Mean-field Hartree-Fock Approximation Research Articles

Self-trapping transition in a two dimensional extended Holstein-Hubbard model: A mean-field study

The self-trapping transition is studied within the framework of the two-dimensional extended Holstein-Hubbard model for both adiabatic and anti-adiabatic cases. A highly accurate phonon state is chosen as the averaging state to obtain an effective electronic Hamiltonian which is solved for the two different Coulomb correlation regimes separately. For weak correlation, the Hartree-Fock mean-field approximation is employed and for strong correlation, the electronic Hamiltonian is mapped on to an effective t−J model which is solved by using the Gutzwiller approximation and the Zubarev Green's function technique. For the entire range of the Coulomb interaction, the self-trapping transition turns out to be continuous for the anti-adiabatic regime and for the adiabatic regime the transition it is found to be discontinuous.

Thermal squeezing and nonlinear spectral shift of magnons in antiferromagnetic insulators

We investigate the effect of magnon–magnon interactions on the dispersion and polarization of magnon modes in collinear antiferromagnetic (AF) insulators at finite temperatures. In two-sublattice AF systems with uniaxial easy-axis and biaxial easy-plane magneto-crystalline anisotropies, we implement a self-consistent Hartree–Fock mean-field approximation to explore the nonlinear thermal interactions. The resulting nonlinear magnon interactions separate into two-magnon intra- and interband scattering processes. Furthermore, we compute the temperature dependence of the magnon bandgap and AF resonance modes due to nonlinear magnon interactions for square and hexagonal lattices. In addition, we study the effect of magnon interactions on the polarization of magnon modes. We find that although the noninteracting eigenmodes in the uniaxial easy-axis case are circularly polarized, but in the presence of nonlinear thermal interactions the U(1) symmetry of the magnon Hamiltonian is broken. The attractive nonlinear interactions squeeze the low energy magnon modes and make them elliptical. In the biaxial easy-plane case, on the other hand, the bare eigenmodes of low energy magnons are elliptically polarized but thermal nonlinear interactions squeeze them further. Direct measurements of the predicted temperature-dependent AF resonance modes and their polarization can be used as a tool to probe the nonlinear interactions. Our findings establish a framework for exploring the effect of thermal magnon interactions in technologically important magnetic systems, such as magnetic stability of recently discovered two-dimensional magnetic materials, coherent transport of magnons, Bose–Einstein condensation of magnons, and magnonic topological insulators.

Brane order and quantum magnetism in modulated anisotropic ladders

Two-leg spin-$1/2$ ladders with anisotropy and two different dimerization patterns are analyzed at zero temperature. This model is equivalent to a modulated interacting (Kitaev) ladder. The Hartree-Fock mean-field approximation reduces the model to a sum of two quadratic effective Majorana Hamiltonians, which are dual to two quantum transverse XY chains. The mapping between the effective Hamiltonian of the ladder and a pair of chains considerably simplifies calculations of the order parameters and analysis of the hidden symmetry breaking. The ground-state phase diagram of the staggered ladder contains nine phases, four of them are conventional antiferromagnets, while the other five possess nonlocal brane orders. Using the dualities and the newly found exact results for the local and string order parameters of the transverse XY chains, we were able to find analytically all the magnetizations and the brane order parameters for the staggered case, as functions of the renormalized couplings of the effective Hamiltonian. The columnar ladder has three ground-state phases and does not possess magnetic long-ranged order. The brane order parameters for these three phases are calculated numerically from the Toeplitz determinants. All brane-ordered phases are spin liquids with identified distinct order parameters, winding numbers, and sets of the Majorana edge modes. Disorder lines and the special points of disentanglement are found in both dimerization patterns. We expect this study to motivate the search for the real spin-Peierls anisotropic ladder compounds which manifest predicted properties.

Mott-Insulator to Peierls Insulator Transition in the Two-Dimensional Holstein-Hubbard Model

We propose a variational study of the two dimensional square lattice to study the phase transition from an antiferromagnetic SDW Mott insulator to the bipolaronic CDW Peierls insulator for the half-filled Holstein Hubbard model. The correlated system is dealt analytically in two different regimes according to the renormalized Coulomb correlation strength. Different unitary transformations are performed to obtain an effective electronic Hamiltonian which is solved by using the mean-field Hartree-Fock approximation for the weakly correlated electrons. For the strong interaction case, the Zuberev’s Green function technique is used to solve the effective t-J model at the mean field limit. The variations of the effective Hubbard hopping parameter and the effective Coulomb interaction strength for different electron–electron and electron–phonon interaction coefficients have been studied. We have found a transition from the Mott-insulator to a Peierls insulator analytically for the two dimensional Holstein Hubbard model.

