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arXiv:0906.2531

Coherent state path integral and nonlinear sigma model for a condensate composed of fermions with precise, discrete steps in the time development

B. Mieck

Published Jun 14, 2009• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0906.2531

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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.
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.
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.
An alternative formulation of the discrete real time coherent state path integral is examined. Following the development of a previous work [T. L. Marchioro II, J. Math Phys. 31, 2935 (1990)] the discrete coherent state action is transformed to second order form. The resulting path integral expression contains an integrand which factors into a Gaussian probability density and an oscillatory term, making the integral directly amenable to Monte Carlo techniques. Several simple examples in one and two spatial dimensions are presented and the possibilities of extending such an approach to general multidimensional systems are discussed.
Two long-standing problems in the construction of coherent state path integrals, the unwarranted assumption of path continuity and the ambiguous definition of the Hamiltonian symbol, are rigorously solved. To this end the fully controlled dual representation familiar from lattice quantum field theories is introduced. Dualization allows for both the step-by-step check in the construction of discrete path integrals and for the identification of the Hamiltonian and Berry phase part of the action, thus making both discrete and continuous path integrals consistently defined. Once the correct action is constructed, we are able to follow the transition to the continuum for the polar form of general Bose-Hubbard models and to provide an exact form of the path integral for general spin systems, where previous works showed its failure for all standard choices of operator ordering.
The paper solves the problem of continuum limit in bosonic thermal coherent-state path integrals. For this purpose, exact discrete versions of the path integral are constructed for three different orderings of the Hamiltonian: normal, anti-normal and symmetric (Weyl order). Subsequently, their different continuum versions are checked on the harmonic oscillator, to choose the symmetric ordering as a possibly correct choice for all polynomial Hamiltonians. Spotted mathematical subtleties in the simple case serve as a clue to the general solution. Finally, a general justification for the symmetric order is provided by deriving the continuum path integral starting from the exact discrete case using a renormalization procedure in the imaginary time frequency domain. While the role of Weyl order has already been found, the paper provides the missing proof of its suitability for every polynomial Hamiltonian and simplifies the previously established construction by referring only to creation and annihilation operators (without position and momentum operators).