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Coherent state path integral and super-symmetry for condensates composed of bosonic and fermionic atoms. (English) Zbl 1141.82308

Summary: 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 with the same interaction potential so that a super-symmetry results between the bosonic and fermionic components of super-fields. Apart from the super-unitary invariance \(U(L | S)\) of the density terms, we specialize on the examination of super-symmetries for pair condensate terms. Effective equations are derived for anomalous terms which are related to the molecular- and BCS- condensate pairs. 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 orthosympletic super-group \(Osp(S,S | 2L)\). A nonlinear sigma model follows from the spontaneous breaking of the ortho-symplectic super-group \(Osp(S,S | 2L)\) to the coset decomposition \(Osp(S,S | 2L) / U(L | S) \otimes U(L | S)\). The invariant subgroup \(U(L | S)\) for the vacuum or background fields is represented by the density terms in the self-energy whereas the super-matrices on the coset space \(Osp(S,S | 2L) / U(L | S)\) describe the anomalous molecular and BCS- pair condensate terms. A change of integration measure is performed for the coset decomposition \(Osp(S,S | 2L) / U(L | S) \otimes U(L | S)\), 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 \(\hat{G}_{OSP/U}^{-1/2}\) of the metric tensor of \(Osp(S,S | 2L) / U(L | S)\) so that the non-Euclidean integration measure with super-Jacobi-determinant \([\text{SDET}\hat{G}_{OSP/U})]^{-1/2}\) (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. The dynamics of the eigenvalues of the coset matrices is determined by sine-Gordon equations which have a similar meaning for the dynamics of the molecular- and BCS-pair condensates as the Gross-Pitaevskii equation for the coherent wave function in BEC phenomena.

MSC:

82B10 Quantum equilibrium statistical mechanics (general)
81R30 Coherent states
81S40 Path integrals in quantum mechanics
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