Gross-Pitaevskii Equation for a System of Randomly Interacting Cold Bosons

Random interaction models have been successful in describing the amorphous properties of solids such as spin-glasses and structural glasses. This modelling approach is applied to a system of zero-spin cold bosons moving in an amorphous environment. The bosons are given to interact according to a non-random hard-core interaction. Additionally the bosons are subjected to a random interaction potential similar to that used for glasses. The approach is to apply a combination of replica methods and field theoretic techniques developed for superfluid Bose systems. This leads to an equation for the low-temperature Bose-Einstein condensate which is derived in the Hartree-Fock approximation. The equation is similar to the Gross-Pitaevskii equation, but the hard-core coupling constant is renormalised by the presence of the random interactions in a position dependent way. The amorphous contribution contains the replica diagonal and off-diagonal Green's functions, for which the Dyson equations are formulated.