remains at the formal level, but we hope that several aspects of the particle-vibration coupling scheme will be clarified.

Physics of even-even superheavy nuclei with 96<Z<110 in the quark-meson-coupling model

The Quark-Meson-Coupling (QMC) model has been applied to the study of the properties of even-even super-heavy nuclei with 96 < Z < 110, over a wide range of neutron numbers. The aim is to identify the deformed shell gaps at N = 152 and N = 162 predicted in macroscopic-microscopic (macro-micro) models, in a model based on the mean-field Hartree-Fock+BCS approximation. The predictive power of the model has been tested on proton and neutron spherical shell gaps in light doubly closed (sub)shell nuclei. In the super-heavy region, the ground state binding energies of 98 < Z < 110 and 146 < N < 160 differ, in the majority of cases, from the measured values by less than 2.5 MeV, with the deviation decreasing with increasing Z and N. The axial quadrupole deformation parameter, calculated over the range of neutron numbers 138 < N < 184, revealed a prolate-oblate coexistence and shape transition around N = 168, followed by an oblate-spherical transition towards the expected N = 184 shell closure in Cm, Cf, Fm and No. The closure is not predicted in Rf, Sg, Hs and Ds as another shape transition to a highly deformed shape in Sg, Hs and Ds for N > 178 appears, while 288Rf (N = 184) remains oblate. The bulk properties predicted by QMC are found to have a limited sensitivity to the deformed shell gaps at N = 152 and 162. However, the evolution of the neutron single-particle spectra with 0 < beta2 < 0.55 gives unambiguous evidence for the location and size of the N = 152 and 162 gaps as a function of Z and N. In addition, the neutron number dependence of neutron pairing energies provides supporting evidence for existence of the energy gaps.

BCS-BEC crossover and superconductor-insulator transition in Hopf-linked Graphene layers: Hopfene

We have proposed a topological carbon allotrope, named Hopfene, which has three-dimensional (3D) arrays of Hopf-links to bind 2D Graphene layers both horizontally and vertically without forming strong σ bonds between layers. Tight-binding calculations show unique band structures of this crystal, which predicts semi-metal characteristics with the existence of both Weyl and Dirac Fermions depending on the Fermi energy. Here, we have theoretically examined superconductivity of Hopfene based on the attractive Hubbard model. Regardless of its simplicity of the model, we found non-trivial competitions between Hartree–Fock mean-field contributions and Cooper-paring interactions to open semiconductor and superconducting energy gaps, respectively. Consequently, the superconducting order parameters are significantly reduced at every quarterly doping concentration, where the system is in the close vicinity of the quantum critical point, and we found superconductor-insulator transition in the strong coupling limit. Upon doping, we confirmed a classical scenario of a smooth crossover from weak coupling Bardeen-Cooper-Schrieffer (BCS) superconductivity to strong coupling Bose–Einstein Condensation (BEC) of preformed pairs by increasing the interaction strength. We think the proposed Hopfene is a useful platform to investigate the impacts of the topological nature of the Fermi surfaces on the superconductivity and other orders, including charge-density-waves and magnetic orders, and possible quantum phase transitions among them.

Renormalizing random-phase approximation by using exact pairing

A fully self-consistent renormalized random-phase approximation is constructed based on the self-consistent Hartree-Fock mean field plus exact pairing solutions (EP). This approach exactly conserves the particle number and restores the energy-weighted sum rule, which is violated in the conventional renormalized particle-hole random-phase approximation for a given multipolarity. The numerical calculations are carried out for several light-, medium-, and heavy-mass nuclei such as $^{22}\mathrm{O}$, $^{60}\mathrm{Ni}$, and $^{90}\mathrm{Zr}$ by using the effective MSk3 interaction. To study the pygmy dipole resonance (PDR), the calculations are also performed for the two light and neutron-rich $^{24,28}\mathrm{O}$ isotopes, whose PDRs are known to be dominant. The results obtained show that the inclusion of ground-state correlations beyond the random-phase approximation (RPA) by means of the occupation numbers obtained from the EP affects the RPA solutions within the whole mass range, although this effect decreases with increasing the mass number. At the same time, the antipairing effect is observed via a significant reduction of pairing in neutron-rich nuclei. The enhancement of PDR is found in most neutron-rich nuclei under consideration within our method.

Relation between fermionic and qubit mean fields in the electronic structure problem.

