Spontaneously broken translational invariance of compactified space

We propose a mechanism to break the translational invariance of compactified space spontaneously. As a simple model, we study a real φ4 model compactified on MD−1⊗S1 in detail, where we impose a nontrivial boundary condition on φ for the S1-direction. It is shown that the translational invariance for the S1-direction is spontaneously broken when the radius R of S1 becomes larger than a critical radius R∗ and also that the model behaves like a φ4 model on a single kink background for R→∞. It is pointed out that spontaneous breakdown of translational invariance is accompanied by that of some global symmetries, in general, in our mechanism.