Critical slowing down of topological modes

We investigate the critical slowing down of the topological modes using local updating algorithms in lattice 2-d CP models. We show that the topological modes experience a critical slowing down that is much more severe than the one of the quasi-Gaussian modes relevant to the magnetic susceptibility, which is characterized by τmag ∼ ξ z with z ≈ 2. We argue that this may be a general feature of Monte Carlo simulations of lattice theories with non-trivial topological properties, such as QCD, as also suggested by recent Monte Carlo simulations of 4-d SU(N) lattice gauge theories. Monte Carlo simulations of critical phenomena in statistical mechanics and of quantum field theories, such as QCD, in the continuum limit are hampered by the problem of critical slowing down (CSD) [1]. The autocorrelation time τ , which is related to the number of iterations needed to generate a new independent configuration, grows with increasing length scale ξ. In simulations of lattice QCD where the upgrading methods are essentially local, it has been observed that the autocorrelation times of topological modes are typically much larger than those of other observables not related to topology, such as Wilson loops and their correlators, see for instance Refs. [2]–[7]. Recent Monte Carlo simulations [5, 6] of the 4-d SU(N) lattice gauge theories (for N = 3, 4, 6) provided evidence of a severe CSD for the topological modes, using a rather standard local overrelaxed upgrading algorithm (constructed taking a mixture of overrelaxed microcanonical and heat-bath updatings). Indeed, the autocorrelation time τQ of the topological charge grows very rapidly with the length scale ξ ≡ σ, where σ is the string tension, showing an apparent exponential behavior τQ ∼ exp(cξ) in the range of values of ξ where data are available. This peculiar effect was not observed in plaquette–plaquette or Polyakov line correlations, suggesting an approximate decoupling between topological modes and non-topological ones, such as those determining the confining properties. The issue of the CSD of topological modes is particularly important for lattice QCD, because it may pose a serious limitation for numerical studies of physical issues related to topological properties, such as the mass and the matrix elements of the η meson, and in general the physics related to the broken U(1)A symmetry. The above-mentioned results suggest that the dynamics of the topological modes in Monte Carlo simulations is rather different from that of quasi-Gaussian modes. CSD of quasi-Gaussian modes for traditional local algorithms, such as standard Metropolis or heat bath, is related to an approximate random-walk spread of information around the lattice. Thus, the corresponding autocorrelation time τ is expected to behave as τ ∼ ξ (an independent configuration is obtained when the information travels a distance of the order of the correlation length ξ, and the information is transmitted from a given site/link to the nearest neighbors). This guess is correct for Gaussian (free-field) models; in general it is expected that τ ∼ ξ, where z is a dynamical critical exponent, and z ≈ 2 for quasi-Gaussian modes. 1 On the Optimized overrelaxation procedures may achieve a reduction of z, although the condition z ≥ 1 holds for local algorithms [8].