Finite size effects in the presence of a chemical potential: A study in the classical non-linear O (2) sigma-model Debasish Banerjee $^{1}$, Shailesh Chandrasekharan1 , 2 ∗ $^{1}$Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005 India $^{2}$Department of Physics, Box 90305, Duke University, Durham, North Carolina 27708, USA. In the presence of a chemical potential, the physics of level crossings leads to singularities at zero tem- perature, even when the spatial volume is finite. These singu larities are smoothed out at a finite temperature but leave behind non-trivial finite size effects which must b e understood in order to extract thermodynamic quantities using Monte Carlo methods, particularly close to critical points. We illustrate some of these is- sues using the classical non-linear O (2) sigma model with a coupling β and chemical potential µ 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 s tudy 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 ( β, µ ) plane. PACS numbers: 11.10.-z, 02.70.Ss, 05.30.-d, 75.10.Hk $^{∗}$Electronic address: debasish@theory.tifr.res.in,sch@phy.duke.edu 1 arXiv:1001.3648v3 [hep-lat] 12 Jul 2010 Abstract