Computation of the effective potential for the two-dimensionalO(N) nonlinear σ-models

A Monte Carlo method for the computation of effective Hamiltonians for theO(N) nonlinear lattice σ-models is described. The procedure is based on simulations of auxiliary statistical mechanical systems with fixed block spins and thus avoids the simulation of systems with large correlation length. The method is applied to study the renormalization group flow of the effective potential for the Ising model, the XY model, and the Heisenberg ferromagnet in two dimensions. Some of the results are compared with second order high temperature expansions. For small block size a very good agreement is found in a large range of the inverse temperature β.