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arXiv:nucl-th/0511043 · Physical Review C

Contour integral method for thermal and quantal fluctuations

K. Kaneko, M. Hasegawa

Published Dec 30, 2005• 10 citations
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Abstract

The partition function by means of the static path approximation (SPA) plus the random-phase approximation (RPA) treatment can be written as a contour integral form without solving the RPA equations for a separable interaction. This method is an efficient way to evaluate numerically the partition function for realistic calculations. As an illustration, we adopt the pairing model at finite temperature.

DOI
10.1103/physrevc.72.061306

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The partition function by means of the static path approximation (SPA) plus the random-phase approximation (RPA) treatment can be written as a contour integral form without solving the RPA equations for a separable interaction. This method is an efficient way to evaluate numerically the partition function for realistic calculations. As an illustration, we adopt the pairing model at finite temperature.
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.
We develop a scheme to exactly evaluate the correlation energy in the random-phase approximation, based on linear response theory [Y. R. Shimizu, J. D. Garrett, R. A. Broglia, M. Gallardo, and E. Vigezzi, Rev. Mod. Phys. 61, 131 (1989)]. It is demonstrated that our formula is equivalent to a contour integral representation recently proposed [F. Donau, D. Almehed, and R. G. Nazmitdinov, Phys. Rev. Lett. 83, 280 (1999)] being numerically more efficient for realistic calculations. Examples are presented for pairing correlations in rapidly rotating nuclei.
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.