The action of the long-range residual force on the expectation value of observables in the nuclear ground states is evaluated by finding optimal values for the coefficients of the canonical transformation which connects the phonon vacuum state with the independent-particle ground-state using the two-level Lipkin-Meshkov-Glick (LMG) model. After estimating the improvements over the predictions of the independent-particle approximation we compare the ground and first excited states wave functions, obtained using the presented approach, with those, obtained using the conventional random phase approximation (RPA) and its extended version. The problem with overbinding of the nuclear ground state calculated using the RPA is shown to be removed if one sticks to the prescriptions of the present approach. The reason being that the latter conforms to the original variational formulation.The action of the long-range residual force on the expectation value of observables in the nuclear ground states is evaluated by finding optimal values for the coefficients of the canonical transformation which connects the phonon vacuum state with the independent-particle ground-state using the two-level Lipkin-Meshkov-Glick (LMG) model. After estimating the improvements over the predictions of the independent-particle approximation we compare the ground and first excited states wave functions, obtained using the presented approach, with those, obtained using the conventional random phase approximation (RPA) and its extended version. The problem with overbinding of the nuclear ground state calculated using the RPA is shown to be removed if one sticks to the prescriptions of the present approach. The reason being that the latter conforms to the original variational formulation.

Journal of Physics Conference SeriesSep 10, 20140 citations

Improved treatment of ground-state correlations: Modified random phase approximation

Nguyen Dinh Dang, Vladimir Zelevinsky

A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.

Physical Review CNov 20, 200138 citations

Ground-state correlations and restoration of broken symmetry to nuclear mean field theory

Abraham Klein, Niels R. Walet, G. Do Dang

We reconsider the long-standing problem that the ground-state correlations predicted by the standard quasiboson (random phase) approximation are too large by a factor of two in the limit of weak residual interaction, and are thus inconsistent with the Pauli exclusion principle. We solve this problem by noting that for the derivation of the equations of the random phase approximation (RPA), which determine a set of matrix elements and excitation energies, it is unnecessary to assume that the ground state is the vacuum of a set of boson excitations, whereas in the faculty calculation of the ground-state correlation energy, this assumption is made. For the study of excitation energies and the restoration of symmetries broken by an initial mean-field solution, we apply a symmetry-preserving form of the equations of motion for fermion particle-hole excitation operators rather than for bosons. We illustrate this for the case of translations. We then describe a separate calculation of the ground-state energy that is consistent with the Pauli principle, but can nevertheless be evaluated in terms of the solutions of the RPA equations of motion. In contrast to some earlier work, we treat direct and exchange terms on an equal footing throughout.We reconsider the long-standing problem that the ground-state correlations predicted by the standard quasiboson (random phase) approximation are too large by a factor of two in the limit of weak residual interaction, and are thus inconsistent with the Pauli exclusion principle. We solve this problem by noting that for the derivation of the equations of the random phase approximation (RPA), which determine a set of matrix elements and excitation energies, it is unnecessary to assume that the ground state is the vacuum of a set of boson excitations, whereas in the faculty calculation of the ground-state correlation energy, this assumption is made. For the study of excitation energies and the restoration of symmetries broken by an initial mean-field solution, we apply a symmetry-preserving form of the equations of motion for fermion particle-hole excitation operators rather than for bosons. We illustrate this for the case of translations. We then describe a separate calculation of the ground-state energy that is consistent with the Pauli principle, but can nevertheless be evaluated in terms of the solutions of the RPA equations of motion. In contrast to some earlier work, we treat direct and exchange terms on an equal footing throughout.

Nuclear Physics ADec 1, 199123 citations

On the interdependence between ground and one-phonon RPA states

Michelangelo Sambataro, J. Suhonen

Working within a schematic model and in a boson formalism, we provide a descrip tion of the ground and first-excited states of a nuclear system in terms of two independent phonon operators. The description reveals itself quite effective in reproducing the energies of the states. A comparison between these two phonon operators allows us to localize a region where a correct description of these states is compatible with the Random Phase Approximation (RPA) requirement of a single phonon. This region is found to coincide approximately with that spanned by the standard RPA. Outside this region, the two phonons start differing rapidly therefore making the application of the RPA scheme impossible. This allows a better understanding of recently performed higher-order RPA calculations.Working within a schematic model and in a boson formalism, we provide a descrip tion of the ground and first-excited states of a nuclear system in terms of two independent phonon operators. The description reveals itself quite effective in reproducing the energies of the states. A comparison between these two phonon operators allows us to localize a region where a correct description of these states is compatible with the Random Phase Approximation (RPA) requirement of a single phonon. This region is found to coincide approximately with that spanned by the standard RPA. Outside this region, the two phonons start differing rapidly therefore making the application of the RPA scheme impossible. This allows a better understanding of recently performed higher-order RPA calculations.

