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Revista Mexicana de Física

The self consistent RPA

J. Dukelsky, Peter Schuck

Published Jan 1, 1998• 0 citations
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The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.
Self-Consistent RPA is extended in a way so that it is compatible with a variational ansatz for the ground-state wave function as a fermionic many-body vacuum. Employing the usual equation-of-motion technique, we arrive at extended RPA equations of the Self-Consistent RPA structure. In principle the Pauli principle is, therefore, fully respected. However, the correlation functions entering the RPA matrix can only be obtained from a systematic expansion in powers of some combinations of RPA amplitudes. We demonstrate for a model case that this expansion may converge rapidly.
The self-consistent random-phase approximation (SCRPA) is extended to nonzero temperature. Within the SCRPA, the collective dynamics of nucleon pairs is built on the occupation numbers that deviate from the Fermi-Dirac distribution as a result of temperature-dependent correlations beyond the RPA. The equations for single-particle occupation numbers are derived using the method of the double-time Green's functions. The approach is applied to the Richardson model of pairing. The results of numerical calculations for energies of states and occupation numbers, as well as for the thermodynamic quantities such as the excitation energy and heat capacity of the system, are compared with those obtained within the conventional RPA and the perturbative thermal SCRPA and with the exact results. We find that, overall, the SCRPA offers a better agreement with the exact results.