The antiferromagnetic phi4 Model, I. The Mean-field Solution
  Certain higher dimensional operators of the lagrangian may render the vacuum inhomogeneous. A rather rich phase structure of the phi4 scalar model in four dimensions is presented by means of the mean-field approximation. One finds para- ferro- ferri- and antiferromagnetic phases and commensurate-incommensurate transitions. There are several particles described by the same quantum field in a manner similar to the species doubling of the lattice fermions. It is pointed out that chiral bosons can be introduced in the lattice regularized theory. 
V. Branchina, H. Mohrbach, J. Polonyi