Self-consistent Hartree-Fock Approximation Research Articles

Magnetic phases from competing Hubbard and extended Coulomb interactions in twisted bilayer graphene

We implement a self-consistent Hartree-Fock approximation based on a microscopic model in real space, which allows us to consider the interplay between the Hubbard and the extended Coulomb interaction in twisted bilayer graphene at the magic angle. These two interactions tend to favor different symmetry breaking patterns, having therefore complementary roles in the regimes where one or the other dominates. We show that, for sufficiently large values of the on-site Hubbard repulsion, magic angle graphene has an antiferromagnetic ground state at the charge neutrality point, while at half-filling of the lowest valence band the state becomes fully spin-polarized. In general, a suitable screening of the extended Coulomb interaction is required to observe the magnetic state in either case, as otherwise the instabilities take place in the charge sector, preferentially in the form of time-reversal, chiral or valley symmetry breaking.

Coupled spin–orbital fluctuations in a three orbital model for 4d and 5d oxides with electron fillings n = 3, 4, 5—application to NaOsO3, Ca2RuO4 and Sr2IrO4

A unified approach is presented for investigating coupled spin–orbital fluctuations within a realistic three-orbital model for strongly spin–orbit coupled systems with electron fillings n = 3, 4, 5 in the t 2g sector of d yz , d xz , d xy orbitals. A generalized fluctuation propagator is constructed which is consistent with the generalized self-consistent Hartree–Fock approximation where all Coulomb interaction contributions involving orbital diagonal and off-diagonal spin and charge condensates are included. Besides the low-energy magnon, intermediate-energy orbiton and spin–orbiton, and high-energy spin–orbit exciton modes, the generalized spectral function also shows other high-energy excitations such as the Hund’s coupling induced gapped magnon modes. We relate the characteristic features of the coupled spin–orbital excitations to the complex magnetic behavior resulting from the interplay between electronic bands, spin–orbit coupling, Coulomb interactions, and structural distortion effects, as realized in the compounds NaOsO3, Ca2RuO4, and Sr2IrO4.

Nonstationary equation for the one-particle wave function of the Bose–Einstein condensate

Based on the self-consistent Hartree–Fock approximation, the nonstationary equation is obtained for the one-particle wave function describing the Bose–Einstein condensate in a rarefied gas of spin-zero bosons. A rarefied gas of bosons is exposed to the static external field, which ensures its finite ground state. The derived equation allows one to correctly determine the ground state energy in the stationary case.

Calculation of two-photon absorption by nanocrystals of CsPbBr3

We present calculations of the two-photon absorption cross section ${\ensuremath{\sigma}}^{(2)}(\ensuremath{\omega})$ for nanocrystals (NCs) of the inorganic perovskite ${\mathrm{CsPbBr}}_{3}$ for photon energies $\ensuremath{\omega}$ ranging from the absorption threshold at $2\ensuremath{\omega}\ensuremath{\approx}2.3\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ up to $2\ensuremath{\omega}=3.1\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$. The calculations employ a $4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{4pt}{0ex}}\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ envelope-function model, with final-state excitons described in a self-consistent Hartree-Fock approximation. The $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ corrections to ${\ensuremath{\sigma}}^{(2)}(\ensuremath{\omega})$ are found to be rather large, giving a reduction of about 30% in the cross section at the largest energies considered. The cross section is shown to be independent of polarization in the effective-mass approximation (EMA), but including $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ corrections leads to a small difference in ${\ensuremath{\sigma}}^{(2)}(\ensuremath{\omega})$ between circular and linear polarization, which rises to about 16% at $2\ensuremath{\omega}=3.1\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$. The theoretical cross section follows closely a power-law dependence on NC size, ${\ensuremath{\sigma}}^{(2)}(\ensuremath{\omega})\ensuremath{\propto}{L}^{\ensuremath{\alpha}}$, with theoretical exponents $\ensuremath{\alpha}=3.4$ (EMA) or $\ensuremath{\alpha}=3.2$ ($4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{0.16em}{0ex}}\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model), in excellent agreement with experiment. The dominant contribution to the exponent $\ensuremath{\alpha}$ is shown to be the number of final-state excitons per unit energy. Measured values of the absolute (normalized) cross section ${\ensuremath{\sigma}}^{(2)}(\ensuremath{\omega})$ show a large spread of values, differing by as much as a factor of 25 for some NC sizes. Our calculations strongly favor a group of measurements at the lower end of the reported range of ${\ensuremath{\sigma}}^{(2)}(\ensuremath{\omega})$.

