Time-dependent Hartree-Fock Approximation Research Articles

Dynamics of many-body delocalization in the time-dependent Hartree–Fock approximation

We explore dynamics of disordered and quasi-periodic interacting lattice models using a self-consistent time-dependent Hartree–Fock (TDHF) approximation, accessing both large systems (up to L=400 sites) and very long times (up to t=105). We find that, in the t→∞ limit, the many-body localization (MBL) is always destroyed within the TDHF approximation. At the same time, this approximation provides important information on the long-time character of dynamics in the ergodic side of the MBL transition. Specifically, for one-dimensional (1D) disordered chains, we find slow power-law transport up to the longest times, supporting the rare-region (Griffiths) picture. The information on this subdiffusive dynamics is obtained by the analysis of three different observables – temporal decay ∼t−β of real-space and energy-space imbalances as well as domain wall melting – which all yield consistent results. For two-dimensional (2D) systems, the decay is faster than a power law, in consistency with theoretical predictions that β grows as logt for the decay governed by rare regions. At longest times and moderately strong disorder, β approaches the limiting value β=1 corresponding to 2D diffusion. In quasi-periodic (Aubry–André) 1D systems, where rare regions are absent, we find considerably faster decay that reaches the ballistic value β=1, which provides further support to the Griffiths picture of the slow transport in random systems.

Dephasing in strongly disordered interacting quantum wires

Many-body localization is a fascinating theoretical concept describing the intricate interplay of quantum interference, i.e., localization, with many-body interaction-induced dephasing. Numerous computational tests and also several experiments have been put forward to support the basic concept. Typically, averages of time-dependent global observables have been considered, such as the charge imbalance. We here investigate within the disordered spinless Hubbard ($t\text{\ensuremath{-}}V$) model how dephasing manifests in time-dependent variances of observables. We find that after quenching a N\'eel state the local charge density exhibits strong temporal fluctuations with a damping that is sensitive to disorder $W$: variances decay in a power-law manner, ${t}^{\ensuremath{-}\ensuremath{\zeta}}$, with an exponent $\ensuremath{\zeta}(W)$ strongly varying with $W$. A heuristic argument suggests the form $\ensuremath{\zeta}\ensuremath{\approx}\ensuremath{\alpha}(W){\ensuremath{\xi}}_{\text{sp}}$, where ${\ensuremath{\xi}}_{\text{sp}}(W)$ denotes the noninteracting localization length and $\ensuremath{\alpha}(W)$ characterizes the multifractal structure of the dynamically active volume fraction of the many-body Hilbert space. In order to elucidate correlations underlying the damping mechanism, exact computations are compared with results from the time-dependent Hartree-Fock approximation. Implications for experimentally relevant observables, such as the imbalance, will be discussed.

Inverse $N$-body scattering with the time-dependent hartree-fock approximation

<p style='text-indent:20px;'>We consider an inverse <inline-formula><tex-math id="M1">\begin{document}$N$\end{document}</tex-math></inline-formula>-body scattering problem of determining two potentials—an external potential acting on all particles and a pair interaction potential—from the scattering particles. This paper finds that the time-dependent Hartree-Fock approximation for a three-dimensional inverse <inline-formula><tex-math id="M2">\begin{document}$N$\end{document}</tex-math></inline-formula>-body scattering in quantum mechanics enables us to recover the two potentials from the scattering states with high-velocity initial states. The main ingredient of mathematical analysis in this paper is based on the asymptotic analysis of the scattering operator defined in terms of a scattering solution to the Hartree-Fock equation at high energies. We show that the leading part of the asymptotic expansion of the scattering operator uniquely reconstructs the Fourier transform of the pair interaction, and the second term of the expansion uniquely reconstructs the <inline-formula><tex-math id="M3">\begin{document}$X$\end{document}</tex-math></inline-formula>-ray transform of the external potential.

Third-order nonlinear effects on femtosecond x-ray-absorption near-edge spectra

Reduced density matrix equations combined with a cluster molecular-orbital method are formulated to describe resonant core-valence electron excitation by an intense femtosecond x-ray pulse. The electron-hole interaction is treated by the time-dependent unrestricted Hartree-Fock approximation supplemented by a screening correction in the Born approximation. Nonlinear $K$-shell absorption spectra of metallic copper are thereby computed. Numerical results indicate that, in the third-order nonlinear regime, where the fraction of excited electrons increases in proportion to the x-ray intensity, the absorption spectra above the $K$ edge undergo shifts toward high-energy side. The origin of this third-order nonlinearity can be traced to a negative renormalization of the core orbital energy due to strong electron-hole attraction, which offers a sharp contrast to the usual optical nonlinearity due simply to the increment of upper-state populations. For higher intensities, broadening of the absorption edge and saturation of absorption are predicted. These trends are qualitatively consistent with nonlinear spectroscopy experiments using the SACLA x-ray free-electron laser.

