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Progress of Theoretical and Experimental Physics

Accuracy of pairing theory for a trapped magnetic dipole fermion gas

M. Tohyama

Published Feb 1, 2016• 0 citations
Publisher

Abstract

Pairing properties of a trapped fermion gas consisting of atoms with magnetic dipole moment are studied using the Hartree–Fock–Bogoliubov theory (HFB) and the results are compared with those obtained from the exact diagonalization approach (EDA). It is shown that HFB underestimates two-body correlations and gives quite different atom–atom correlations than the results in EDA. Extended time-dependent HFB (ETDHFB), which is derived from the truncation of equations of motion for reduced density matrices, is also applied and it is found that the difficulties of HFB can be remedied by ETDHFB.

DOI
10.1093/ptep/ptw001

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Pairing properties of a trapped fermion gas consisting of atoms with magnetic dipole moment are studied using the Hartree–Fock–Bogoliubov theory (HFB) and the results are compared with those obtained from the exact diagonalization approach (EDA). It is shown that HFB underestimates two-body correlations and gives quite different atom–atom correlations than the results in EDA. Extended time-dependent HFB (ETDHFB), which is derived from the truncation of equations of motion for reduced density matrices, is also applied and it is found that the difficulties of HFB can be remedied by ETDHFB.
An extension of the time-dependent Hartree-Fock-Bogoliubov theory (ETDHFB) which includes higher-order effects such as screening of the pairing correlation is proposed. ETDHFB is applied to a pairing model and a fermion system trapped in a harmonic potential to test its feasibility by comparison with exact solutions. With the use of perturbative expressions for the pairing tensor and the two-body density matrix derived from ETDHFB, the screening effect is investigated for atomic fermion systems and isotopes of tin nuclei. It is found that the screening effect on the pairing correlation is not significant.
The ground states and collective excitations of trapped Fermion gases consisting of atoms with magnetic dipole moment are studied using a time-dependent density-matrix approach. The advantages of the density-matrix approach are that one-body and two-body observables are directly calculated using one-body and two-body density matrices and that it has a clear relation to the Hartree-Fock (HF) and time-dependent HF theory. The HF calculations show the magnetization of the gases when the dipole-dipole interaction is strong. It is shown that the tensor properties of the dipole-dipole interaction are revealed in the excitation modes associated with spin degrees of freedom.
The extension of density functional theory (DFT) to include pairing correlations without formal violation of the particle-number conservation condition is described. This version of the theory can be considered as a foundation of the application of existing DFT plus pairing approaches to atoms, molecules, ultracooled and magnetically trapped atomic Fermi gases, and atomic nuclei where the number of particles is conserved exactly. The connection with Hartree-Fock-Bogoliubov (HFB) theory is discussed, and the method of quasilocal reduction of the nonlocal theory is also described. This quasilocal reduction allows equations of motion to be obtained which are much simpler for numerical solution than the equations corresponding to the nonlocal case. Our theory is applied to the study of some even Sn isotopes, and the results are compared with those obtained in the standard HFB theory and with the experimental ones.
Ground-state Hartree-Fock-Bogoliubov (HFB) theory is applied to imbalanced spin-1/21/21/2 one-dimensional (1D) Fermi systems that are spatially confined by either a harmonic or a hard-wall trapping potential. It has been hoped that such systems, which can be realized using ultracold atomic gases, would exhibit the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. The HFB formalism generalizes the standard Bogoliubov quasiparticle transformation by allowing for Cooper pairing to exist between all possible single-particle states and accounts for the effects of the inhomogeneous trapping potential as well as the mean-field Hartree potential. This provides an unbiased framework to describe inhomogeneous densities and pairing correlations in the FFLO state of a confined 1D gas. In a harmonic trap, numerical minimization of the HFB ground-state energy yields a spatially oscillating order parameter reminiscent of the FFLO state. However, we find that this state has almost no imprint in the local fermion densities (consistent with experiments that found no evidence of the FFLO phase). In contrast, for a hard-wall geometry, we find a strong signature of the spatial oscillations of the FFLO pairing amplitude reflected in the local in situ densities. In the hard-wall case, the excess spins are strongly localized near regions where there is a node in the pairing amplitude, creating an unmistakable crystalline modulation of the density.
Using the Kadanoff-Baym nonequilibrium Green's-function formalism, we derive the self-consistent Hartree-Fock-Bogoliubov (HFB) collisionless kinetic equations and the associated equation of motion for the condensate wave function for a trapped Bose-condensed gas. Our work generalizes earlier work by Kane and Kadanoff for a uniform Bose gas. We include the off-diagonal (anomalous) pair correlations, and thus we have to introduce an off-diagonal distribution function in addition to the normal (diagonal) distribution function. This results in two coupled kinetic equations. If the off-diagonal distribution function can be neglected as a higher-order contribution, we obtain the semiclassical kinetic equation recently used by Zaremba, Griffin, and Nikuni (based on the simpler Popov approximation). We discuss the static local equilibrium solution of our coupled HFB kinetic equations within the semiclassical approximation. We also verify that a solution is the rigid in-phase oscillation of the equilibrium condensate and noncondensate density profiles, oscillating with the trap frequency.
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 (TDHF)+BCS(\mathrm{TDHF})+\mathrm{BCS}(TDHF)+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 TDHF+BCS\mathrm{TDHF}+\mathrm{BCS}TDHF+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 UUU and VVV 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.