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 supergroup Osp(S, S|2L). A nonlinear sigma model follows from the spontaneous breaking of the orthosymplectic super-group Osp(S, S|2L) to the coset decomposition Osp(S, S|2L)\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|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) ⊗ 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 Ĝ−1/2 Osp\U of the metric tensor of Osp(S, S|2L)\U (L|S) so that the non-Euclidean integration measure with super-Jacobi-determinant [SDET( ĜOsp\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.
A method is described for a proper numerical simulation of various random potentials and is used to compare the distributions of the potentials, projected onto the fist two Landau bands. In the case of a slowly varying potential (correlation length >> cyclotron radius) we find a universal distribution, proving an expected universality for the delocalization transition, whereas for a short-range potential the distributions in the higher Landau bands become nonuniversal. Numerical simulations in the first Landau level support nonuniversal localization properties for very short-range correlations. * * *I would like to thank H. A. WEIDENMULLER for valuable discussions and advice and the Heidelberger Akademie der Wissenschaften for its financial support.
A coherent state path integral is considered for bosons with an ensemble average of a random potential and with an additional, repulsive interaction in the context of BEC under inclusion of specially prepared disorder. The essential normalization of the coherent state path integral, as a generating function of observables, is obtained from the non-equilibrium time contour for 'forward' and 'backward' propagation so that a time contour metric has to be taken into account in the ensemble average with the random potential. Therefore, the respective symmetries for the derivation of a nonlinear sigma model follow from the involved time contour metric which leads to a coset decomposition Sp(4) / U(2) ⊗ U(2) of the symplectic group Sp(4) with the subgroup U(2) for the unitary invariance of the densityrelated vacuum or ground state; the corresponding spontaneous symmetry breaking gives rise to anomalous-or 'Nambu'doubled field degrees of freedom within self-energy matrices which are finally regarded by remaining coset matrices. The notion of a 'return probability', according to the original 'Anderson-localization', is thus naturally contained within coherent state path integrals of a non-equilibrium contour time for equivalent 'forward' and 'backward' propagation.The two Gaussian distributions for random potentials V I ( x), V II ( x, t) are determined by the second moments (1.4,1.5) for static and dynamic disorder, respectively, and vanishing mean values. Both distributions are delta-function like, concerning the spatially 'contact' Kronecker-delta and concerning the white-noise delta-function of time. Moreover, we emphasize the two different normalizations of second moments in (1.4,1.5) which are important in subsequent transformations and derivations for a proper, finite scaling of energy ranges within the nonlinear sigma models (cf. the second moments and their normalization in random matrix theories). Therefore, there occur two different disorder parameters R I , R II 2.1 Precise time steps with shifts '∆t p ' of the complex conjugated fields 'ψ * x (t p +∆t p )'
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