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The thermal field theory is applied to fermionic superfluids by doubling thedegrees of freedom of the BCS theory. We construct the two-mode states and thecorresponding Bogoliubov transformation to obtain the BCS thermal vacuum. Theexpectation values with respect to the BCS thermal vacuum produce thestatistical average of the thermodynamic quantities. The BCS thermal vacuumallows a quantum-mechanical perturbation theory with the BCS theory serving asthe unperturbed state. We evaluate the leading-order corrections to the orderparameter and other physical quantities from the perturbation theory. A directevaluation of the pairing correlation as a function of temperature shows thepseudogap phenomenon results from the perturbation theory. The BCS thermalvacuum is shown to be a generalized coherent and squeezed state. Thecorrespondence between the thermal vacuum and purification of the densitymatrix allows a unitary transformation, and we found the geometric phase in theparameter space associated with the transformation.

BCS thermal vacuum of fermionic superfluids and its perturbation theory