Projection after variation in the finite-temperature Hartree-Fock-Bogoliubov approximation

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.

Figure 1

Particle-number PAV thermal energy in the rotating frame from the explicit projection (solid blue line) and the Pfaffian formula (red dashed line) as a function of inverse temperature $\beta$. Open circles show the exact diagonalization results. The unprojected HFB thermal energy (green dashed-dotted line) is also shown for comparison. All energies are in units of the pairing strength $G$.