Thermal resonating Hartree-Bogoliubov theory based on the projection method

We propose a rigorous thermal resonating mean-field theory (Res-MFT). A state is approximated by superposition of multiple MF wavefunctions (WFs) composed of non-orthogonal Hartree-Bogoliubov (HB) WFs. We adopt a Res-HB subspace spanned by Res-HB ground and excited states. A partition function (PF) in a SO(2N) coherent state representation |g> (N:Number of single-particle states) is expressed as Tr(e^{-\beta H})=2^{N-1} \int <g|e^{-\beta H}|g>dg (\beta=1/k_BT). Introducing a projection operator P to the Res-HB subspace, the PF in the Res-HB subspace is given as Tr(Pe^{-\beta H}), which is calculated within the Res-HB subspace by using the Laplace transform of e^{-\beta H} and the projection method. The variation of the Res-HB free energy is made, which leads to a thermal HB density matrix W_{Res}^{thermal} expressed in terms of a thermal Res-FB operator F_{Res}^{thermal} as W_{Res}^{thermal}={1_{2N}+exp(\beta F_{Res}^{thermal})}^{-1}. A calculation of the PF by an infinite matrix continued fraction is cumbersome and a procedure of tractable optimization is too complicated. Instead, we seek for another possible and more practical way of computing the PF and the Res-HB free energy within the Res-MFT.