Nonlinear sigma model for a condensate composed of fermionic atoms

A nonlinear sigma model is derived for the time development of a Bose–Einstein condensate composed of fermionic atoms. Spontaneous symmetry breaking of a Sp(2) symmetry in a coherent state path integral with anticommuting fields yields Goldstone bosons in a Sp(2) U(2) coset space. After a Hubbard–Stratonovich transformation from the anticommuting fields to a local self-energy matrix with anomalous terms, the assumed short-ranged attractive interaction reduces this symmetry to a SO(4) U(2) coset space with only one complex Goldstone field for the singlet pairs of fermions. This bosonic field for the anomalous term of fermions is separated in a gradient expansion from the density terms. The U(2) invariant density terms are considered as a background field or unchanged interacting Fermi sea in the spontaneous symmetry breaking of the SO(4) invariant action and appear as coefficients of correlation functions in the nonlinear sigma model for the Goldstone boson. The time development of the condensate composed of fermionic atoms results in a modified Sine–Gordon equation