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We propose a rigorous thermal resonating mean-field theory (Res-MFT). A stateis approximated by superposition of multiple MF wavefunctions (WFs) composed ofnon-orthogonal Hartree-Bogoliubov (HB) WFs. We adopt a Res-HB subspace spannedby Res-HB ground and excited states. A partition function (PF) in a SO(2N)coherent state representation |g> (N:Number of single-particle states) isexpressed 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 theRes-HB subspace is given as Tr(Pe^{-\beta H}), which is calculated within theRes-HB subspace by using the Laplace transform of e^{-\beta H} and theprojection method. The variation of the Res-HB free energy is made, which leadsto a thermal HB density matrix W_{Res}^{thermal} expressed in terms of athermal Res-FB operator F_{Res}^{thermal} asW_{Res}^{thermal}={1_{2N}+exp(\beta F_{Res}^{thermal})}^{-1}. A calculation ofthe PF by an infinite matrix continued fraction is cumbersome and a procedureof tractable optimization is too complicated. Instead, we seek for anotherpossible and more practical way of computing the PF and the Res-HB free energywithin the Res-MFT.

Thermal resonating Hartree-Bogoliubov theory based on the projection method