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 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.
Abstract We consider bosonic atoms with a repulsive contact interaction in a trap potential for a Bose‐Einstein condensation (BEC) and additionally include a random potential. The ensemble averages for two models of static (I) and dynamic (II) disorder are performed and investigated in parallel. The bosonic many body systems of the two disorder models are represented by coherent state path integrals on the Keldysh time contour which allow exact ensemble averages for zero and finite temperatures. These ensemble averages of coherent state path integrals therefore present alternatives to replica field theories or super‐symmetric averaging techniques. Hubbard‐Stratonovich transformations (HST) lead to two corresponding self‐energies for the hermitian repulsive interaction and for the non‐hermitian disorder‐interaction. The self‐energy of the repulsive interaction is absorbed by a shift into the disorder‐self‐energy which comprises as an element of a larger symplectic Lie algebrasp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product ofM×sp(2); ‘M’ is the number of discrete time intervals of the disorder‐self‐energy in the generating function ). After removal of the remaining Gaussian integral for the self‐energy of the repulsive interaction, the first order variations of the coherent state path integrals magnified image result in the exact mean field or saddle point equations, solely depending on the disorder‐self‐energy matrix. These equations can be solved by continued fractions and are reminiscent to the `Nambu‐Gorkov' Green function formalism in superconductivity because anomalous terms or pair condensates of the bosonic atoms are also included into the selfenergies. The derived mean field equations of the models with static (I) and dynamic (II) disorder are particularly applicable for BEC ind= 3 spatial dimensions because of the singularity of the density of states at vanishing wavevector. However, one usually starts out from restricted applicability of the mean field approach ford= 2; therefore, it is also pointed out that one should consider different HST's ind= 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decompositionSp(4M)\U(2M) into densities and anomalous terms of the total disorder‐self‐energysp(4M) for deriving a nonlinear sigma model.
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 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 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.
Coherent state path integrals are applied to a many-body problem for non-relativistic electrons in a central potential and an external magnetic field; however, in comparison to previous coherent state path integrals, we definitely fix the symmetry quantum numbers to specific values with second quantized field operators which are restricted by delta functions of a Dirac identity in a trace representation of anti-commuting coherent states. We perform an anomalous doubling for the delta functions of the two-particle parts so that a Hubbard-Stratonovich transformation can be taken for corresponding self-energies with a coset decomposition. The given field theory can be extended to an ensemble average over the external magnetic field so that mean eigenvalue densities and eigenvalue correlations can be obtained.
The goal of this diploma thesis is to develop a field theory, for a specific class of disordered bosonic systems, in order to study, whether disordered bosons generically \nhave universal statistical properties, similar to fermions. \nNow, if disordered bosons show system independent statistical features, the evident \nquestion is whether the bosonic universality classes are different from the fermionic \nones. For fermions, the classification in terms of symmetric spaces is a largely algebraical one, which happens in a complexified setting before compact or non-compact \nreal forms are specified. Hence one can rightly state that all effective bosonic non-linear \nσ-models are already classified within the ten fold way together with the fermionic ones. \nAnd indeed, in the system studied in [LSZ06], well known universal GUE statistics \nwere found in the bulk and Lück finds GOE statistics in [Lü09], both for systems of \ndisordered bosons. \nBut for bosons non-linear σ-models are not everything. A subtle point about noninteracting, i.e. quadratic, bosonic systems is stability. One single particle state \nof negative energy for a bosonic system immediately leads to an unbounded many \nparticle Hamiltonian. For fermionic systems, a finite number of such single particle \nstates is of no concern, due to the Pauli principle. This important difference leads \nto the well studied Gaussian ensembles not providing feasible distributions of bosonic \nHamiltonians. If, however, more complicated probability distributions are taken into \naccount, one has to be prepared to face a more complicated effective model than a \npure non-linear σ-model in the end. In this sense, an interesting question is whether \nbosonic systems if they show universal behaviour lead to novel universality classes. \nA hint to those might be the unusual density of states near zero energy that was \nfound in [LSZ06] or [GC02] and [GA04]. On physical grounds, one can expect the \ncommutation relations of bosons to have important effects at low frequencies, which \nfits to the observations of [GC02] and [GA05]. \nIn this thesis, we will develop and study a specific model, which is the next step \nafter the work of Lück, Sommers and Zirnbauer, [LSZ06], towards a more realistic \ndescription of a disordered system of bosonic degrees of freedom. As in [LSZ06], our \nmodel is still purely random, i.e. there is no limit of an underlying pure system and \nwe also stay with a model without global symmetries, such as time reversal or charge \nconjugation. But we turn from the homogeneous, zero-dimensional model to a spatially \nextended one and implement the physically important feature of locality. Furthermore, \nwe add an additional parameter to tune the number of modes per volume. In particular \nwe can hereby enforce a macroscopical number of zero modes. The goal is to study the \ndensity of states of the new model and therefore to develop an effective field theory. \nThroughout we will pay special attention to the region near zero frequency and watch \nout for unusual, possibly universal features.
The SU(2) coherent-state path-integral representation is used to derive a fairly explicit expression for the partition function Z0[\ensuremath{\psi}\ensuremath{\rightarrow}] of noninteracting quantum spins in the presence of an arbitrary time-dependent external field \ensuremath{\psi}\ensuremath{\rightarrow}. By performing then the standard Gaussian average of Z0[\ensuremath{\psi}\ensuremath{\rightarrow}] over \ensuremath{\psi}\ensuremath{\rightarrow} the partition function of the ferromagnetically coupled Heisenberg model is represented in a way which allows one to simplify the derivation of results such as obtained by Leibler and Orland [Ann. Phys. (N.Y.) 132, 277 (1981)] and to generalize them to arbitrary spin quantum number.