Antiferromagnetic φ4 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 φ⁴ 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.