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

TL;DR

This paper investigates finite size effects in the classical non-linear O(2) sigma-model with a chemical potential, using the worm algorithm to study energy levels and phase diagram features despite the sign problem.

Contribution

It introduces an effective quantum mechanical approach to analyze finite size effects in the O(2) sigma-model with chemical potential, overcoming the sign problem.

Findings

Finite size effects due to chemical potential are characterized.

Energy levels of up to four particles are computed as a function of box size.

A part of the phase diagram in the (β, μ) plane is uncovered.

Abstract

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.