  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.
