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Nonuniform symmetry breaking in noncommutative $λΦ^4$ theory

The spontaneous symmetry breaking in noncommutative $λΦ^4$ theory has been analyzed by using the formalism of the effective action for composite operators in the Hartree-Fock approximation. It turns out that there is no phase transition to a constant vacuum expectation of the field and the broken phase corresponds to a nonuniform background. By considering $<ϕ(x)>=A \cos(\vec Q \cdot \vec x)$ the generated mass gap depends on the angles among the momenta $\vec k$ and $\vec Q$ and the noncommutativity parameter $\vecθ$. The order of the transition is not easily determinable in our approximation.