Topological critical slowing down: variations on a toy model

Abstract

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.

18 pages, 14 eps figures. Some changes and references added. To be published by Phys Rev E

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Cited by in corpus (20)

• Flow-based generative models for Markov chain Monte Carlo in lattice field theory
• Equivariant flow-based sampling for lattice gauge theory
• Phase Transitions in Particle Physics -- Results and Perspectives from Lattice Quantum Chromo-Dynamics
• Topology in full QCD at high temperature: a multicanonical approach
• Efficient Modelling of Trivializing Maps for Lattice ${\varphi }^4$ Theory Using Normalizing Flows: A First Look at Scalability
• Topological susceptibility of $N_f=2+1$ QCD from staggered fermions spectral projectors at high temperatures
• ${\chi }_{\text{top}}(T\gg T_{\text{c}})$ in pure-glue QCD through reweighting
• Large-$N$ $SU(N)$ Yang-Mills theories with milder topological freezing
• Topological properties of $C{P}^{N-1}$ models in the large-$N$ limit
• Large-$N$ expansion and $\theta$-dependence of $2d$ $C{P}^{N-1}$ models beyond the leading order
• Towards reduction of autocorrelation in HMC by machine learning
• The topological susceptibility of two-dimensional $U(N)$ gauge theories
• Lattice determination of the topological susceptibility slope ${\chi }^{\prime }$ of $2d{\mathrm{CP}}^{N-1}$ models at large $N$
• Topological susceptibility of $2d{\mathrm{CP}}^1$ or $\mathrm{O}(3)$ non-linear $\sigma$-model: is it divergent or not?
• Open-Boundary Conditions in the Deconfined Phase
• Reduced critical slowing down for statistical physics simulations
• Compact Gauge Fields on Causal Dynamical Triangulations: a 2D case study
• Pair Approximating the Action For Molecular Rotations in Path Integral Monte Carlo
• Scaling flow-based approaches for topology sampling in $\mathrm{SU}(3)$ gauge theory
• Path Integral Monte Carlo in the Angular Momentum Basis for a Chain of Planar Rotors