Time-dependent Hartree-Fock-Bogoliubov Research Articles

Time-dependent Hartree–Fock–Bogoliubov method for molecular systems: An alternative excited-state methodology including static electron correlation

TheHartree–Fock–Bogoliubov (HFB) method, an efficient way to treat static electron correlations, has been applied to molecular systems in the ground state. In this letter, we propose a time-dependent (TD) HFB method for treating the excited states of molecules with static correlations at a reasonable computational cost. To avoid electron number violations, we also consider the canonical basis variant of the TDHFB method proposed by Ebata et al. Results of calculations for the elongated hydrogen molecule reveal that the TDHFB method can partially incorporate two-electron excited determinants.

The time-dependent Hartree–Fock–Bogoliubov equations for Bosons

We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation.

Spin-isospin structure of the nuclear scissors mode

The fine structure of the scissors mode is investigated within the time dependent Hartree-Fock-Bogoliubov (TDHFB) approach. The solution of TDHFB equations by the Wigner function moments (WFM) method predicts a splitting of the scissors mode into three intermingled branches. Together with the conventional scissors mode two new modes arise due to spin degrees of freedom. They generate significant $M1$ strength below the conventional energy range. The results of calculations of scissors resonances in rare earths and actinides by WFM and quasiparticle-phonon nuclear model methods are compared with experimental data. A remarkable coherence of both methods together with experimental data is observed.

Real space density-constrained time-dependent Hartree-Fock-Bogoliubov method and pairing effects in the fusion of calcium isotopes

To describe the fusion cross sections with a method that includes the superfluidity and to understand the impact of pairing on both the fusion barrier and cross sections, we propose a real-space particle- and pairing-density-constrained scheme of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) method and a definition of TDHFB particle current. Using the Gogny-TDHFB code, this scheme is applied to investigate the fusion reactions of $^{40}\mathrm{Ca}+^{40}\mathrm{Ca}, ^{40}\mathrm{Ca}+^{44}\mathrm{Ca}$, and $^{40}\mathrm{Ca}+^{48}\mathrm{Ca}$ by comparison. The results show that this density-constrained TDHFB scheme is reliable for analysis of the collision processes involved with superfluid nuclei.

1S0 Pairing Gaps, Chemical Potentials and Entrainment Matrix in Superfluid Neutron-Star Cores for the Brussels–Montreal Functionals

Temperature and velocity-dependent 1S0 pairing gaps, chemical potentials and entrainment matrix in dense homogeneous neutron–proton superfluid mixtures constituting the outer core of neutron stars, are determined fully self-consistently by solving numerically the time-dependent Hartree–Fock–Bogoliubov equations over the whole range of temperatures and flow velocities for which superfluidity can exist. Calculations have been made for npeμ in beta-equilibrium using the Brussels–Montreal functional BSk24. The accuracy of various approximations is assessed and the physical meaning of the different velocities and momentum densities appearing in the theory is clarified. Together with the unified equation of state published earlier, the present results provide consistent microscopic inputs for modeling superfluid neutron-star cores.

Dynamical Hartree–Fock–Bogoliubov Approximation of Interacting Bosons

We consider a many-body Bosonic system with pairwise particle interaction given by $$N^{3\beta -1}v(N^\beta x)$$ where $$0<\beta <1$$ and v a non-negative spherically symmetric function. Our main result is the extension of the local-in-time Fock space approximation of the exact dynamics of squeezed states proved in Grillakis and Machedon (Commun Partial Differ Equ 42(1):24–67, 2017) for $$0<\beta <\frac{2}{3}$$ to a global-in-time approximation for $$0<\beta <1$$ . Our work can also be viewed as a generalization of the results in Boccato et al. (Ann Henri Poincaré 18(1):113–191, 2017) to a more general set of initial data that includes coherent states along with an improved error estimate. The key ingredients in establishing the Fock space approximation are the work of Grillakis and Machedon on the the local well-posedness theory (Grillakis and Machedon in Commun Partial Differ Equ 44(12):1431–1465, 2019), some recent established global estimate in Chong et al. (Commun Partial Differ Equ 56:1–41, 2021), and our quantitative results on the uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov system.

Neutron Tunneling: A New Mechanism to Power Explosive Phenomena in Neutron Stars, Magnetars, and Neutron Star Mergers

Abstract Neutron tunneling between neutron-rich nuclei in inhomogeneous dense matter encountered in neutron star crusts can release enormous energy on a short timescale to power explosive phenomena in neutron stars. In this work, we clarify aspects of this process that can occur in the outer regions of neutron stars when oscillations or cataclysmic events increase the ambient density. We use a time-dependent Hartree–Fock–Bogoliubov formalism to determine the rate of neutron diffusion and find that large amounts of energy can be released rapidly. The roles of nuclear binding, two-body interaction, and pairing in neutron diffusion times are investigated. We consider a one-dimensional quantum diffusion model and extend our analysis to study the impact of diffusion in three dimensions. We find that these novel neutron transfer reactions can generate energy in the amount of ≃ 1040–1044 erg under suitable conditions and assumptions.

