We demonstrate that the standard O(n) symmetric phi(4) field theory does not correctly describe the leading finite-size effects near the critical point of spin systems with periodic boundary conditions on a d-dimensional lattice with d > 4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For n --> infinity and n = 1, explicit results are given for the susceptibility and for the Binder cumulant. They imply that these quantities do not have the universal properties predicted previously and that recent analyses of Monte ...
