Lattice $ϕ^4$ theory of finite-size effects above the upper critical dimension

Abstract

We present a perturbative calculation of finite-size effects near $T_c$ of the $ϕ^4$ lattice model in a $d$-dimensional cubic geometry of size $L$ with periodic boundary conditions for $d > 4$. The structural differences between the $ϕ^4$ lattice theory and the $ϕ^4$ field theory found previously in the spherical limit are shown to exist also for a finite number of components of the order parameter. The two-variable finite-size scaling functions of the field theory are nonuniversal whereas those of the lattice theory are independent of the nonuniversal model parameters.One-loop results for finite-size scaling functions are derived. Their structure disagrees with the single-variable scaling form of the lowest-mode approximation for any finite $ξ/L$ where $ξ$ is the bulk correlation length. At $T_c$, the large-$L$ behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to $T_c$ of the lattice model, such as $T_{max}(L)$ of the maximum of the susceptibility $χ$, are found to scale asymptotically as $T_c - T_{max}(L) \sim L^{-d/2}$, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict $χ_{max} \sim L^{d/2}$ asymptotically. On a quantitative level, the asymptotic amplitudes of this large -$L$ behavior close to $T_c$ have not been observed in previous MC simulations at $d = 5$ because of nonnegligible finite-size terms $\sim L^{(4-d)/2}$ caused by the inhomogeneous modes. These terms identify the possible origin of a significant discrepancy between the lowest-mode approximation and previous MC data. MC data of larger systems would be desirable for testing the magnitude of the $L^{(4-d)/2}$ and $L^{4-d}$ terms predicted by our theory.