Topological susceptibility on the lattice: The two-dimensional O(3)σmodel

As a test of the procedures used in lattice gauge theories, we study the topological susceptibility $\chi$ in a two-dimensional O(3) $\sigma$ or ${\mathrm{CP}}^1$ model. We determine $\chi$ by defining the density of topological charge as a local operator on the lattice. Following the prescriptions of field theory we perform the additive and multiplicative renormalizations needed to extract $\chi$ from Monte Carlo data. We also determine $\chi$ by the cooling method, finding consistent results. A combined use of cooling and field theory again gives the same result and insight into the renormalization mechanism. Finally we give a direct determination, by Monte Carlo techniques, of both the multiplicative and additive renormalizations, by heating configurations with a definite number of instantons. The results are consistent with perturbation theory.