Timescales of Quantum Equilibration, Dissipation and Fluctuation in Nuclear Collisions.

Understanding the dynamics of equilibration processes in quantum systems as well as their interplay with dissipation and fluctuation is a major challenge in quantum many-body theory. The timescales of such processes are investigated in collisions of atomic nuclei using fully microscopic approaches. Results from time-dependent Hartree-Fock and time-dependent random-phase approximation calculations are compared for 13 systems over a broad range of energies. The timescale for full mass equilibration (∼2×10^{-20} s) is found to be much larger than timescales for neutron-to-proton equilibration, kinetic energy, and angular momentum dissipations which are on the order of 10^{-21} s. Fluctuations of mass numbers in the fragments and correlations between their neutron and proton numbers build up within only a few 10^{-21} s. This indicates that dissipation is basically not impacted by mass equilibration, but is mostly driven by the exchange of nucleons between the fragments.

Same Charge Polaron and Bipolaron Scattering on Conducting Polymers

We study the dynamics of scattering of two positive polarons moving toward each other with parallel spins and of a polaron and a bipolaron both with positive charges and different velocities, in single chains of cis-polyacetylene and polyparaphenylene. We use the Su-Schrieffer-Heeger model with Hubbard extensions solved within of the time-dependent Hartree-Fock approximation to account for electron-electron interactions, developed over a hexagonal lattice. The main results are elastic scattering in most cases. This behavior changes to transmission of the polaron through the bipolaron when the ratio of their velocities in the same direction is in the range of 3:1 to 5:1 when no external electric field is present. The velocities ratio changes to of 6:1 to 8:1 under external electric field. This suggests that these systems may present an abrupt increase in the charge transport, under specific conditions, in an otherwise smoothly and monotonically increasing relation to the applied field. Their effective masses in these systems are also determined: the polaron effective mass is 4-8 times the rest mass of a free electron, and that of the bipolaron is 15-30 times the electron mass.

Efficient calculation of beyond RPA correlation energies in the dielectric matrix formalism.

We present efficient methods to calculate beyond random phase approximation (RPA) correlation energies for molecular systems with up to 500 atoms. To reduce the computational cost, we employ the resolution-of-the-identity and a double-Laplace transform of the non-interacting polarization propagator in conjunction with an atomic orbital formalism. Further improvements are achieved using integral screening and the introduction of Cholesky decomposed densities. Our methods are applicable to the dielectric matrix formalism of RPA including second-order screened exchange (RPA-SOSEX), the RPA electron-hole time-dependent Hartree-Fock (RPA-eh-TDHF) approximation, and RPA renormalized perturbation theory using an approximate exchange kernel (RPA-AXK). We give an application of our methodology by presenting RPA-SOSEX benchmark results for the L7 test set of large, dispersion dominated molecules, yielding a mean absolute error below 1 kcal/mol. The present work enables calculating beyond RPA correlation energies for significantly larger molecules than possible to date, thereby extending the applicability of these methods to a wider range of chemical systems.

Exploring Zeptosecond Quantum Equilibration Dynamics: From Deep-Inelastic to Fusion-Fission Outcomes in ^{58}Ni+^{60}Ni Reactions.

Energy dissipative processes play a key role in how quantum many-body systems dynamically evolve toward equilibrium. In closed quantum systems, such processes are attributed to the transfer of energy from collective motion to single-particle degrees of freedom; however, the quantum many-body dynamics of this evolutionary process is poorly understood. To explore energy dissipative phenomena and equilibration dynamics in one such system, an experimental investigation of deep-inelastic and fusion-fission outcomes in the ^{58}Ni+^{60}Ni reaction has been carried out. Experimental outcomes have been compared to theoretical predictions using time dependent Hartree-Fock and time dependent random phase approximation approaches, which, respectively, incorporate one-body energy dissipation and fluctuations. Excellent quantitative agreement has been found between experiment and calculations, indicating that microscopic models incorporating one-body dissipation and fluctuations provide a potential tool for exploring dissipation in low-energy heavy ion collisions.

