Renormalization group flow and fixed point of the lattice topological charge in the 2-d O(3) sigma-model

Abstract

We study the renormalization group evolution up to the fixed point of the lattice topological susceptibility in the 2-d O(3) non-linear sigma-model. We start with a discretization of the continuum topological charge by a local charge density, polynomial in the lattice fields. Among the different choices we propose also a Symanzik--improved lattice topological charge. We check step by step in the renormalization group iteration the progressive dumping of quantum fluctuations, which are responsible for the additive and multiplicative renormalizations of the lattice topological susceptibility with respect to the continuum definition. We find that already after three iterations these renormalizations are negligible and an excellent approximation of the fixed point is achieved. We also check by an explicit calculation that the assumption of slowly varying fields in iterating the renormalization group does not lead to a good approximation of the fixed point charge operator.

arXiv comment29 pages (25 PostScript file text + 4 PostScript figures); revised version with new title, new abstract, added conclusions and other minor changes

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• The Schwinger Model with Perfect Staggered Fermions
• Topological susceptibility of $2d{\mathrm{CP}}^1$ or $\mathrm{O}(3)$ non-linear $\sigma$-model: is it divergent or not?
• The clash of positivities in topological density correlators
• $\theta$-dependence in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models
• Optimised Dirac Operators on the Lattice: Construction, Properties and Applications
• Instanton classical solutions of SU(3) fixed point actions on open lattices
• Topological Susceptibility of the 2d O(3) Model under Gradient Flow