Critical and crossover behavior in the double-Gaussian model on a lattice

The double-Gaussian model, as recently introduced by Baker and Bishop, is studied in the context of a lattice-dynamics Hamiltonian belonging to the familiar ${\phi }^4$ class. Advantage is taken of the partition-function factorability (into Ising and Gaussian components) to place bounds on the Ising-class critical temperature for various lattice dimensions and all degrees of displaciveness in the bare Hamiltonian. Further, a simple criterion for a noncritical and nonuniversal crossover from order-disorder to Gaussian behavior is evaluated in numerical detail. In one and two dimensions these critical and crossover properties are compared with predictions based on real-space decimation renormalization-group flows, as previously exploited in the ${\phi }^4$ model by Beale et al. The double-Gaussian model again introduces some unique analytical advantages.