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

We present a perturbative calculation of finite-size effects near T_c of the φ⁴ lattice model in a d-dimensional cubic geometry of size L with periodic boundary conditions for d>4. The structural differences between the φ⁴ lattice theory and the φ⁴ 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) ~L^-d/2, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict χ_max~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 ~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.