Time-dependent Hartree-Fock-Bogoliubov Research Articles

Quantal corrections to mean-field dynamics including pairing

Extending the stochastic mean-field model by including pairing, an approach is proposed for describing evolutions of complex many-body systems in terms of an ensemble of Time-Dependent Hartree-Fock Bogoliubov trajectories which is determined by incorporating fluctuations in the initial state. Non-linear evolution of the initial fluctuations provides an approximate description of quantal correlations and fluctuations of collective observables. Since the initial-state fluctuations break the particle-number symmetry, the dynamical description in which pairing correlations play a crucial role is greatly improved as compare to the mean-field evolution. The approach is illustrated for a system of particles governed by a pairing Hamiltonian.

Elementary Excitations of a Higgs–Yukawa System

This work investigates the physics of elementary excitations for the so-called relativistic quantum scalar plasma system, also known as the Higgs–Yukawa system. Following the Nemes–Piza–Kerman–Lin many-body procedure, the random-phase approximation (RPA) equations were obtained for this model by linearizing the time-dependent Hartree–Fock–Bogoliubov equations of motion around equilibrium. The resulting equations have a closed solution, from which the spectrum of excitation modes are studied. We show that the RPA oscillatory modes give the one-boson and two-fermion states of the theory. The results indicate the existence of bound states in certain regions in the phase diagram. Applying these results to recent Large Hadron Collider observations concerning the mass of the Higgs boson, we determine limits for the intensity of the coupling constant g of the Higgs–Yukawa model, in the RPA mean-field approximation, for three decay channels of the Higgs boson. Finally, we verify that, within our approximations, only Higgs bosons with masses larger than 190 GeV/\(c^2\) can decay into top quarks.

Landau Damping of Collective Mode in a Quasi-One-Dimensional Repulsive Bose—Einstein Condensate

We investigate the Landau damping of the collective mode in a quasi-one-dimensional repulsive Bose—Einstein condensate by using the self-consistent time-dependent Hartree—Fock—Bogoliubov approximation. We put forward a new method to calculate the Landau damping rate of the collective mode in the condensate and discuss the dependence of the Landau damping on temperature, on transverse trapping frequency, on atom number in the condensate, and on length of the system. Different from the usual calculation method for the three dimension system, our new calculation method is an interactive one by considering the practical relaxation of the elementary excitation. With little approximation, our theoretical calculation results agree with the experimental ones. Comparing with the usual calculation method, our theory is helpful to deduce the inter-particle interactions in damping phenomenon.

Linear responses in time-dependent Hartree-Fock-Bogoliubov method with Gogny interaction

A numerical method to integrate the time-dependent Hartree-Fock Bogoliubov (TDHFB) equations with Gogny interaction is proposed. The feasibility of the TDHFB code is illustrated by the conservation of the energy, particle numbers, and center-of-mass in the small amplitude vibrations of oxygen 20. The TDHFB code is applied to the isoscalar quadrupole and/or isovector dipole vibrations in the linear (small amplitude) region in oxygen isotopes (masses A = 18,20,22 and 24), titanium isotopes (A = 44,50,52 and 54), neon isotope (A = 26), and magnesium isotopes (A = 24 and 34). The isoscalar quadrupole and isovector dipole strength functions are calculated from the expectation values of the isoscalar quadrupole and isovector dipole moments.

Pairing dynamics in particle transport

We analyze the effect of pairing on particle transport in time-dependent theories based on the Hartree-Fock-Bogoliubov (HFB) or BCS approximations. The equations of motion for the HFB density matrices are unique and the theory respects the usual conservation laws defined by commutators of the conserved quantity with the Hamiltonian. In contrast, the theories based on the BCS approximation are more problematic. In the usual formulation of time-dependent Hartree-Fock $(\mathrm{TDHF})+\mathrm{BCS}$, the equation of continuity is violated and one sees unphysical oscillations in particle densities. This can be ameliorated by freezing the occupation numbers during the evolution in $\mathrm{TDHF}+\mathrm{BCS}$, but there are other problems with the BCS approximation that make it doubtful for reaction dynamics. We also compare different numerical implementations of the time-dependent HFB equations. The equations of motion for the $U$ and $V$ Bogoliubov transformations are not unique, but it appears that the usual formulation is also the most efficient. Finally, we compare the time-dependent HFB solutions with numerically exact solutions of the two-particle Schr\"odinger equation. Depending on the treatment of the initial state, the HFB dynamics produces a particle emission rate at short times similar to that of the Schr\"odinger equation. At long times, the total particle emission can be quite different, due to inherent mean-field approximation of the HFB theory.