Charge and pairing dynamics in the attractive Hubbard model: Mode coupling and the validity of linear-response theory

Pump-probe experiments have turned out as a powerful tool in order to study the dynamics of competing orders in a large variety of materials. The corresponding analysis of the data often relies on standard linear-response theory generalized to non-equilibrium situations. Here we examine the validity of such an approach within the attractive Hubbard model for which the dynamics of pairing and charge-density wave orders is computed using the time-dependent Hartree-Fock approximation (TDHF). Our calculations reveal that the `linear-response assumption' is justified for small to moderate non-equilibrium situations (i.e., pump pulses) when the symmetry of the pump-induced state differs from that of the external field. This is the case, when we consider the pairing response in a charge-ordered state or the charge-order response in a superconducting state. The situation is very different when the non-equilibrium state and the external probe field have the same symmetry. In this case, we observe significant changes of the response in magnitude but also due to mode coupling when moving away from an equilibrium state, indicating the failure of the linear-response assumption.

Polaron and bipolaron stability on paraphenylene polymers.

Polyparaphenylene is the prototypical conjugated polymer containing phenyl rings and its properties are good references for a family of derived polymers. We investigate the structure, stability, and dynamics of polarons and bipolarons in polyparaphenylene chains under an applied electric field. To do this, we use a bidimensional SSH Hamiltonian model with the Hubbard extension, i.e., with local and nearest-neighbor Coulomb interaction, which has been designed to work with general hexagonal lattices, from which polyparaphenylene can be seen as a prominent case. Using the time-dependent Hartree-Fock approximation, we calculate the structural characteristics, the maximum field strength, supported before polarons and bipolarons gets unstable, and the maximum velocity achieved by these charge carriers. We obtained the polaron and bipolaron terminal velocity to be 0.51 Å/fs and 1.15 Å/fs, respectively. The maximum field strength determined by our calculations is 0.54 mV/Å and 0.80 mV/Å, respectively. Our results are in good agreement with other theoretical methods and experiments.

Nonequilibrium itinerant-electron magnetism: A time-dependent mean-field theory

We study the dynamical magnetic susceptibility of a strongly correlated electronic system in the presence of a time-dependent hopping field, deriving a generalized Bethe-Salpeter equation which is valid also out of equilibrium. Focusing on the single-orbital Hubbard model within the time-dependent Hartree-Fock approximation, we solve the equation in the non-equilibrium adiabatic regime, obtaining a closed expression for the transverse magnetic susceptibility. From this, we provide a rigorous definition of non-equilibrium (time-dependent) magnon frequencies and exchange parameters, expressed in terms of non-equilibrium single-electron Green functions and self-energies. In the particular case of equilibrium, we recover previously known results.

Collective bulk and edge modes through the quantum phase transition in graphene atν=0

Undoped graphene in a strong, tilted magnetic field exhibits a radical change in conduction upon changing the tilt-angle, which can be attributed to a quantum phase transition from a canted antiferromagnetic (CAF) to a ferromagnetic (FM) bulk state at filling factor $\nu=0$. This behavior signifies a change in the nature of the collective ground state and excitations across the transition. Using the time-dependent Hartree-Fock approximation, we study the collective neutral (particle-hole) excitations in the two phases, both in the bulk and on the edge of the system. The CAF has gapless neutral modes in the bulk, whereas the FM state supports only gapped modes in its bulk. At the edge, however, only the FM state supports gapless charge-carrying states. Linear response functions are computed to elucidate their sensitivity to the various modes. The response functions demonstrate that the two phases can be distinguished by the evolution of a local charge pulse at the edge.

Nuclear Pasta at Finite Temperature with the Time-Dependent Hartree-Fock Approach

We present simulations of neutron-rich matter at sub-nuclear densities, like supernova matter. With the time-dependent Hartree-Fock approximation we can study the evolution of the system at temperatures of several MeV employing a full Skyrme interaction in a periodic three-dimensional grid [1].The initial state consists of α particles randomly distributed in space that have a Maxwell-Boltzmann distribution in momentum space. Adding a neutron background initialized with Fermi distributed plane waves the calculations reflect a reasonable approximation of astrophysical matter.The matter evolves into spherical, rod-like, connected rod-like and slab-like shapes. Further we observe gyroid-like structures, discussed e.g. in [2], which are formed spontaneously choosing a certain value of the simulation box length. The ρ-T-map of pasta shapes is basically consistent with the phase diagrams obtained from QMD calculations [3]. By an improved topological analysis based on Minkowski functionals [4], all observed pasta shapes can be uniquely identified by only two valuations, namely the Euler characteristic and the integral mean curvature.In addition we propose the variance in the cell-density distribution as a measure to distinguish pasta matter from uniform matter.

