Finite density $\mathbf{O}(\mathbf{3})$ non-linear sigma model and low energy physics

We present lattice results for simulations of the $O(3)$ non-linear sigma model at finite chemical potential. The complex action problem is overcome by a dual variable representation of the model. We discuss two aspects of the theory at finite density: 1) The relation of the finite density data to scattering phases and wave-functions. 2) The phase-structure of the theory as a function of chemical potential and the possibility of a Kosterlitz-Thouless phase transition.