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In the presence of a chemical potential, the physics of level crossings leadsto singularities at zero temperature, even when the spatial volume is finite.These singularities are smoothed out at a finite temperature but leave behindnon-trivial finite size effects which must be understood in order to extractthermodynamic quantities using Monte Carlo methods, particularly close tocritical points. We illustrate some of these issues using the classicalnon-linear O(2) sigma model with a coupling $\beta$ and chemical potential$\mu$ on a 2+1 dimensional Euclidean lattice. In the conventional formulationthis model suffers from a sign problem at non-zero chemical potential and hencecannot be studied with the Wolff cluster algorithm. However, when formulated interms of world-line of particles, the sign problem is absent and the model canbe studied efficiently with the "worm algorithm". Using this method we studythe finite size effects that arise due to the chemical potential and develop aneffective quantum mechanical approach to capture the effects. As a side resultwe obtain energy levels of up to four particles as a function of the box sizeand uncover a part of the phase diagram in the $(\beta,\mu)$ plane.

Finite size effects in the presence of a chemical potential: A study in the classical non-linear O(2) sigma-model