For quantum computing applications, the electronic Hamiltonian for the electronic structure problem needs to be unitarily transformed into a qubit form. We found that mean-field procedures on the original electronic Hamiltonian and on its transformed qubit counterpart can give different results. We establish conditions of when fermionic and qubit mean fields provide the same or different energies. In cases when the fermionic mean-field (Hartree-Fock) approach provides an accurate description (electronic correlation effects are small), the choice of molecular orbitals for the electron Hamiltonian representation becomes the determining factor in whether the qubit mean-field energy will be equal to or higher than that of the fermionic counterpart. In strongly correlated cases, the qubit mean-field approach has a higher chance to undergo symmetry breaking and lower its energy below the fermionic counterpart.

Theoretical study of modified electron band dispersion and density of states due to high frequency phonons in graphene-on-substrates

We propose here a theoretical model for the study of band gap opening in graphene-on- polarizable substrate taking the effect of electron–electron and electron–phonon (EP) interactions at high frequency phonon vibrations. The Hamiltonian consists of hopping of electrons upto third nearest- neighbors and the effect substrate, where A sublattice site is raised by energy [Formula: see text] and B sublattice site is suppressed by energy [Formula: see text], hence producing a band gap energy of [Formula: see text]. Further, we have considered Hubbard type electron–electron repulsive interactions at A and B sublattices, which are considered within Hartree–Fock meanfield approximation. The electrons in the graphene plane interact with the phonon’s present in the polarized substrate in the presence of phonon vibrational energy within harmonic approximation. The temperature-dependent electron occupancies are computed numerically and self-consistently for both spins at both the sublattice sites. By using these electron occupancies, we have calculated the electron band dispersion and density of states (DOS), which are studied for the effects of EP interaction, high phonon frequency, Coulomb energy and substrate induced gap.

Phase transitions of the Kane–Mele–Hubbard model with a long-range hopping

The interacting Kane–Mele model with a long-range hopping is studied using analytical method. The original Kane–Mele model is defined on a honeycomb lattice. In the work, we introduce a four-lattice-constant range hopping and the on-site Hubbard interaction into the model and keep its lattice structure unchanged. From the single-particle energy spectrum, we obtain the critical strength of the long-range hopping tL at which the topological transition occurs in the non-interacting limit of the model and our results show that it is independent of the spin–orbit coupling. After introducing the Hubbard interaction, we investigate the Mott transition and the magnetic transition of the generalized strongly correlated Kane–Mele model using the slave-rotor mean field theory and Hartree–Fock mean field theory respectively. In the small long-range hopping region, it is a correlated quantum spin Hall state below the Mott transition, while a topological Mott insulator above the Mott transition. By comparing the energy band of spin degree of freedom with the one of electrons in non-interacting limit, we find a condition for the tL-driven topological transition. Under the condition, critical values of tL at which the topological transition occurs are obtained numerically from seven self-consistency equations in both regions below and above the Mott transition. Influences of the interaction and the spin–orbit coupling on the topological transition are discussed in this work. Finally, we show complete phase diagrams of the generalized interacting topological model at some strength of spin–orbital coupling.

Weak-interaction rates in stellar conditions

Weak-interaction rates, including β-decay and electron captures, are studied in several mass regions at various densities and temperatures of astrophysical interest. In particular, we study odd-A nuclei in the pf-shell region, which are involved in presupernova formations. Weak rates are relevant to understand the late stages of the stellar evolution, as well as the nucleosynthesis of heavy nuclei. The nuclear structure involved in the weak processes is studied within a quasiparticle proton-neutron random-phase approximation with residual interactions in both particle-hole and particle-particle channels on top of a deformed Skyrme Hartree-Fock mean field with pairing correlations. First, the energy distributions of the Gamow-Teller strength are discussed and compared with the available experimental information, measured under terrestrial conditions from charge-exchange reactions. Then, the sensitivity of the weak-interaction rates to both astrophysical densities and temperatures is studied. Special attention is paid to the relative contribution to these rates of thermally populated excited states in the decaying nucleus and to the electron captures from the degenerate electron plasma.

Bubble nuclei within the self-consistent Hartree-Fock mean field plus pairing approach

The depletion of the nuclear density at its center, called the nuclear bubble, is studied within the Skyrme Hartree-Fock mean field consistently incorporating the superfluid pairing. The latter is obtained within the finite-temperature Bardeen-Cooper-Schrieffer theory and within the approach using the exact pairing. The numerical calculations are carried out for $^{22}\mathrm{O}$ and $^{34}\mathrm{Si}$ nuclei, whose bubble structures, caused by a very low occupancy of the $2{s}_{1/2}$ level, were previously predicted at $T=0$. Among 24 Skyrme interactions under consideration, the MSk3 is the only one which reproduces the experimentally measured occupancy of the $2{s}_{1/2}$ proton level as well as the binding energy, and consequently produces the most pronounced bubble structure in $^{34}\mathrm{Si}$. As compared to the approaches employing the same BSk14 interaction, our approach with exact pairing predicts a pairing effect which is stronger in $^{22}\mathrm{O}$ and weaker in $^{34}\mathrm{Si}$. The increase in temperature depletes the bubble structure and completely washes it out when the temperature reaches a critical value, at which the factor measuring the depletion of the nucleon density vanishes.

