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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 wormalgorithm to complex Langevin and multiparameter reweighting. Using the wormalgorithm, the thermodynamics of the model is investigated, and continuumresults 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 SilverBlaze phenomenon is reproduced. Regarding the complex Langevin, we test variousimplementations of discretizing the complex Langevin equation. We found thatthe exponentialized Euler discretization of the Langevin equation gives wrongresults for the action and the density at low $T/m$. By performing continuumextrapolation we found that this discrepancy does not disappear and dependsslightly on temperature. The discretization with spherical coordinates performsimilarly at low $\mu/T$, but goes wrong also at some higher temperatures athigh $\mu/T$. However, a third discretization that uses a constraining force toachieve the $\phi^2 = 1$ condition gives correct results for the action, butwrong results for the density at low $\mu/T$.

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