Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

We study three possible ways to circumvent the sign problem in the O(3) nonlinear sigma model in 1+1 dimensions. We compare the results of the worm algorithm to complex Langevin and multiparameter reweighting. Using the worm algorithm, the thermodynamics of the model is investigated, and continuum results are shown for the pressure at different $\mu/T$ values in the range $0-4$. By performing $T=0$ simulations using the worm algorithm the Silver Blaze phenomenon is reproduced. Regarding the complex Langevin, we test various implementations of discretizing the complex Langevin equation. We found that the exponentialized Euler discretization of the Langevin equation gives wrong results for the action and the density at low $T/m$. By performing continuum extrapolation we found that this discrepancy does not disappear and depends slightly on temperature. The discretization with spherical coordinates perform similarly at low $\mu/T$, but goes wrong also at some higher temperatures at high $\mu/T$. However, a third discretization that uses a constraining force to achieve the $\phi^2 = 1$ condition gives correct results for the action, but wrong results for the density at low $\mu/T$.