Coherent state path integral and super-symmetry for condensates composed of bosonic and fermionic atoms

Details Coherent state path integral and super-symmetry for condensates composed of bosonic and fermionic atoms Coherent state path integral and super-symmetry for condensates composed of bosonic and fermionic atoms

2007-02-08

arxiv.org/abs/cond-mat/0702223v4

Fortschr. Phys. ("Progress of Physics") Vol. 55, (No. 9-10), pp. 989-1120 (2007)

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

A super-symmetric coherent state path integral on the Keldysh time contour is considered for bosonic and fermionic atoms which interact among each other with a common short-ranged two-body potential. We investigate the symmetries of Bose-Einstein condensation for the equivalent bosonic and fermionic constituents and specialize on the examination of super-symmetries for pair condensate terms. A Hubbard-Stratonovich transformation from 'Nambu'-doubled super-fields leads to a generating function with super-matrices for the self-energy whose manifold is given by the ortho-symplectic super-group Osp(S,S|2L). Effective equations are derived for anomalous terms which are related to the molecular- and BCS- condensate pairs. A change of integration measure for the coset decomposition Osp(S,S|2L)/U(L|S)xU(L|S) is performed, including a separation of density and anomalous parts of the self-energy with a gradient expansion for the Goldstone modes. The independent anomalous fields in the actions can be transformed by the inverse square root of the metric tensor of Osp(S,S|2L)/U(L|S) so that the coset integration measure with the super-Jacobi-determinant can be removed from the coherent state path integral and Gaussian-like integrations remain. The variations of the independent coset fields in the effective actions result in classical field equations for a nonlinear sigma model with the anomalous terms.

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