  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.
