Random field effects in a Bose system: Critical behaviour and crossover phenomena in the Hartree limit

The low temperature grand canonical critical properties of a d-dimensional n-vector Bose system in the presence of a random field, which behaves like [h∗khk]av ∽ kθ (θ ⩾ 0), are investigated with the use of replica trick and the Hartree approximation. With a boson free particle spectrum, which behaves like kσ (0 < σ ⩽ 2), several situations occur, both at zero and non-zero temperature regimes, which are studied in detail for different values of d, σ, θ. In particular, some questions concerning the effective lowering of the space dimensionality and the violation of the usual scaling laws when the random field is present are also checked. Furthermore crossover phenomena appear where, alternatively, the random field intensity and the temperature assume the role of crossover parameter. It is shown that these crossover processes can be described in terms of crossover scaling functions and effective critical exponents in conformity with the standard crossover theory.