Cutoff and lattice effects in the theory of confined systems

Abstract

We study cutoff and lattice effects in the O(n) symmetric $\phi^4$ theory for a $d$-dimensional cubic geometry of size $L$ with periodic boundary conditions. In the large-N limit above $T_c$, we show that $\phi^4$ field theory at finite cutoff $\Lambda$ predicts the nonuniversal deviation $\sim (\Lambda L)^{-2}$ from asymptotic bulk critical behavior that violates finite-size scaling and disagrees with the deviation $\sim e^{-cL}$ that we find in the $\phi^4$ lattice model. The exponential size dependence requires a non-perturbative treatment of the $\phi^4$ model. Our arguments indicate that these results should be valid for general $n$ and $d > 2$.