Self-consistent pairing interaction and collective motion in nuclei: a semiclassical approach

The solutions of the semiclassical time–dependent Hartree–Fock-Bogoliubov equations of motion are studied in a linear approximation in which the pairing field is allowed to oscillate and to become complex. The pairing field fluctuations are derived from the self–consistent relation (the gap equation of the BCS type), while the static pairing field is approximated with a phenomenological constant Δ. The self–consistent pairing-field fluctuations introduce possibility of new collective modes of the system. We have found out the dispersion relation which determines possible collective modes of the system generated by the pairing interaction. The obvious solution of our dispersion relation at ω=0 corresponds to the Anderson–Goldstone–Nambu mode and is related to gauge symmetry. The solutions of dispersion relation have been studied in a simple model, in which nuclei are represented as homogeneous spheres of symmetric nuclear matter characterized by parameters (size, density, pairing gap) typical of heavy nuclei. We have found that the dispersion relation has approximate solution at ħω≈2Δ for the monopole channel. Our semiclassical approach is valid for both weak and strong pairing. Here we focus on weak pairing which is suitable for systems with size and pairing parameter appropriated to nuclei.

Linear response calculation using the canonical-basis TDHFB with a schematic pairing functional

A canonical-basis formulation of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory is obtained with an approximation that the pair potential is assumed to be diagonal in the time-dependent canonical basis. The canonical-basis formulation significantly reduces the computational cost. We apply the method to linear-response calculations for even-even nuclei. E1 strength distributions for proton-rich Mg isotopes are systematically calculated. The calculation suggests strong Landau damping of giant dipole resonance for drip-line nuclei.

Effects of self-consistency in semiclassical pairing theory

An extended Vlasov equation, derived from the time-dependent Hartree–Fock–Bogoliubov equation of motion, is used to study the effects of pairing on the nuclear density-density linear response function in semiclassical approximation. Within the approximations adopted here, the fluctuations of the pairing field are purely imaginary and play a crucial role in eliminating a spurious mode that gives fluctuations of the number of particle and in reproducing the correct value of the energy-weighted sum rule. The semiclassical approach seems to have some difficulty in describing pairing vibrations. Further work on this problem is needed.

Extended Fluid–Dynamics and Collective Motion of Two Trapped Fermion Species with Pairing Interactions

We extend our earlier fluid–dynamical description of fermion superfluids incorporating the particle energy flow together with the equation of motion for the internal kinetic energy of the pairs. The formal scheme combines a set of equations similar to those of classical hydrodynamics with the equations of motion for the anomalous density and for its related momentum density and kinetic energy density. This dynamical frame represents a second order truncation of an infinite hierarchy of equations of motion isomorphic to the full time dependent Hartree–Fock–Bogoliubov equations in coordinate representation. We analyze the equilibrium solutions and fluctuations for a homogeneous, unpolarized fermion system of two species, and show that the collective spectrum presents the well-known Anderson–Bogoliubov low energy mode of homogeneous superfluids and a pairing vibration near the gap energy.

Canonical-basis time-dependent Hartree-Fock-Bogoliubov theory and linear-response calculations

We present simple equations for a canonical-basis (Cb) formulation of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. The equations are obtained from the TDHFB theory with an approximation that the pair potential is assumed to be diagonal in the Cb. The Cb formulation significantly reduces the computational cost. We apply the method to linear-response calculations for even-even light nuclei and demonstrate its capability and accuracy by comparing our results with recent calculations of the quasiparticle random-phase approximation with Skyrme functionals. We show systematic studies of $E1$ strength distributions for Ne and Mg isotopes. The evolution of the low-lying pygmy strength seems to be determined by the interplay of several factors, which include the neutron excess, the separation energy, the neutron-shell effects, the deformation, and the pairing.

