from a sign problem exactly like in QCD. Not surprisingly, the phase diagram of the condensed matter arising in the O (2) non-linear sigma model has not been studied from first princi ples. On the other hand non-relativistic bosonic lattice models, especially in the Hamiltonian formulation have been studied for many years by the condensed matter community. Here one naturally constructs the field theory with bosonic world lines and there is no sign p roblem when one introduces a chemical potential. Thus, it is natural that a world-line approach could solve the corresponding sign problem even for a relativistic field theory. This was sh own explicitly for both the linear sigma model [8] and the O (2) non-linear sigma model [9]. 

While the world-line representation for bosonic lattice fie ld theories was well known for many years, the main advance in the field that improved our abilityto perform a first principles calcula- tion in the presence of a chemical potential, was the discovery of an efficient Monte Carlo algo- rithm called the “worm algorithm” [10]. Variants of this alg orithm in the name of “directed loop algorithm” [11, 12] have been used to solve a variety of model s that arise in the strong coupling limit of lattice gauge theories [13–16]. The worm algorithmhas also been found to be an efficient approach to study a wider class of fermionic field theories inthe loop representation in two dimen- sions where fermion sign problems are absent [17–20] and wea k coupling Abelian lattice gauge theory [21]. A combination of the worm algorithm and the determinantal algorithm was recently developed to solve the lattice Thirring model in the fermion bag formulation in higher dimensions [22]. All these developments should allow us to explore the physics of a chemical potential using first principles in a variety of lattice models with interest ing symmetries. 

In this work, we explore the O (2) non-linear sigma model in the presence of a chemical poten- tial and show that interesting finite size effects naturallyarise due to the level crossing phenomena. Understanding these effects is important to extract the thermodynamic limit and thus uncover the ( β, µ ) phase diagram, where β is the coupling and µ is the chemical potential. Our studies should be useful for future studies since the finite size effects we u ncover is a universal feature. Our work also provides accurate results that can be used to compare with results from other methods, like the complex Langevin method, which are being explored as a solution to the sign problem in general [23–25]. Our work is organized as follows: In sect ion II we discuss our model and observables in order to set up the notation. In section III we discuss the finite size effects that arise in the presence of a chemical potential and develop an effective quantum mechanical description that captures these effects in section IV. Sections V and VI contain our results obtained using the worm algorithm. In particular we show that the observed finit e size effects are described well by 