Effect of finite temperature and external magnetic field on non-equilibrium transport in a single molecular transistor with quantum dissipation: Anderson-Holstein-Caldeira-Leggett model

A magnetic field and temperature dependent quantum transport is investigated in a single molecular transistor. Our system is modelled by an onsite electron-electron correlation, polaronic interaction in the quantum dot and the coupling between quantum dot local phonon and the substrate phonon which produces a damping effect on the dynamics of the quantum dot. We use the Anderson-Holstein Hamiltonian along with the Caldeira-Leggett model to tackle the dissipation and polaronic effects and mean-field Hartree-Fock approximation to treat the onsite Coulomb correlation. We calculate transport properties such as spectral density function, tunnelling current, differential conductance employing finite-temperature Keldysh non-equilibrium Green function method under the influence of a finite temperature and an external magnetic field. It is shown that temperature has an intervening effect on the spin-resolved transport properties which can be used as a tuneable spin-filter.

Influence of superconductivity on the magnetic moment of quantum impurity embedded in BCS superconductor

We study the influence of superconductivity on the formation of the localized magnetic moment for a single-level quantum impurity embedded in an s-wave Bardeen–Cooper–Schrieffer (BCS) superconducting medium, modeled by single-impurity Anderson Hamiltonian. We have combined Bogoliubov transformation with Green’s function method within self-consistent Hartree–Fock mean field approximation to analyze the conditions necessary in metal (in the superconducting) for the formation of the magnetic moment at the impurity site for the low-frequency limit |ω| ≪ Δsc as well as for the finite superconducting gap Δsc. We have compared these results with other theoretical results and with the single-level quantum impurity embedded in the normal metallic host. Further, we analyze the spectral density of the quantum impurity embedded in a superconducting host to study the sub-gap states as a function of impurity parameters.

Extended Falicov-Kimball model: Exact solution for finite temperatures

The extended Falicov-Kimball model is analyzed exactly in the ground state at half filling in the limit of large dimensions. In the model the on-site and the intersite density-density interactions between all particles are included. We determined the model's phase diagram and found a discontinuous transition between two different charge-ordered phases. Our analytical calculations show that the ground state of the system is insulating for any nonzero values of the interaction couplings. We also show that the dynamical mean-field theory and the static broken-symmetry Hartree-Fock mean-field approximation give the same results for the model at zero temperature. In addition, we prove using analytical expressions that at infinitesimally small, but finite, temperatures the system can be metallic.

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.

Extended Falicov-Kimball model: Exact solution for the ground state

The extended Falicov-Kimball model is analyzed exactly in the ground state at half filling in the limit of large dimensions. In the model the on-site and the intersite density-density interactions between all particles are included. We determined the model's phase diagram and found a discontinuous transition between two different charge-ordered phases. Our analytical calculations show that the ground state of the system is insulating for any nonzero values of the interaction couplings. We also show that the dynamical mean-field theory and the static broken-symmetry Hartree-Fock mean-field approximation give the same results for the model at zero temperature. In addition, we prove using analytical expressions that at infinitesimally small, but finite, temperatures the system can be metallic.

Interaction-induced phase transitions of type-II Weyl semimetals

The study of Weyl semimetals (WSMs) lies at the forefront of the nontrivial topological phenomena in condensed-matter physics. In this work, we study the effect of on-site repulsive Hubbard interaction on the WSM system with a nonzero tilt at half filling. Within the Hartree-Fock mean-field approximation, we treat the Hubbard interaction self-consistently and find that the Fock exchange field vanishes, while the Hartree field can renormalize the topological mass, the tilt, and the Fermi velocity of the Weyl cones. When the renormalized tilt is larger than the renormalized Fermi velocity, the Hubbard interaction will induce the quantum phase transition from a type-I WSM to a type-II WSM. We then provide the interaction-induced phase diagrams of WSMs in different parametric spaces, in which the antiferromagnetic order at strong interaction is also considered. In addition, we analyze another model hosting two pairs of Weyl nodes, and similar results are obtained. The implications of these results are discussed.

