Finite-Size Effects in the ${\varphi }^4$ Field Theory Above the Upper Critical Dimension

Abstract

We demonstrate that the standard O(n) symmetric ${\varphi }^4$ field theory does not correctly describe the leading finite-size effects near the critical point of spin systems 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 $n\to \infty$ and $n=1$ explicit results are given for the susceptibility and for the Binder cumulant. They imply that recent analyses of Monte-Carlo results for the five-dimensional Ising model are not conclusive.

Comments: 4 pages, latex, 1 figure