Europhysics Letters (EPL)Oct 15, 19982 citations

Test of the proton-neutron random-phase approximation method within an extended Lipkin-type model

Sabin Stoica, I. Mihut, J. Suhonen

An extended Lipkin-Meshkov-Glick model for testing the proton-neutron random-phase approximation $(pn\mathrm{RPA})$ method is developed, taking into account explicitly proton and neutron degrees of freedom. Besides the proton and neutron single-particle terms two types of residual proton-neutron interactions, one simulating a particle-particle and the other a particle-hole interaction, are included in the model Hamiltonian so that the model is exactly solvable in an isospin $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ basis. The behavior of the first excited (collective) state obtained by (i) exact diagonalization of the Hamiltonian matrix and (ii) with the $\mathrm{pn}\mathrm{RPA}$ is studied as a function of the model parameters and the two results are compared with each other. Furthermore, charge-changing operators simulating nuclear beta decay and their action on eigenfunctions of the model Hamiltonian are defined and transition amplitudes of them are calculated using exact, the Tamm-Dancoff, and $\mathrm{pn}\mathrm{RPA}$ eigenfunctions.An extended Lipkin-Meshkov-Glick model for testing the proton-neutron random-phase approximation $(pn\mathrm{RPA})$ method is developed, taking into account explicitly proton and neutron degrees of freedom. Besides the proton and neutron single-particle terms two types of residual proton-neutron interactions, one simulating a particle-particle and the other a particle-hole interaction, are included in the model Hamiltonian so that the model is exactly solvable in an isospin $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ basis. The behavior of the first excited (collective) state obtained by (i) exact diagonalization of the Hamiltonian matrix and (ii) with the $\mathrm{pn}\mathrm{RPA}$ is studied as a function of the model parameters and the two results are compared with each other. Furthermore, charge-changing operators simulating nuclear beta decay and their action on eigenfunctions of the model Hamiltonian are defined and transition amplitudes of them are calculated using exact, the Tamm-Dancoff, and $\mathrm{pn}\mathrm{RPA}$ eigenfunctions.

Physical Review CJun 1, 20015 citations

Scalar ground-state observables in the random phase approximation

Calvin W. Johnson, Ionel Stetcu

We calculate the ground-state expectation value of scalar observables in the matrix formulation of the random phase approximation (RPA). Our expression, derived using the quasiboson approximation, is a straightforward generalization of the RPA correlation energy. We test the reliability of our expression by comparing against full diagonalization in $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces. In general the RPA values are an improvement over mean-field (Hartree-Fock) results, but are not always consistent with shell-model results. We also consider exact symmetries broken in the mean-field state and whether or not they are restored in RPA.We calculate the ground-state expectation value of scalar observables in the matrix formulation of the random phase approximation (RPA). Our expression, derived using the quasiboson approximation, is a straightforward generalization of the RPA correlation energy. We test the reliability of our expression by comparing against full diagonalization in $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces. In general the RPA values are an improvement over mean-field (Hartree-Fock) results, but are not always consistent with shell-model results. We also consider exact symmetries broken in the mean-field state and whether or not they are restored in RPA.

arXiv:nucl-th/0208025Physical Review CDec 4, 20027 citations

An extended Lipkin-Meshkov-Glick model for testing the proton-neutron random-phase approximation

(pn\mathrm{RPA})

method is developed, taking into account explicitly proton and neutron degrees of freedom. Besides the proton and neutron single-particle terms two types of residual proton-neutron interactions, one simulating a particle-particle and the other a particle-hole interaction, are included in the model Hamiltonian so that the model is exactly solvable in an isospin $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ basis. The behavior of the first excited (collective) state obtained by (i) exact diagonalization of the Hamiltonian matrix and (ii) with the

\mathrm{pn}\mathrm{RPA}

is studied as a function of the model parameters and the two results are compared with each other. Furthermore, charge-changing operators simulating nuclear beta decay and their action on eigenfunctions of the model Hamiltonian are defined and transition amplitudes of them are calculated using exact, the Tamm-Dancoff, and

\mathrm{pn}\mathrm{RPA}

eigenfunctions.