Triplet of Nuclear Scissors Modes

The fine structure of the scissors mode is investigated within the Time Dependent Hartree-Fock-Bogoliubov (TDHFB) approach. The solution of TDHFB equations by the Wigner Function Moments (WFM) method predicts a splitting of the scissors mode into three intermingled branches. Together with the conventional scissors mode two new modes arise due to spin degrees of freedom. They generate significant $M1$ strength below the conventional energy range. The results of calculations of scissors resonances in Rare Earths and Actinides by WFM and QPNM methods are compared with experimental data. A remarkable coherence of both methods together with experimental data is observed.

Effects of quantum fluctuations on the dynamics of dipolar Bose polarons

We study the dynamics of dipolar Bose polarons in the presence of normal and anomalous fluctuations using the time-dependent Hartree–Fock-Bogoliubov theory. The density profiles of the condensate, the anomalous component and the impurity are deeply analyzed. The time evolution of the width and the center-of-mass oscillation of such quantities are also highlighted. We calculate corrections due to quantum fluctuations and impurity to the chemical potential and the radius of the condensate, and of the anomalous component, in the weak coupling regime using the Thomas–Fermi approximation. Effects of the dipole–dipole interaction, impurity–host interaction and the anomalous fluctuation on the width and on the breathing frequencies of the impurity are discussed by variational and numerical means.

Non-perturbative collective inertias for fission: A comparative study

The non-perturbative method to compute Adiabatic Time Dependent Hartree Fock Bogoliubov (ATDHFB) collective inertias is extended to the Generator Coordinate Method (GCM) in the Gaussian overlap approximation (GOA) including the case of density dependent forces. The two inertias schemes are computed along the fission path of the 234U and compared with the perturbative results. We find that the non-perturbative schemes predict very similar collective inertias with a much richer structure than the one predicted by perturbative calculations. Moreover, the non-perturbative inertias show an extraordinary similitude with the exact GCM inertias computed numerically from the energy overlap. These results indicate that the non-perturbative inertias provide the right structure as a function of the collective variable and only a phenomenological factor is required to mock up the exact GCM inertia, bringing new soundness to the microscopic description of fission.

Examining empirical evidence of the effect of superfluidity on the fusion barrier

Background: Several Time-Dependent Hartree-Fock-Bogoliubov (TDHFB) calculations predict that the super- fluidity enhances the fluctuations of the fusion barrier. This effect is not fully understood and not yet revealed experimentally. Purpose: The goal of this study is to investigate empirically the effect of the superfluidity on the fusion barrier width. Method: First, the local regression method is introduce and used to determine the barrier distribution more precisely. A second method that requires only the calculation of an integral of the cross section is developed to determine accurately the fluctuations of the barrier. A benchmark is done between this two methods and with the fitting method usually used. This integral method showing a better agreement in a test case, it is applied systematically in a selection of 115 fusion reactions. Results: The fluctuations of the barrier for superfluid systems are on average larger than for magic or semi-magic nuclei. This is due to the deformation effects and the effect of the superfluidity. To disentangle those two effects, we compare the experimental width to the width estimated from a model that takes into account the tunneling, the deformation and the vibration effect. The deviation of the experimental width from this theory for reaction between superfluid nuclei shows that the superfluidity enhance the fusion barrier width. Conclusions: This analysis shows that the predicted effect of the superfluidity on the width of the barrier is real and is of the order of 1 MeV.

Uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov equations in $$\mathbb {R}^{1+1}$$ R 1 + 1

In this article, we prove the global well-posedness of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) equations in $\mathbb{R}^{1+1}$ with two-body interaction potentials of the form $N^{-1}v_N(x) = N^{\beta-1} v(N^\beta x)$ where $v$ is a sufficiently regular radial function $v \in L^1(\mathbb{R})\cap C^\infty(\mathbb{R})$. In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon, Comm. PDEs., (2017), we are able to show for any scaling parameter $\beta>0$ the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of $N$, cf. (Bach et al. in arXiv:1602.05171).

Comparative study of the requantization of the time-dependent mean field for the dynamics of nuclear pairing

To describe quantal collective phenomena, it is useful to requantize the time-dependent mean-field dynamics. We study the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory for the two-level pairing Hamiltonian, and compare results of different quantization methods. The one constructing microscopic wave functions, using the TDHFB trajectories fulfilling the Einstein-Brillouin-Keller quantization condition, turns out to be the most accurate. The method is based on the stationary-phase approximation to the path integral. We also examine the performance of the collective model which assumes that the pairing gap parameter is the collective coordinate. The applicability of the collective model is limited for the nuclear pairing with a small number of single-particle levels, because the pairing gap parameter represents only a half of the pairing collective space.

Quantum and thermal fluctuations in two-component Bose gases

We study the effects of quantum and thermal fluctuations on Bose-Bose mixtures at finite temperature employing the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. The theory governs self-consistently the motion of the condensates, the noncondensates, and of the anomalous components on an equal footing. The finite temperature criterion for the phase separation is established. We numerically analyze the temperature dependence of different densities for both miscible and immiscible mixtures. We show that the degree of the overlap between the two condensates and the thermal clouds is lowered and the relative motion of the centers of mass of the condensed and thermal components is strongly damped due to the presence of the pair anomalous fluctuations. Our results are compared with previous theoretical and experimental findings. On the other hand, starting from our TDHFB equations, we develop a random-phase theory for the elementary excitations in a homogeneous mixture. We find that the normal and anomalous fluctuations may lead to enhance the excitations and the thermodynamics of the system.