Size scaling of the exchange interaction in the quantum Hall effect regime

We use the self-consistent Hartree-Fock approximation for numerically addressing the integer quantum Hall effect (IQHE) regime in terms of many-body physics at higher Landau levels (LLs). We investigate the dependence of many-particle interactions on the lateral size of the electron system. We use the exchange enhancement of the $g$-factor for spin-polarized Landau levels as an indicator for the strength of the exchange interaction. The driving force for the $g$-factor enhancement is a Hund's rule behavior for the occupation of spin-split Landau levels that lowers the many-particle ground state energy by arranging as many spins in parallel as possible. By increasing the total number of electrons and total number of available states per LL, it can therefore be expected that the exchange-enhanced spin gap should increase as well. In contrast to the dependence on the magnetic field, an increase of the total number of states by simply increasing the system size at constant magnetic field shows a clear saturation behavior above a lateral system size of 1000 nm. The importance of this result is underlined by an extended introduction, which demonstrates the permanent dominance of many-body interactions in all transport regimes of the IQHE. A modeling of IQHE systems therefore has to include many-body interactions, and our results open a pathway towards many-body modeling of quantum Hall systems of macroscopic size.

Fully self-consistent calculation of β-decay half-lives within Skyrme energy density functional * *Supported by the National Natural Science Foundation of China (11975096, 11805280, 11775014, 11635003, 11161130520), National Basic Research Program of China (2010CB832903), European Commission's 7th Framework Programme (Fp7-PEOPLE-2010-IRSES) under Grant Agreement Project (269131), and Continuous Basic Scientific Research Project (WDJC-2019-13)

-decay half-lives of some magic and semi-magic nuclei have been studied in a fully self-consistent Skyrme Hartree-Fock (HF) plus charge-exchange random phase approximation (RPA). The self-consistency is addressed, in that the same Skyrme energy density functional is adopted in the calculation of ground states and Gamow-Teller excited states. First, the impact of terms on the -decay half-lives is investigated by using the SGII interaction, revealing a large influence. Subsequently, numerical calculations are performed for the selected nuclei with Skyrme energy density functionals SGII, LNS, SKX, and SAMi. Finally, comparisons to available experimental data and predictions of different theoretical models are discussed.

Bosonization of the Q=0 continuum of Dirac fermions

We develop a bosonization formalism that captures non-perturbatively the interaction effects on the $\mathbf{Q}=0$ continuum of excitations of nodal fermions above one dimension. Our approach is a natural extension of the classic bosonization scheme for higher dimensional Fermi surfaces to include the $\mathbf{Q}=0$ neutral excitations that would be absent in a single-band system. The problem is reduced to solving a boson bilinear Hamiltonian. We establish a rigorous microscopic footing for this approach by showing that the solution of such boson bilinear Hamiltonian is exactly equivalent to performing the infinite sum of Feynman diagrams associated with the Kadanoff-Baym particle-hole propagator that arises from the self-consistent Hartree-Fock approximation to the single particle Green's function. We apply this machinery to compute the interaction corrections to the optical conductivity of 2D Dirac Fermions with Coulomb interactions reproducing the results of perturbative renormalization group at weak coupling and extending them to the strong coupling regime.

Time-reversal symmetry breaking versus chiral symmetry breaking in twisted bilayer graphene

By applying a self-consistent Hartree-Fock approximation, we show that the mechanism of dynamical symmetry breaking can account for the insulating phase that develops about the charge neutrality point of twisted bilayer graphene around the magic angle. ($i$) If the Coulomb interaction is screened by metallic gates, the opening of a gap between the lowest-energy valence and conduction bands proceeds through the breakdown of chiral symmetry at strong coupling. Increasing the dielectric screening, however, we find a critical coupling at which chiral symmetry breaking is suppressed, triggering a very strong signal for time-reversal symmetry breaking with Haldane mass. ($ii$) If the long-range tail of the Coulomb interaction is not screened, we see the appearance of yet a different dominant pattern at strong coupling, which is characterized by breaking the time-reversal invariance but with opposite flux in the two sublattices of each carbon layer, with the consequent valley symmetry breaking. In this case a gap is also opened in the Dirac cones, but superposed to the splitting of the degeneracy of the low-energy bands at the $K$ points of the moir\'e Brillouin zone.