Shell model and Hartree-Fock calculations of longitudinal and transverse electroexcitation of positive and negative parity states in O17

The nuclear structure of the $^{17}\mathrm{O}$ nucleus has been investigated using shell model and self-consistent Hartree-Fock calculations. In particular, elastic and inelastic electron scattering form factors, energy levels, and transition probabilities are calculated for positive and negative low-lying states. Two different shell model spaces have been used for this purpose. The first one is the psdpn model space for positive parity states and the second one is ${p}_{1/2}sd$ model space for negative parity states. For all selected excited states, Skyrme interactions are adopted to generate from them a one-body potential in Hartrre-Fock theory to calculate the single-particle matrix elements. The deduced results are discussed for the longitudinal and transverse form factors and compared with the available experimental data. It has been confirmed that combining the shell model plus Hartree-Fock mean field method with the Skyrme interaction can accommodate very well the nuclear excitation properties, and work better for low lying states than for higher excitations. Furthermore, the combination can be used to reproduce the positive and negative parity states after choosing the suitable model space, effective two-body interaction, and parameterization to reach highly descriptive and predictive results.

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.

Attosecond photoionization dynamics in neon

We study the role of electron-electron correlation in the ground-state of Ne, as well as in photoionization dynamics induced by an attosecond XUV pulse. For a selection of central photon energies around 100 eV, we find that while the mean-field time-dependent Hartree-Fock method provides qualitatively correct results for the total ionization yield, the photoionization cross section, the photoelectron momentum distribution as well as for the time-delay in photoionization, electron-electron correlation is important for a quantitative description of these quantities.

Charge Separation in Donor–C60 Complexes with Real-Time Green Functions: The Importance of Nonlocal Correlations

We use the nonequilibrium Green function (NEGF) method to perform real-time simulations of the ultrafast electron dynamics of photoexcited donor-C60 complexes modeled by a Pariser-Parr-Pople Hamiltonian. The NEGF results are compared to mean-field Hartree-Fock (HF) calculations to disentangle the role of correlations. Initial benchmarking against numerically highly accurate time-dependent density matrix renormalization group calculations verifies the accuracy of NEGF. We then find that charge-transfer (CT) excitons partially decay into charge separated (CS) states if dynamical nonlocal correlation corrections are included. This CS process occurs in ∼10 fs after photoexcitation. In contrast, the probability of exciton recombination is almost 100% in HF simulations. These results are largely unaffected by nuclear vibrations; the latter become however essential whenever level misalignment hinders the CT process. The robust nature of our findings indicates that ultrafast CS driven by correlation-induced decoherence may occur in many organic nanoscale systems, but it will only be correctly predicted by theoretical treatments that include time-nonlocal correlations.

Self-consistent Hartree–Fock RPA calculations in 208Pb

The nuclear structure of 208Pb is studied in the framework of the self-consistent random phase approximation (SCRPA). The Hartree–Fock mean field and single particle states are used to implement a completely SCRPA with Skyrme-type interactions. The Hamiltonian is diagonalised within a model space using five Skyrme parameter sets, namely LNS, SkI3, SkO, SkP and SLy4. In view of the huge number of the existing Skyrme-force parameterizations, the question remains which of them provide the best description of data. The approach attempts to accurately describe the structure of the spherical even–even nucleus 208Pb. To illustrate our approach, we compared the binding energy, charge density distribution, excitation energy levels scheme with the available experimental data. Moreover, we calculated isoscalar and isovector monopole, dipole, and quadrupole transition densities and strength functions.

Excitonic mass gap in uniaxially strained graphene

We study the conditions for spontaneously generating an excitonic mass gap due to Coulomb interactions between anisotropic Dirac fermions in uniaxially strained graphene. The mass gap equation is realized as a self-consistent solution for the self-energy within the Hartree-Fock mean-field and static random phase approximations. It depends not only on momentum, due to the long-range nature of the interaction, but also on the velocity anisotropy caused by the presence of uniaxial strain. We solve the nonlinear integral equation self-consistently by performing large scale numerical calculations on variable grid sizes. We evaluate the mass gap at the charge neutrality (Dirac) point as a function of the dimensionless coupling constant and anisotropy parameter. We also obtain the phase diagram of the critical coupling, at which the gap becomes finite, against velocity anisotropy. Our numerical study indicates that with an increase in uniaxial strain in graphene the strength of critical coupling decreases, which suggests anisotropy supports formation of excitonic mass gap in graphene.