Transfer probabilities for the reactions O14,20+O20 in terms of multiple time-dependent Hartree-Fock-Bogoliubov trajectories

The transfer reaction between two nuclei in the superfluid phase is studied with the Time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. In order to restore the symmetry of the relative gauge angle a set of independent TDHFB evolutions is done. Then the transfer probability is computed using a triple projection method. This method is first tested to determine the transfer probabilities on a toy model and compared to the exact solution. It is then applied to the reactions $^{20}$O+$^{20}$O and $^{14}$O+$^{20}$O in a realistic framework with a Gogny interaction.

Collective excitations of a dilute Bose gas at finite temperature: time dependent Hartree–Fock–Bogoliubov theory

Using the time-dependent Hartree–Fock–Bogoliubov approach, where the condensate is coupled with the thermal cloud and the anomalous density, we study the equilibrium and the dynamical properties of three-dimensional quantum-degenerate Bose gas at finite temperature. Effects of the anomalous correlations on the condensed fraction and the critical temperature are discussed. In uniform Bose gas, useful expressions for the Bogoliubov excitations spectrum, the first and second sound, the condensate depletion and the superfluid fraction are derived. Our results are tested by comparing the findings computed by quantum Monte Carlo simulations. We present also a systematic investigation of the collective modes of a Bose condensate confined in an external trap. Our predictions are in qualitative agreement with previous experimental and theoretical results. We show in particular that our theory is capable of explaining the so-called anomalous behavior of the mode.

Multidimensional Skyrme-density-functional Study of the Spontaneous Fission of $^{238}$U

We determined the spontaneous fission lifetime of 238U by a minimization of the action integral in a three-dimensional space of collective variables. Apart from the mass-distribution multipole moments Q20 (elongation) and Q30 (left–right asymmetry), we also considered the pairing-fluctuation parameter λ2 as a collective coordinate. The collective potential was obtained self-consistently using the Skyrme energy density functional SkM*. The inertia tensor was obtained within the nonperturbative cranking approximation to the adiabatic time-dependent Hartree–Fock–Bogoliubov approach. As a result, the pairing-fluctuation parameter λ2 allowed us to control the pairing gap along the fission path, which significantly changed the spontaneous fission lifetime.

Self-localized state and solitons in a Bose-Einstein-condensate–impurity mixture at finite temperature

We study the properties of a Bose-Einstein-condensate (BEC)--impurity mixture at finite temperature employing the time-dependent Hartree-Fock Bogoliubov (TDHFB) theory which is a set of coupled nonlinear equations of motion for the condensate and its normal and anomalous fluctuations on the one hand and for impurity on the other. The numerical solutions of these equations in the static quasi-one-dimensional regime show that the thermal cloud and the anomalous density are deformed as happens to the condensate and the impurity becomes less localized at nonzero temperatures. Effects of the BEC fluctuations on the self-trapping state are studied in homogeneous weakly interacting BEC-impurity at low temperature. The self-trapping threshold is also determined in such a system. The formation of solitons in the BEC-impurity mixture at finite temperature is investigated.

Quadrupole shape dynamics from the viewpoint of a theory of large-amplitude collective motion

Low-lying quadrupole shape dynamics is a typical manifestation of large-amplitude collective motion in finite nuclei. To describe the dynamics on a microscopic foundation, we have formulated a consistent scheme in which the Bohr collective Hamiltonian for the five-dimensional quadrupole shape variables is derived on the basis of the time-dependent Hartree–Fock–Bogoliubov theory. It enables us to incorporate the Thouless–Valatin effect on the shape inertial functions, which has been neglected in previous microscopic Bohr Hamiltonian approaches. Quantitative successes are illustrated for the low-lying spectra in Se, Mg and Cr, which display shape-coexistence, shape-mixing and shape-transitional behavior.

Time-dependent Hartree-Fock-Bogoliubov calculations using a Lagrange mesh with the Gogny interaction

A new numerical method of calculating the Hartree-Fock-Bogoliubov (HFB) and time-dependent HFB (TDHFB) with the Gogny interaction is proposed. The three-dimensional harmonic-oscillator (3DHO) basis functions are replaced by one-dimensional spatial grid points of Lagrange mesh plus two-dimensional harmonic-oscillator basis functions in the perpendicular plane to the direction of the spatial grid points [Lagrange mesh and harmonic oscillator (LMHO)]. By using the LMHO, the calculations of the HFB and TDHFB are carried out in the typical nuclei of ${}^{20}\mathrm{O}$ and ${}^{34}\mathrm{Mg}$ as an illustration of the feasibility of the TDHFB formulation with the LMHO. The strength functions of the quadrupole vibrations ($K=0$) are obtained and are compared with the ones calculated with the 3DHO.