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arXiv:cond-mat/0702223 · Fortschritte der Physik

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

B. Mieck

Published Jul 31, 2007• 8 citations
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Abstract

Abstract 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 invarianceU(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‐groupOsp(S,S| 2L). A nonlinear sigma model follows from the spontaneous breaking of the ortho‐symplectic super‐groupOsp(S,S| 2L) to the coset decompositionOsp(S,S| 2L) \U(L|S)⊗U(L|S). The invariant subgroupU(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 spaceOsp(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 decompositionOsp(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$\hat{G}_{Osp\backslash U}^{-1/2}$ of the metric tensor ofOsp(S,S| 2L) \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.

DOI
10.1002/prop.200710392

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Abstract 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 invarianceU(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‐groupOsp(S,S| 2L). A nonlinear sigma model follows from the spontaneous breaking of the ortho‐symplectic super‐groupOsp(S,S| 2L) to the coset decompositionOsp(S,S| 2L) \U(L|S)⊗U(L|S). The invariant subgroupU(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 spaceOsp(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 decompositionOsp(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 rootG^Osp\U−1/2\hat{G}_{Osp\backslash U}^{-1/2}G^Osp\U−1/2​ of the metric tensor ofOsp(S,S| 2L) \U(L|S) so that the non‐Euclidean integration measure with super‐Jacobi‐determinant[SDET(G^Osp\U)]1/2[{\rm SDET} (\hat{G}_{Osp\backslash U})]^{1/2}[SDET(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.
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A coherent state path integral is considered for fermions with precise, discrete time separation. We derive in detail a nonlinear sigma model of a pair condensate concerning the precise, discrete time step order which is usually abbreviated by the appealing (but in fact non-existent) hermitian form of the fields. This problem is ubiquitous in quantum many-body physics and already occurs in the original Dyson equation. However, one has only to include a few amendments concerning the precise, discrete time step development of coherent state path integrals, compared to previous, abridged approaches, so that one can accomplish the exact treatment of a spontaneous symmetry breaking with a coset decomposition for pair condensates and density parts.
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) x U(2) of the symplectic group Sp(4) with the subgroup U(2) for the unitary invariance of the density-related 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.
A Bose–Einstein condensate in a double-well potential is considered by using the Feynman path integral theory combined with a variational approach. The system consists of N interacting bosons confined in the double well which is taken as a harmonic potential with a Gaussian barrier. The calculation is carried out within the first cumulant approximation measured with respect to the harmonic action containing variational parameters. Assuming a separable expression for the trial action corresponds to the usual mean field approximation. Performing the variational calculations, we obtain analytical results for the ground state energy and its condensate wavefunction. Using a projection onto the odd contributions, the first excited state energy and condensate wavefunction are determined, too. We find very good agreement with a full numerical solution of the Gross–Pitaevskii equation over the full range of potential parameters.
We study the spontaneous symmetry breaking (SSB) of a superfluid Bose-Fermi (BF) mixture in a double-well potential (DWP). The mixture is described by the Gross-Pitaevskii equation (GPE) for the bosons, coupled to an equation for the order parameter of the Fermi superfluid, which is derived from the respective density functional in the unitarity limit (a similar model applies to the BCS regime, too). Straightforward SSB in the degenerate Fermi gas loaded into a DWP is impossible, as it requires an attractive self-interaction, and the intrinsic nonlinearity in the Fermi gas is repulsive. Nonetheless, we demonstrate that the symmetry breaking is possible in the mixture with attraction between fermions and bosons, like 40K^{40}\mathrm{K}40K and 87Rb^{87}\mathrm{Rb}87Rb. Numerical results are represented by dependencies of asymmetry parameters for both components on particle numbers of the mixture, NF{N}_{\mathrm{F}}NF​ and NB{N}_{\mathrm{B}}NB​, and by phase diagrams in the (NF,NB)({N}_{\mathrm{F}},{N}_{\mathrm{B}})(NF​,NB​) plane, which displays regions of symmetric and asymmetric ground states. The dynamical picture of the SSB, induced by a gradual transformation of the single-well potential into the DWP, is reported too. An analytical approximation is proposed for the case when the GPE for the boson wave function may be treated by means of the Thomas-Fermi (TF) approximation. Under a special linear relationship between NF{N}_{\mathrm{F}}NF​ and NB{N}_{\mathrm{B}}NB​, the TF approximation allows us to reduce the model to a single equation for the fermionic function, which includes competing repulsive and attractive nonlinear terms. The latter one directly displays the mechanism of the generation of the effective attraction in the Fermi superfluid, mediated by the bosonic component of the mixture.
We apply Feynman's path integral theory to study Bose-Einstein condensates in a double well potential and in a random potential. The main idea of this thesis is to perform the mean field approximation in two body interaction by the path integral approach. Bose particles are confined in the double well which is taken as a harmonic potential with Gaussian or cosine barrier. Performing the variational calculations we obtain analytical results of the ground state energy, wave functions, and overlap integrals. The overlap integrals enable us to calculate the tunneling rates which are in good agreement with numerical values for the two-mode model of Gross-Pitaevskii's equation. We also consider the model of a Bose system consisting of N particles with finite interacting pair potentials under the influence of random potentials. The porosity or nonhomogeneity of the system can be represented by the density of porosity with amplitude of the fluctuation, and a correlation length. We obtain analytical results of the effective mass and the ground state energy which are in good agreement with that of Bugoliubov approach. The propagator of the system is used to calculate the partition function, specific heat, the critical temperature, condensate density, and the superfluid density of the system.