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We present a perturbative calculation of finite-size effects near $T_c$ ofthe $\phi^4$ lattice model in a $d$-dimensional cubic geometry of size $L$ withperiodic boundary conditions for $d > 4$. The structural differences betweenthe $\phi^4$ lattice theory and the $\phi^4$ field theory found previously inthe spherical limit are shown to exist also for a finite number of componentsof the order parameter. The two-variable finite-size scaling functions of thefield theory are nonuniversal whereas those of the lattice theory areindependent of the nonuniversal model parameters.One-loop results forfinite-size scaling functions are derived. Their structure disagrees with thesingle-variable scaling form of the lowest-mode approximation for any finite$\xi/L$ where $\xi$ is the bulk correlation length. At $T_c$, the large-$L$behavior becomes lowest-mode like for the lattice model but not for thefield-theoretic model. Characteristic temperatures close to $T_c$ of thelattice model, such as $T_{max}(L)$ of the maximum of the susceptibility$\chi$, 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 Isingmodel. We also predict $\chi_{max} \sim L^{d/2}$ asymptotically. On aquantitative level, the asymptotic amplitudes of this large -$L$ behavior closeto $T_c$ have not been observed in previous MC simulations at $d = 5$ becauseof nonnegligible finite-size terms $\sim L^{(4-d)/2}$ caused by theinhomogeneous modes. These terms identify the possible origin of a significantdiscrepancy between the lowest-mode approximation and previous MC data. MC dataof larger systems would be desirable for testing the magnitude of the$L^{(4-d)/2}$ and $L^{4-d}$ terms predicted by our theory.Comment: Accepted in Int. J. Mod. Phys.

Lattice φ4 Theory of Finite-Size Effects Above the Upper Critical Dimension

Lattice φ⁴ Theory of Finite-Size Effects Above the Upper Critical Dimension