Quenching factor of Gamow-Teller and spin dipole giant resonances

Gamow-Teller (GT) and spin-dipole (SD) strength distributions of four doubly magic nuclei $^{48}$Ca, $^{90}$Zr, $^{132}$Sn and $^{208}$Pb are studied by the self-consistent Hartree-Fock plus random phase approximation (RPA) method. The Skyrme forces SAMi and SAMi-T without/with tensor interactions are adopted in our calculations. The calculated strengths are compared with available experimental data. The RPA results of GT and SD strengths of all four nuclei show fine agreement with observed GT and SD resonances in energy. A small GT peak below the main GT resonance is better described by the Skyrme interaction SAMi-T with the tensor terms. The quenching factors for GT and SD are extracted from the comparisons between RPA results and experimental strengths. It is pointed out that the quenching effect on experimental SD peaks is somewhat modest compared with that on GT peaks in the four nuclei.

Spin transport in an electrically driven magnon gas near Bose-Einstein condensation: Hartree-Fock-Keldysh theory

An easy-plane ferromagnetic insulator in a uniform external magnetic field and in contact with a phonon bath and a normal metal bath is studied theoretically in the presence of dc spin current injection via the spin Hall effect in the metal. The Keldysh path integral formalism is used to model the magnon gas driven into a nonequilibrium steady state by mismatched bath temperatures and/or electrical injection, and we analyze the magnon system in the normal (uncondensed) state, but close to the instability to Bose-Einstein condensation (BEC), within the self-consistent Hartree-Fock approximation. We find that the steady state magnon distribution function generally has a non-thermal form that cannot be described by a single effective chemical potential and effective temperature. We also show that the BEC instability in the electrically-driven magnon system is signaled by a sign change in the imaginary part of the poles for long-wavelength magnon modes and by the divergence of the nonequilibrium magnon distribution function. In the presence of two bath temperatures, we find that the correlation length of the superfluid order parameter fluctuations exhibits nontrivial finite temperature crossover behaviors that are richer than the standard crossover behaviors obtained for the vacuum-superfluid transition in an equilibrium dilute Bose gas. We study the consequences of these thermal crossovers on the magnon spin conductivity and obtain an inverse square-root divergence in the spin conductivity in the vicinity of the electrically-induced BEC instability. A spintronics device capable of testing our spin transport predictions is discussed.

Dipole moments in strongly disordered electron glass

We study the effect of long-range Coulomb interaction on the dipole matrix elements in strongly disordered electron glass. The average dipole matrix element is computed numerically by a self-consistent Hartree-Fock approximation. For distances greater than the localization length the majority of dipole matrix elements decay exponentially with a dipole size. Mott states give rise to dipoles with a size much longer than the localization length.

On the ground-state energy of a finite inhomogeneous degenerate Bose gas

The ground-state energy of a finite inhomogeneous system of bosons located in a scalar external field was found within the framework of the self-consistent Hartree-Fock approximation, on the basis of the representation of the second quantization without using the formalism of anomalous averages. The ground-state wave function corresponds to the stationary Gross-Pitaevskii equation for the Bose-Einstein condensate wave function. It was shown that the ground-state energy can be found using the energy determined by the stationary Gross-Pitaevskii equation only for a system that satisfies the thermodynamic limit.

Manifestation of many-body interactions in the integer quantum Hall effect regime

We use the self-consistent Hartree-Fock approximation for numerically addressing the integer quantum Hall (IQH) regime in terms of many-body physics at higher Landau levels (LL). The results exhibit a strong tendency to avoid the simultaneous existence of partly filled spin-up and spin-down LLs. Partly filled LLs appear as a mixture of coexisting regions of full and empty LLs. We obtain edge stripes with approximately constant filling factor $\nu$ close to half-odd filling at the boundaries between the regions of full and empty LLs, which we explain in terms of the $g$-factor enhancement as a function of a locally varying $\nu$ across the compressible stripes.The many-particle interactions follow a behaviour as it would result from applying Hund's rule for the occupation of the spin split LLs. The screening of the disorder and edge potential appears significantly reduced as compared to screening based on a Thomas-Fermi approximation. For addressing carrier transport, we use a non-equilibrium network model (NNM) that handles the lateral distribution of the experimentally injected non-equilibrium chemical potentials $\mu$.

