Topological charge on the lattice: The O(3) case

We discuss the topological classification of the field configurations on the lattice. The regulator may act as a source of topological defects of the order of the lattice spacing we introduce a family of operators Q_L (block topological charge) that discriminates against instantons of sizes less than or equal to L. We prove that the correct charge implies first taking the limit a → 0 with L fixed and then L → 0. Numerical simulations à la Monte Carlo for O(3) in two dimensions are discussed. The observed density of small instantons agrees very well with theoretical calculations in the continuum theory.