Finite-size scaling above the upper critical dimension revisited: the case of the five-dimensional Ising model

Abstract

Monte-Carlo results for the moments $\langle M^k\rangle$ of the magnetization distribution of the nearest-neighbor Ising ferromagnet in a Ld geometry, where L ($4 \leq L \leq 22$) is the linear dimension of a hypercubic lattice with periodic boundary conditions in d=5 dimensions, are analyzed in the critical region and compared to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod. Phys. C 9, 1007 (1998)]. We show that this finite-size scaling theory (formulated in terms of two scaling variables) can account for the longstanding discrepancies between Monte-Carlo results and the so-called "lowest-mode" theory, which uses a single scaling variable $tL^{d/2}$ where $t=T/T_{\rm c}-1$ is the temperature distance from the critical temperature, only to a very limited extent. While the CD theory gives a somewhat improved description of corrections to the "lowest-mode" results (to which the CD theory can easily be reduced in the limit $t \to 0$, $L \to \infty$, $tL^{d/2}$ fixed) for the fourth-order cumulant, discrepancies are found for the susceptibility ($L^d \langle M^2 \rangle$). Reasons for these problems are briefly discussed