Existence problem of proton semi-bubble structure in the 21 + state of 34Si

The fully self-consistent Hartree-Fock (HF) plus random phase approximation (RPA) based on Skyrme-type interaction is used to study the existence problem of proton semi-bubble structure in the $2_1^+$ state of $^{34}$Si. The experimental excitation energy and the B(E2) strength of the $2_1^+$ state in $^{34}$Si can be reproduced quite well. The tensor effect is also studied. It is shown that the tensor interaction has a notable impact on the excitation energy of the $2_1^+$ state and a small effect on the B(E2) value. Besides, its effect on the density distributions in the ground and $2_1^+$ state of $^{34}$Si is negligible. Our present results with T36 and T44 show that the $2_1^+$ state of $^{34}$Si is mainly caused by proton transiton from $\pi 1d_{5/2}$ orbit to $\pi 2s_{1/2}$ orbit, and the existence of a proton semi-bubble structure in this state is very unlikely.

On the ground state energy of the inhomogeneous Bose gas

Within the self-consistent Hartree–Fock approximation, an explicit in this approximation expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose–Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose–Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross–Pitaevski approach, is found. The obtained form of kinetic energy is valid beyond the Hartree–Fock approximation and can be applied for arbitrary strong interparticle interaction.

Phonon induced optical gain in a current carrying two-level quantum dot

In this work we consider a current carrying two level quantum dot(QD) that is coupled to a single mode phonon bath. Using self-consistent Hartree-Fock approximation, we obtain the I-V curve of QD. By considering the linear response of our system to an incoming classical light, we see that depending on the parametric regime, the system could have weak or strong light absorption or may even show lasing. This lasing occurs at high enough bias voltages and is explained by a population inversion considering side bands, while the total electron population in the higher level is less than the lower one. The frequency at which we have the most significant lasing depends on the level spacing and phonon frequency and not on the electron-phonon coupling strength.

On statistical theory of rarefied gas in the Coulomb model of matter. Particle identity and effective interaction potential of initial atoms

The free energy of a Coulomb system of interacting electrons and nuclei is studied in the adiabatic approximation for nuclei with allowance for particle identity effects. An explicit expression is found for the effective short-range interaction potential of “identical” initial atoms in the self-consistent Hartree–Fock approximation for the electron subsystem and the first-order perturbation theory in a small parameter defined via the ratio of the atomic size to the mean distance between initial atoms.

Another approach for obtaining the excitation spectra in degenerate Bose gases with delta-shaped interaction potentials

A formalism is proposed for describing an equilibrium Bose gas based on consideration of a macroscopic but finite volume. The presence of a Bose-Einstein condensate is taken into account through an off-diagonal long-range order in the transition to the thermodynamic limit. On this basis, degenerate Bose gases with delta-shaped interaction potentials are described in terms of a self-consistent Hartree-Fock approximation. Explicit expressions are obtained for the energy spectra of single-particle and collective excitations.

On the thermodynamics of degenerate Bose gas with delta-shaped interaction potential

Within the self-consistent Hartree–Fock approximation, the equilibrium weakly nonideal Bose gas with a delta-shaped interaction potential in the presence of the Bose–Einstein condensate is considered without using quasi-averages. On this basis, using the virial theorem and diagram techniques of the perturbation theory for the equilibrium system in a macroscopic volume, the equation of state providing the isothermal compressibility finiteness, including the Bose–Einstein condensate domain of existence, is obtained.

On the Ground-State Energy and Local Pressure of an Inhomogeneous Bose Gas

Within the self-consistent Hartree-Fock approximation, an explicit expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose-Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose-Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross-Pitaevski approach, is found. This form is valid beyond the Hartree-Fock approximation and can be applied for arbitrary strong interparticle interaction.