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Physical Review C

Thermal response of states of approximate angular momentum in the finite temperature restricted Hartree-Fock approximation

R. M. Quick, H. G. Miller

Published Oct 1, 1986• 5 citations
Publisher

Abstract

The thermal response of states of approximate angular momentum in $^{24}\mathrm{Mg}$ with ${J}^{\ensuremath{\pi}}$${=8}^{+}$ has been calculated in the finite temperature restricted Hartree-Fock approximation. Full angular momentum projection has been circumvented by the use of constraints on the total angular momentum J${^}^{2}$ and the z component J${^}_{z}$. States with an abnormal occupation of the single particle orbitals at T=0 exhibit a discontinuity in the thermal response of their variational energy.

DOI
10.1103/physrevc.34.1458

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The thermal response of states of approximate angular momentum in 24Mg^{24}\mathrm{Mg}24Mg with {J}^{\ensuremath{\pi}}$${=8}^{+} has been calculated in the finite temperature restricted Hartree-Fock approximation. Full angular momentum projection has been circumvented by the use of constraints on the total angular momentum J{^}^{2} and the z component J{^}_{z}. States with an abnormal occupation of the single particle orbitals at T=0 exhibit a discontinuity in the thermal response of their variational energy.
Applying thermofield dynamics, we reformulate the exact quantum number projection in the finite-temperature Hartree-Fock-Bogoliubov theory. Explicit formulas are derived for the simultaneous projection of particle number and angular momentum, in parallel to the zero-temperature case. We also propose a practical method for the variation-after-projection calculation, by approximating entropy consistently with the Peierls inequality. Using quantum number projection in finite-temperature mean-field theory will be useful for studying the effects of quantum fluctuations associated with the conservation laws on thermal properties of nuclei.
With proper care on the boundary conditions, the total orbital angular momentum L z at finite temperatures is microscopically calculated for the Anderson-Brinkman-Morel state in a cylinder with axial symmetry about the z axis, i.e. the textures considered by McClure and Takagi at T =0. The pairing in this geometry occurs between m and 1- m one-particle states, where m denotes the axial quantum number. It is found that L z decreases from its zero temperature value ℏ2N\frac{\hbar}{2} {\cal N}2ℏ​N, with N the total particle number, in the same way as the components of the superfluid density tensor. Responsible for this reduction is identified to be the phase shift between m and 1- m quasiparticle states caused by the difference in the centrifugal potentials. This fact implies that the plane-wave representation is not suitable for the problem.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, we extend the Pfaffian formula for the many-body traces of HFB density operators introduced by L. M. Robledo in Ref. [1] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. We validate our method through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei 162Dy,148Sm^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}162Dy,148Sm, and 150Sm^{150}\mathrm{Sm}150Sm. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, I extend the Pfaffian formula for the many-body traces of HFB density operators introduced by Robledo [L. M. Robledo, Phys. Rev. C. 79, 021302(R) (2009)] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. The method is validated through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.