Topological critical slowing down: Variations on a toy model

Abstract

Numerical simulations of lattice quantum field theories whose continuum counterparts possess classical solutions with nontrivial 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 propose an alternative algorithm which completely solves the freezing problem, but unfortunately is specifically tailored for this particular model and not easily exportable to more complex systems.

Recent Progress on the QCD Phase Diagram

“…Interesting algorithmic developments have been reported since LATTICE 2017 to measure χ t to very high temperatures [28,29]. Since topological tunnelings become rarer as one goes to higher temperatures, one has to sample a large number of configurations to measure χ t making the problem computationally challenging.…”

Topological susceptibility of two-dimensional U(N) gauge theories

“…It is however well known that Monte Carlo algorithms typically used in numerical simulations suffer from a severe critical slowing down as the continuum limit is approached, with autocorrelation times of topological observables that grow about exponentially in the inverse of the lattice spacing [10][11][12][13]. This led to the development of new algorithms, specifically devised to improve the sampling of topologically nontrivial configuration [7,8,[14][15][16][17][18][19][20][21][22][23].…”

Topology in full QCD at high temperature: a multicanonical approach

“…iii) On the other hand, when trying to get closer to the continuum limit, a different challenge emerges: because of the topological nature of the problem, standard updating algorithms fail to correctly sample the distribution of Q and get trapped in path integral sectors with fixed topology. This freezing of topological charge leads to a severe critical slowing down of numerical simulations [44][45][46][47][48];…”

“…The main purpose of this paper is to make progress towards an independent determination of χ(T ) in the continuum limit, trying to solve at least problem i) from first principles and without any extra assumption. To that purpose, we exploit a reweighting technique which combines ideas typical of multicanonical simulations [50] and of metadynamics [51][52][53], and has already proved to be extremely efficient in the toy model of the 1D quantum rotor [48], where it permits to correctly sample the distribution P (Q) for Q 2 going down by several orders of magnitude, and more recently in pure gauge theories [54].…”