Lee-Yang cluster expansion approach to the BCS-BEC crossover: BCS and BEC limits

It is shown that a cluster expansion technique, which is usually applied in the high-temperature regime to calcutate virial coefficients, can be applied to evaluate the superfluid transition temperature of the BCS-BEC crossover \`a la Lee and Yang. The transition temperature is identified with the emergence of the singularity in the sum of a certain infinite series of cluster functions. In the weak-coupling limit, we reproduce the Thouless criterion and the number equation of Nozi\`eres and Schmitt-Rink, and hence the transition temperature of the BCS theory. In the strong-coupling limit, we reproduce the transition temperature of Bose-Einstein condensation of non-interacting tightly bound dimers.