Interaction-induced insulating states in multilayer graphenes

We explore the electronic ground states of Bernal-stacked multilayer graphenes using the Hartree-Fock mean-field approximation and the full-parameter band model. We find that the electron-electron interaction tends to open a band gap in multilayer graphenes from bilayer to 8-layer, while the nature of the insulating ground state sensitively depends on the band parameter $\gamma_2$, which is responsible for the semimetallic nature of graphite. In 4-layer graphene, particularly, the ground state assumes an odd-spatial-parity staggered phase at $\gamma_2 = 0$, while an increasing, finite value of $\gamma_2$ stabilizes a different state with even parity, where the electrons are attracted to the top layer and the bottom layer. The two phases are topologically distinct insulating states with different Chern numbers, and they can be distinguished by spin or valley Hall conductivity measurements. Multilayers with more than five layers also exhibit similar ground states with potential minima at the outermost layers, although the opening of a gap in the spectrum as a whole is generally more difficult than in 4-layer because of a larger number of energy bands overlapping at the Fermi energy.

Dielectric constant of graphene-on-polarized substrate: A tight-binding model study

We report here a microscopic tight-binding theoretical study of the dynamic dielectric response of graphene-on-polarizable substrate with impurity. The Hamiltonian consists of first, second and third nearest-neighbour electron hopping interactions besides doping and substrate-induced effects on graphene. We have introduced electron–electron correlation effect at A and B sublattices of graphene which is considered within Hartree–Fock mean-field approximation. The electron occupancies at both sublattices are calculated and solved self-consistently and numerically for both up- and down-spin orientations. The polarization function appearing in the dielectric function is a two-particle Green’s function which is calculated by using Zubarev’s Green’s function technique. The temperature and optical frequency-dependent dielectric function is evaluated and compared with experimental data by varying Coulomb correlation energy, substrate-induced gap and impurity concentrations.

Origin of magnetoelectric effect inCo4Nb2O9andCo4Ta2O9: The lessons learned from the comparison of first-principles-based theoretical models and experimental data

We report results of joint experimental and theoretical studies on magnetoelectric (ME) compounds Co4Nb2O9 and Co4Ta2O9. On the experimental side, we present results of the magnetization and dielectric permittivity measurements in the magnetic field. On the theoretical side, we construct the low-energy Hubbard-type model for the magnetically active Co 3d bands in the Wannier basis, using the input of first-principles electronic structure calculations, solve this model in the mean-field Hartree-Fock approximation, and evaluate the electric polarization in terms of the Berry phase theory. Both experimental and theoretical results suggest that Co4Ta2O9 is magnetically softer than Co4Nb2O9. Therefore, it is reasonable to expect that the antiferromagnetic structure of Co4Ta2O9 can be easier deformed by the external magnetic field, yielding larger polarization. This trend is indeed reproduced by our theoretical calculations, but does not seem to be consistent with the experimental behavior of the polarization and dielectric permittivity. Thus, we suggest that there should be a hidden mechanism controlling the ME coupling in these compounds, probably related to the magnetic striction or a spontaneous change of the magnetic structure, which breaks the inversion symmetry. Furthermore, we argue that unlike in other ME systems (e.g. Cr2O3), in Co4Nb2O9 and Co4Ta2O9 there are two crystallographic sublattices, which contribute to the ME effect. These contributions are found to be of the opposite sign and tend to compensate each other. The latter mechanism can be also used to control and reverse the electric polarization in these compounds.

Shell model and Hartree–Fock calculations for some exotic nuclei

Mass density distributions, the associated nuclear radii and elastic electron scattering form factors of light exotic nuclei, [Formula: see text]Li, [Formula: see text]Be, [Formula: see text]Be and 8B have been calculated using shell model (SM) and Hartree–Fock (HF) methods. We consider truncated spsdpf no core SM and WBP two-body effective interaction to give the SM wave functions. The single-particle matrix elements have been calculated with Skyrme-Hartree–Fock (SHF) potential with different parametrizations. It is shown that the calculated densities and form factors are in fine agreement with experimental data. This agreement can be interpreted as the adequacy of the HF mean-field approximation for exotic nuclei.

Probe-type of superconductivity by impurity in materials with short coherence length: the s-wave and η-wave phases study

The effects of a single non-magnetic impurity on superconducting states in the Penson–Kolb–Hubbard model have been analyzed. The investigations have been performed within the Hartree–Fock mean field approximation in two steps: (i) the homogeneous system is analysed using the Bogoliubov transformation, whereas (ii) the inhomogeneous system is investigated by self-consistent Bogoliubov-de Gennes equations (with the exact diagonalization and the kernel polynomial method). We analysed both signs of the pair hopping, which correspond to s-wave and η-wave superconductivity. Our results show that an enhancement of the local superconducting gap at the impurity-site occurs for both cases. We obtained that Cooper pairs are scattered (at the impurity site) into the states which are from the neighborhoods of the states, which are commensurate ones with the crystal lattice. Additionally, in the η-phase there are peaks in the local-energy gap (in momentum space), which are connected with long-range oscillations in the spatial distribution of the energy gap, superconducting order parameter (SOP), as well as effective pairing potential. Our results can be contrasted with the experiment and predicts how to experimentally differentiate these two different symmetries of SOP by the scanning tunneling microscopy technique.

