  We numerically study the spectral properties, the entanglement and the
zero-temperature phase structure at nonvanishing chemical potential of the O(3)
nonlinear sigma model. Using matrix product states, a particular kind of
one-dimensional tensor network state, we show that we are able to reach the
asymptotic scaling regime and to reproduce the analytical predictions for the
mass gap at vanishing chemical potential. In addition, we study the scaling of
the entanglement entropy towards the continuum limit obtaining a central charge
consistent with 2. Moreover, our approach does not suffer from the sign problem
and we also explore the phase structure of the model for nonzero chemical
potential and map out the location of the transitions between different charge
sectors with high precision.
