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Cutoff and lattice effects in the $\varphy^4$ theory of confined systems

We study cutoff and lattice effects in the O(n) symmetric $ϕ^4$ 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 $ϕ^4$ field theory at finite cutoff $Λ$ predicts the nonuniversal deviation $\sim (ΛL)^{-2}$ from asymptotic bulk critical behavior that violates finite-size scaling and disagrees with the deviation $\sim e^{-cL}$ that we find in the $ϕ^4$ lattice model. The exponential size dependence requires a non-perturbative treatment of the $ϕ^4$ model. Our arguments indicate that these results should be valid for general $n$ and $d > 2$.