Premium
Coherent state path integral and super‐symmetry for condensates composed of bosonic and fermionic atoms
Author(s) -
Mieck B.
Publication year - 2007
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.200710392
Subject(s) - physics , mathematical physics , path integral formulation , quantum mechanics , homogeneous space , coset , coherent states , symplectic geometry , unitary transformation , sigma model , invariant (physics) , nonlinear system , quantum , combinatorics , pure mathematics , geometry , mathematics
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 | 2 L ). A nonlinear sigma model follows from the spontaneous breaking of the ortho‐symplectic super‐group Osp ( S , S | 2 L ) to the coset decomposition Osp ( S , S | 2 L ) \ U ( L | S )⊗ 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 | 2 L ) \ 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 | 2 L ) \ U ( L | S ) ⊗ 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 $\hat{G}_{Osp\backslash U}^{-1/2}$ of the metric tensor of Osp ( S , S | 2 L ) \ U ( L | S ) so that the non‐Euclidean integration measure with super‐Jacobi‐determinant $[{\rm SDET} (\hat{G}_{Osp\backslash U})]^{1/2}$ 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.