Iterating block spin transformations of the O(3) nonlinear sigma model

We study the iteration of block spin transformations in the O(3) symmetric nonlinear sigma model on a two-dimensional square lattice with the help of the Monte Carlo method. In contrast with the classical Monte Carlo renormalization group approach, we do attempt to explicitly compute the block spin effective actions. Using two different methods for the determination of effective couplings, we study the renormalization group flow for various parametrization and truncation schemes. The largest ansatz for the effective action contains thirteen coupling constants. Actions on the renormalized trajectory should describe theories with no lattice artifacts, even at a small correlation length. However, tests with the step scaling function of Luscher, Weisz, and Wolff reveal that our truncated effective actions show sizable scaling violations indicating that the Ansatze are still too small.