  We analyze a complex scalar field with phi-4 interaction and a chemical
potential mu on the lattice. An exact flux representation of the partition sum
is used which avoids the complex action problem and based on a generalized worm
algorithm we can run Monte Carlo simulations at arbitrary densities. We study
thermodynamical quantities as a function of the chemical potential mu for zero-
and finite temperature. It is shown that at zero temperature thermodynamical
observables are independent of mu up to a critical value mu_c (Silver Blaze
phenomenon). In a spectroscopy calculation we cross-check that mu_c agrees with
the mass m of the scalar field. The Silver Blaze region ends in a second order
phase transition and we show that for low temperatures the second order phase
boundary persists and separates a pseudo Silver Blaze region from a condensed
phase with strong mu-dependence.
