Ensemble averaged coherent state path integral for disordered bosons with a repulsive interaction (Derivation of mean field equations)

Abstract

AbstractWe 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 algebra sp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product of M × 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 in d = 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 for d = 2; therefore, it is also pointed out that one should consider different HST's in d = 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decomposition Sp (4M)\U (2M) into densities and anomalous terms of the total disorder‐self‐energy sp(4M) for deriving a nonlinear sigma model.