Kinetic energy estimates for the accuracy of the time-dependent Hartree–Fock approximation with Coulomb interaction

We study the time evolution of a system of N spinless fermions in R3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree–Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system.

Extension of the continuum time-dependent Hartree-Fock method to proton states.

This paper deals with the solution of the spherically symmetric time-dependent Hartree-Fock approximation applied to nuclear giant monopole resonances in the small amplitude regime. The problem is spatially unbounded as the resonance state is in the continuum. The practical requirement to perform the calculation in a finite-sized spatial region yields an artificial boundary, which is not present physically. The question of how to ensure the boundary does not interfere with the internal solution, while keeping the overall calculation time low is studied. Here we propose an absorbing boundary condition scheme to handle the conflict. The derivation, via a Laplace transform method, and implementation is described. An inverse Laplace transform required by the absorbing boundaries is calculated using a method of nonlinear least squares. The accuracy and efficiency of the scheme is tested and results presented to support the case that they are an effective way of handling the artificial boundary.

Time-dependent Hartree-Fock approach to nuclear “pasta” at finite temperature

We present simulations of neutron-rich matter at subnuclear densities, like supernova matter, with the time-dependent Hartree-Fock approximation at temperatures of several MeV. The initial state consists of $\ensuremath{\alpha}$ particles randomly distributed in space that have a Maxwell-Boltzmann distribution in momentum space. Adding a neutron background initialized with Fermi distributed plane waves the calculations reflect a reasonable approximation of astrophysical matter. This matter evolves into spherical, rod-like, and slab-like shapes and mixtures thereof. The simulations employ a full Skyrme interaction in a periodic three-dimensional grid. By an improved morphological analysis based on Minkowski functionals, all eight pasta shapes can be uniquely identified by the sign of only two valuations, namely the Euler characteristic and the integral mean curvature. In addition, we propose the variance in the cell density distribution as a measure to distinguish pasta matter from uniform matter.

Time-Dependent Hartree-Fock Approach to Nuclear Pasta at Finite Temperature

We present simulations of neutron-rich matter at subnuclear densities, like supernova matter, with the time-dependent Hartree-Fock approximation at temperatures of several MeV. The initial state consists of α particles randomly distributed in space that have a Maxwell-Boltzmann distribution in momentum space. Adding a neutron background initialized with Fermi distributed plane waves the calculations reflect a reasonable approximation of astrophysical matter. This matter evolves into spherical, rod-like, and slab-like shapes and mixtures thereof. The simulations employ a full Skyrme interaction in a periodic three-dimensional grid. By an improved morphological analysis based on Minkowski functionals, all eight pasta shapes can be uniquely identified by the sign of only two valuations, namely the Euler characteristic and the integral mean curvature.

Influence of Coulomb Blockade on Wave Packet Dynamics in Nanoscale Structures

Influence of Coulomb blockade on electron scattering by a quantum dot has been theoretically investigated using a multielectron wave packet simulation technique based on the time-dependent Hartree–Fock approximation. In our simulation, the bound states of electrons in the dot are self-consistently determined. We confirmed that Koopman's theorem keeps its validity only for weak Coulomb interactions. Moreover, we show that the maximum number of electrons trapped in the dot does depend on the strength of Coulomb interactions. Consequently, the transmission and reflection probabilities of an incident wave packet toward the dot are strongly influenced by the number of trapped electrons in the dot.

Continuum time-dependent Hartree-Fock method for giant resonances in spherical nuclei

This paper deals with the solution of the spherically symmetric time-dependent Hartree-Fock approximation applied in the case of nuclear giant monopole resonances in the small and large amplitude regimes. The problem is spatially unbounded as the resonance state is in the continuum. The practical requirement to perform the calculation in a finite-sized spatial region results in a difficulty with the spatial boundary conditions. Here we propose an absorbing boundary condition scheme to handle the conflict. The derivation, via a Laplace transform method, and implementation is described. The accuracy and efficiency of the scheme is tested and the results presented to support the case that they are a effective way of handling the artificial boundary.

Efficient method for computing the Thouless-Valatin inertia parameters

Starting from the adiabatic time-dependent Hartree-Fock approximation (ATDHF), we propose an efficient method to calculate the Thouless-Valatin moments of inertia for the nuclear system. The method is based on the rapid convergence of the expansion of the inertia matrix. The accuracy of the proposed method is verified in the rotational case by comparing the results with the exact Thouless-Valatin moments of inertia calculated using the self-consistent cranking model. The proposed method is computationally much more efficient than the full ATDHF calculation, yet it retains a high accuracy of the order of 1%.