# About $\phi^4$ model

In many books the $\phi^4$ model can produce a topological soliton called kink. Are they right? In the case of sine-Gordon model you can have a topological soliton due to you can express the Lagrangian in terms of two scalar fields taking values in a circle. As the field is constrained on a manifold the model could have stability. I am not sure if $\phi^4$ model could be viewed as a field constrained on a manifold also.

So my questions are:

• Is it right to talk about topological solitons or kinks in $\phi^4$ model?

• Has the $\phi^4$ model an alternative formulation as a constrained field with target space a circle or any other manifold?

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–  Qmechanic Feb 5 '12 at 22:10

The kinks of $\phi^4$ are when the mass squared is negative and there is phi to minus phi symmetry. Then there are two symmetric vacua, and the kink interpolates between them. You can identify field values phi and minus phi to get a simple orbifold. In this case, a kink is always followed by an anti-kink, because of the boundary conditions.