Nuclear scissors with pairing and continuity equation

The coupled dynamics of the isovector and isoscalar giant quadrupole resonances and low lying modes (including scissors) are studied with the help of the Wigner Function Moments (WFM) method generalized to take into account pair correlations. Equations of motion for collective variables are derived on the basis of the Time Dependent Hartree-Fock-Bogoliubov (TDHFB) equations in the harmonic oscillator model with quadrupole-quadrupole (QQ) residual interaction and a Gaussian pairing force. Special care is taken of the continuity equation.

Phase space moments with pair correlations on the example of nuclear scissors

The coupled dynamics of the isovector and isoscalar giant quadrupole resonances and low lying modes (including scissors) are studied with the help of the Wigner Function Moments (WFM) method generalized to take into account pair correlations. Equations of motion for relevant collective variables are derived on the basis of the Time Dependent Hartree-Fock-Bogoliubov (TDHFB) equations. Especial care is taken of the continuity equation. The inclusion of pair correlations leads also to the appearance of the isoscalar low lying mode.

Bragg spectroscopy of a strongly interacting Bose–Einstein condensate

We study Bragg spectroscopy of a strongly interacting Bose–Einstein condensate using time-dependent Hartree–Fock–Bogoliubov theory. We include approximatively the effect of the momentum-dependent scattering amplitude which is shown to be the dominant factor in determining the spectrum for large momentum Bragg scattering. The condensation of the Bragg scattered atoms is shown to alter significantly the observed excitation spectrum by creating a novel pairing channel of mobile pairs.

Kinetic equation for finite systems of fermions with pairing

The solutions of the Wigner-transformed time-dependent Hartree–Fock–Bogoliubov equations are studied in the constant-Δ approximation. This approximation is known to violate particle-number conservation. As a consequence, the density fluctuation and the longitudinal response function given by this approximation contain spurious contributions. A simple prescription for restoring both local and global particle-number conservation is proposed. Explicit expressions for the eigenfrequencies of the correlated systems and for the density response function are derived and it is shown that the semiclassical analogous of the quantum single-particle spectrum has an excitation gap of 2Δ, in agreement with the quantum result. The collective response is studied for a simplified form of the residual interaction.

Generalized coherent state representation of Bose-Einstein condensates

We show that the quantum many-body state of Bose-Einstein condensates (BEC) consistent with the time-dependent Hartree-Fock-Bogoliubov (TDHFB) equations is a generalized coherent state (GCS). At zero temerature, the non-condensate density and the anomalous non-condensate correlation are not independent, allowing us to elimiate one of the three variables in the TDHFB.

Mass parameters for large amplitude collective motion: A perturbative microscopic approach

The adiabatic limit of the time-dependent Hartree–Fock–Bogoliubov approximation is studied in the canonical basis for a rather wide class of collective modes. As a first step towards a fully selfconsistent solution, we study mass parameters where the effects due to the time-odd part of the mean fields are neglected, but where the static rearrangement of the generalized cranking operator is taken into account. Corresponding inertia parameters are compared with those resulting from currently used ansatz in microscopic accounts of the low energy nuclear collective dynamics. For the particular case of the axial quadrupole motion and using the Gogny D1S effective interaction, we show that our mass parameters exhibit strong variations as functions of the deformation whose origins are traced back to the rearrangement of both the Hartree–Fock and pairing fields. The physical consequences of these variations depend of the time scales of the studied collective motions and are illustrated in the context of superdeformed states in the A∼150 region and of the lifetime of a fission isomer in the actinide region.

Time-dependent Hartree-Fock-Bogoliubov description of the pairing interaction

The time-dependent Hartree-Fock-Bogoliubov (TDHFB) method is applied to a system with a pairing interaction. A semiclassical prescription is used for the evaluation of the two-particle-transfer matrix elements. A comparison of matrix elements and excitation energies obtained in the TDHFB approach and the exact ones shows a very good agreement for large degeneracies.