Model study of the effect of Coulomb interaction on band gap of graphene-on-substrates

Graphene develops modulated gap, when it is placed on different substrates. In order to study the effect of Coulomb interaction on the gaps, we propose here a tight-binding model taking nearest-neighbor hopping integrals in the presence of Coulomb interactions on two inequivalent sublattices of honeycomb lattice of graphene. Here Coulomb interaction is treated within a Hartree–Fock mean-field approximation and difference in electron occupation numbers is computed numerically and self-consistently. It is observed that the system develops ferromagnetism at A-site atoms as well as B-site atoms. However this ferromagnetisms in two sub-lattices are antiferromagnetically ordered. The Coulomb interaction develops a gap near ’K’ point in reciprocal space. The evolution of this gap is investigated in the electron density of states, energy band dispersion and electron specific heat of graphene.

Self-trapping phase diagram for the strongly correlated extended Holstein-Hubbard model in two-dimensions

The two-dimensional extended Holstein-Hubbard model is investigated in the strong correlation regime to study the nature of self-trapping transition and the polaron phase diagram in the absence of superconductivity. Using a series of canonical transformations followed by zero-phonon averaging the extended Holstein-Hubbard model is converted into an effective extended Hubbard model which is subsequently transformed into an effective t-J model in the strong correlation limit. This effective t-J model is finally solved using the mean-field Hartree-Fock approximation to show that the self-trapping transition is continuous in the anti-adiabatic limit while it is discontinuous in the adiabatic limit. The phase diagrams for the localization-delocalization transition, namely the phase line and the phase surface separating the small polaron and large polaron states are also shown.

Orbital magnetization of insulating perovskite transition-metal oxides with a net ferromagnetic moment in the ground state

Modern theory of the orbital magnetization is applied to the series of insulating perovskite transition metal oxides (orthorhombic YTiO$_3$, LaMnO$_3$, and YVO$_3$, as well as monoclinic YVO$_3$), carrying a net ferromagnetic (FM) moment in the ground state. For these purposes, we use an effective Hubbard-type model, derived from the first-principles electronic-structure calculations and describing the behavior of magnetically active states near the Fermi level. The solution of this model in the mean-field Hartree-Fock approximation with the relativistic spin-orbit coupling typically gives us a distribution of the local orbital magnetic moments, which are related to the site-diagonal part of the density matrix $\hat{\cal D}$ by the "standard" expression $\boldsymbol{\mu}^0 = - \mu_{\rm B} \mathrm{Tr} \{ \hat{\textbf{L}} \hat{\cal D} \}$ and which are usually well quenched by the crystal field. In this work, we evaluate "itinerant" corrections $\Delta \boldsymbol{\cal M}$ to the net FM moment, suggested by the modern theory. We show that these corrections are small and in most cases can be neglected. Nevertheless, the most interesting aspect of our analysis is that, even for these compounds, which are typically regarded as normal Mott insulators, the "itinerant" corrections reveal a strong $\textbf{k}$-dependence in the reciprocal space, following the behavior of Chern invariants. Therefore, the small value of $\Delta \boldsymbol{\cal M}$ is the result of strong cancelation of relatively large contributions, coming from different parts of the Brillouin zone. We discuss details as well as possible implications of this cancelation, which depends on the crystal structure as well as the type of the magnetic ground state.

Mobility edge phenomenon in a Hubbard chain: A mean field study

We show that a tight-binding one-dimensional chain composed of interacting and non-interacting atomic sites can exhibit multiple mobility edges at different values of carrier energy in presence of external electric field. Within a mean field Hartree-Fock approximation we numerically calculate two-terminal transport by using Green's function formalism. Several cases are analyzed depending on the arrangements of interacting and non-interacting atoms in the chain. The analysis may be helpful in designing mesoscale switching devices.

Localization–delocalization transition in a two-dimensional Holstein–Hubbard model

The nature of self-trapping transition is investigated within the framework of an extended Holstein–Hubbard model in two dimensions using a variational method. We perform a series of canonical transformations including phonon coherence effect that partly depends on the electron density and is partly independent and also the effect of on-site and nearest-neighbor phonon correlations to obtain an effective extended Hubbard model which is finally studied using the mean-field Hartree–Fock approximation. The mean-field solution of the effective extended Hubbard model suggests that the transition from the large polaron state to a small polaron state in the anti-adiabatic regime is continuous, while in the adiabatic limit it predicts a discontinuous transition.