Study of symmetry breaking in a relativistic Bose gas using the contraction algorithm

A relativistic Bose gas at finite density suffers from a sign problem that makes direct numerical simulations infeasible. One possible solution to the sign problem is to reexpress the path integral in terms of Lefschetz thimbles. Using this approach we study the relativistic Bose gas both in the symmetric phase (low density) and the spontaneously broken phase (high density). In the high-density phase we break explicitly the symmetry and determine the dependence of the order parameter on the breaking. We study the relative contributions of the dominant and subdominant thimbles in this phase. We find that the subdominant thimble only contributes substantially when the explicit symmetry breaking is small, a regime that is dominated by finite volume effects. In the regime relevant for the thermodynamic limit, this contribution is negligible.