Investigating the critical slowing down of QCD simulations

Abstract

Simulations of QCD are known to suffer from serious critical slowing down towards the continuum limit. This is particularly prominent in the topological charge. We investigate the severeness of the problem in the range of lattice spacings used in contemporary simulations and propose a method to give more reliable error estimates.

Non-renormalizability of the HMC algorithm

“…The generalized HMC algorithm [14][15][16] integrates a stochastic version of the molecular-dynamics 1 Equation (1.1) is expected to hold on the infinite lattice and on finite lattices with open boundary conditions [10]. On lattices with periodic boundary conditions, topology-changing tunneling transitions can give rise to very large autocorrelation times [7][8][9][10]. Such transitions are lattice artifacts and are therefore not included in the analysis that leads to eq.…”

Autocorrelations in Hybrid Monte Carlo Simulations

“…The reason behind this choice is that even though in eq. 2.4 all modes contribute to Γ for a given observable, if the integrator preserves a symmetry of the discretized action (in our case parity), it is possible to show that contributions come only from algorithmic modes in the same sector as the observable under study [2]. As shown in Fig.…”

“…In the following we present a brief overview of the concepts and definitions used. For a more detailed summary and explanation we refer to [2] and references therein.…”

“…On each of these shorter histories we have then calculated the auto-correlation function. To test the method we have assumed that τ * is known (either from a previous study or from a model as proposed in [2]). In practice we have used the value of τ * obtained from the auto-correlation function of Q 2 on the entire history.…”

Light quarks on the lattice: methods and results for pion physics

“…However, it turns out that in the range of current lattice simulations, the cost of doing so rises with roughly the tenth power of the inverse lattice spacing, because observables like the topological charge show a very severe critical slowing down [7]. This challenge remains to be solved during the next years for trustworthy computations.…”