BCS to Bose crossover: Broken-symmetry state

A functional integral formulation, used previously to calculate ${\mathrm{T}}_{\mathrm{c}}$ and describe normal state properties of the BCS-Bose crossover, is extended to T<${\mathrm{T}}_{\mathrm{c}}$. The saddle point approximation is shown to be qualitatively correct for T≪${\mathrm{T}}_{\mathrm{c}}$ for all couplings, in contrast to the situation above ${\mathrm{T}}_{\mathrm{c}}$. Several features of the crossover are described. The difference between the T=0 ``pair size'' and the (prefactor of the T dependent) Ginzburg-Landau coherence length is pointed out: the two quantities are the same only in the BCS limit. The evolution of the collective modes from the BCS to the Bose regime is discussed together with the mixing of the amplitude and phase in the absence of a particle-hole symmetry.