Symmetry breaking induced by charge density and the entropy of interacting fields

We study interacting complex scalar field theories with global U(1) symmetry and concave potentials. It is usually assumed that spontaneous symmetry breaking is excluded for such interaction. However, we show that degenerate ground states appear when the system is considered as a charged medium, which we take to be so large that it makes sense to speak of a uniform, finite, charge density. This of course implies that we are considering as ground states solutions that select a particular Lorentz frame. The consequent symmetry breaking is accompanied by the usual Goldstone modes. It makes topological solitons possible in 1+1 dimensions. Further, a new kind of nontopological solitons appears, again in 1+1 dimensions. These are embedded in a uniformly charged background. Unlike the Friedberg-Lee-Sirlin solitons, those studied here do not require a complicatedly shaped potential to exist. Although Derricks theorem, which forbids higher-dimensional solitons, cannot be proved in the present context, it appears that such solitons are still forbidden in the presence of finite charge density. When the field is confined to a box, the frequency spectrum is, classically, a continuum. This is in sharp contrast to the situation for linear fields. However, semiclassical quantization, or the requirement that charge be quantized, both make the spectrum discrete. We show by general arguments that the energy spectrum (distinct from the frequency spectrum for nonlinear fields) for the interacting field in a box must have widely spaced levels. For the case of a quartic potential we compute the energy levels exactly in 1+1 dimensions, and verify this conclusion directly. The interacting scalar field thus complies in detail with the bound on specific entropy proposed by one of us earlier as applicable to all finite physical systems.