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Physical Review C

Canonical treatment of fluctuations and random phase approximation correlations at finite temperature

N. Canosa, R. Rossignoli, Peter A. Ring

Published Jan 1, 1999• 11 citations
Publisher

Abstract

The static path plus random phase approximation (SPA+RPA) to the partition function is formulated exactly in a canonical ensemble, for the case of a density decomposition of an arbitrary two-body Hamiltonian. The canonical mean field plus RPA approach is also discussed. Numerical results are shown for a quadrupole interaction, where excellent agreement with the exact canonical results for a light nucleus is obtained. An effective canonical mean field plus RPA approach is in this case also developed, which avoids the sharp phase transition of the ordinary mean field and provides a reliable estimate of full SPA+RPA results.

DOI
10.1103/physrevc.59.185

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The static path plus random phase approximation (SPA+RPA) to the partition function is formulated exactly in a canonical ensemble, for the case of a density decomposition of an arbitrary two-body Hamiltonian. The canonical mean field plus RPA approach is also discussed. Numerical results are shown for a quadrupole interaction, where excellent agreement with the exact canonical results for a light nucleus is obtained. An effective canonical mean field plus RPA approach is in this case also developed, which avoids the sharp phase transition of the ordinary mean field and provides a reliable estimate of full SPA+RPA results.
We discuss the application of the static path plus random-phase approximation (SPA+RPA)(\mathrm{SPA}+\mathrm{RPA})(SPA+RPA) and the ensuing mean field+RPA\text{field}+\mathrm{RPA}field+RPA treatment to the evaluation of entanglement in composite quantum systems at finite temperature. These methods involve just local diagonalizations and the determination of the generalized collective vibrational frequencies. As an illustration, we evaluate the pairwise entanglement in a fully connected XXZXXZXXZ chain of nnn spins at finite temperature in a transverse magnetic field bbb. It is shown that already the mean field+RPA\text{field}+\mathrm{RPA}field+RPA provides an accurate analytic description of the concurrence below the mean field critical region $(\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c})$, exact for large nnn, whereas the full SPA+RPA\mathrm{SPA}+\mathrm{RPA}SPA+RPA is able to improve results for finite systems in the critical region. It is proven as well, that for T>0T>0T>0 weak entanglement also arises when the ground state is separable $(\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c})$, with the limit temperature for pairwise entanglement exhibiting quite distinct regimes for $\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c}$ and $\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c}$.
A self-consistent version of the thermal random phase approximation (TSCRPA) is developed within the Matsubara Green’s function formalism. The TSCRPA is applied to the many-level pairing model, and the normal phase of the system is considered. The TSCRPA results are compared with the exact ones calculated for the grand canonical ensemble. Advantages of the TSCRPA over the thermal mean-field approximation and the standard thermal random phase approximation are demonstrated. Results for correlation functions, excitation energies, etc., as a function of temperature are presented.
Analogies and differences between the random phase approximation (RPA) and the Gaussian approximation at finite temperature are discussed. A comparison is made for Hartree and Hartree-Fock temperature dependent mean fields. Some ambiguities, in the definition of the Hartree mean field and in the relative correction, are analyzed. In a solvable model with a repulsive potential, the Gaussian approximation and the RPA, both built from the Hartree mean field, are found more reliable than the RPA built from a Hartree-Fock mean field.
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.