Topological critical slowing down: variations on a toy model

Numerical simulations of lattice quantum field theories whose continuum counterparts possess classical solutions with non-trivial topology face a severe critical slowing down as the continuum limit is approached. Standard Monte-Carlo algorithms develop a loss of ergodicity, with the system remaining frozen in configurations with fixed topology. We analyze the problem in a simple toy model, consisting of the path integral formulation of a quantum mechanical particle constrained to move on a circumference. More specifically, we implement for this toy model various techniques which have been proposed to solve or alleviate the problem for more complex systems, like non-abelian gauge theories, and compare them both in the regime of low temperature and in that of very high temperature. Among the various techniques, we consider also a new algorithm which completely solves the freezing problem, but unfortunately is specifically tailored for this particular model and not easily exportable to more complex systems.