Superfluid equation of state of cold fermionic gases in the Bose-Einstein regime

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We present an exact many-body theory of ultracold fermionic gases for the Bose-Einstein condensation (BEC) regime of the BEC-BCS crossover. This is a purely fermionic approach which treats explicitely and systematically the dimers formed in the BEC regime as made of two fermions. We consider specifically the zero temperature case and calculate the first terms of the expansion of the chemical potential in powers of the density n. We derive first the mean-field contribution, which has the expected standard expression when it is written in terms of the dimer-dimer scattering length aₘ. We go next in the expansion to the Lee-Huang-Yang order, proportional to n³/². We find the far less obvious result that it retains also the same expression in terms of aₘ as for elementary bosons. The composite nature of the dimers appears only in the next term proportional to n².