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Register a new userSolving Dirac equations on a 3D lattice with inverse Hamiltonian and spectral methods
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A new method to solve the Dirac equation on a 3D lattice is proposed, in which the variational collapse problem is avoided by the inverse Hamiltonian method and the fermion doubling problem is avoided by performing spatial derivatives in momentum space with the help of the discrete Fourier transform, i.e., the spectral method. This method is demonstrated in solving the Dirac equation for a given spherical potential in 3D lattice space. In comparison with the results obtained by the shooting method, the differences in single particle energy are smaller than $10^{-4}$~MeV, and the densities are almost identical, which demonstrates the high accuracy of the present method. The results obtained by applying this method without any modification to solve the Dirac equations for an axial deformed, non-axial deformed, and octupole deformed potential are provided and discussed.
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We solve a singe-particle Dirac equation with Woods-Saxon potentials using an iterative method in the coordinate space representation. By maximizing the expectation value of the inverse of the Dirac Hamiltonian, this method avoids the variational collapse, in which an iterative solution dives into the Dirac sea. We demonstrate that this method works efficiently, reproducing the exact solutions of the Dirac equation.
An efficient method, preconditioned conjugate gradient method with a filtering function (PCG-F), is proposed for solving iteratively the Dirac equation in 3D lattice space for nuclear systems. The filtering function is adopted to avoid the variational collapsed problem and a momentum-dependent preconditioner is introduced to promote the efficiency of the iteration. The PCG-F method is demonstrated in solving the Dirac equation with given spherical and deformed Woods-Saxon potentials. The solutions given by the inverse Hamiltonian method in 3D lattice space and the shooting method in radial coordinate space are reproduced with a high accuracy. In comparison with the existing inverse Hamiltonian method, the present PCG-F method is much faster in the convergence of the iteration, in particular for deformed potentials. It may also provide a promising way to solve the relativistic Hartree-Bogoliubov equation iteratively in the future.
An efficient solution of the Dirac Hamiltonian flow equations has been proposed through a novel expandsion with the inverse of the Dirac effective mass. The efficiency and accuracy of this new expansion have been demonstrated by reducing a radial Dirac Hamiltonian with large scalar and vector potentials to two nonrelativistic Hamiltonians corresponding to particles and antiparticles, respectively. By solving the two nonrelativistic Hamiltonians, it is found that the exact solutions of the Dirac equation, for both particles and antiparticles, can be reproduced with a high accuracy up to only a few lowest order terms in the expansion. This could help compare and bridge the relativistic and nonrelativistic nuclear energy density functional theories in the future.
The stability of the linear chain structure of three $alpha$ clusters for $^{12}$C against the bending and fission is investigated in the cranking covariant density functional theory, in which the equation of motion is solved on a 3D lattice with the inverse Hamiltonian and the Fourier spectral methods. Starting from a twisted three $alpha$ initial configuration, it is found that the linear chain structure is stable when the rotational frequency is within the range of $sim$2.0 MeV to $sim$2.5 MeV. Beyond this range, the final states are not stable against fission. By examining the density distributions and the occupation of single-particle levels, however, these fissions are found to arise from the occupation of unphysical continuum with large angular momenta. To properly remove these unphysical continuum, a damping function for the cranking term is introduced. Eventually, the stable linear chain structure could survive up to the rotational frequency $sim$3.5 MeV, but the fission still occurs when the rotational frequency approaches to $sim$4.0 MeV.
The toroidal states in $^{28}$Si with spin extending to extremely high are investigated with the cranking covariant density functional theory on a 3D lattice. Thirteen toroidal states with spin $I$ ranging from 0 to 56$hbar$ are obtained, and their stabilities against particle emissions are studied by analyzing the density distributions and potentials. The excitation energies of the toroidal states at $I=28$, 36, 44$hbar$ reasonably reproduce the observed three resonances extracted from the 7-$alpha$ de-excitation of $^{28}$Si. The $alpha$ clustering of these toroidal states is supported by the $alpha$-localization function.
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