  In the presence of a chemical potential, the physics of level crossings leads
to singularities at zero temperature, even when the spatial volume is finite.
These singularities are smoothed out at a finite temperature but leave behind
non-trivial finite size effects which must be understood in order to extract
thermodynamic quantities using Monte Carlo methods, particularly close to
critical points. We illustrate some of these issues using the classical
non-linear O(2) sigma model with a coupling $\beta$ and chemical potential
$\mu$ on a 2+1 dimensional Euclidean lattice. In the conventional formulation
this model suffers from a sign problem at non-zero chemical potential and hence
cannot be studied with the Wolff cluster algorithm. However, when formulated in
terms of world-line of particles, the sign problem is absent and the model can
be studied efficiently with the "worm algorithm". Using this method we study
the finite size effects that arise due to the chemical potential and develop an
effective quantum mechanical approach to capture the effects. As a side result
we obtain energy levels of up to four particles as a function of the box size
and uncover a part of the phase diagram in the $(\beta,\mu)$ plane.
