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

TL;DR

This study analyzes Monte Carlo data for the five-dimensional Ising model's magnetization moments, comparing them to a recent finite-size scaling theory, revealing its strengths and limitations in explaining critical phenomena above the upper critical dimension.

Contribution

The paper revisits finite-size scaling above the upper critical dimension, testing a two-variable scaling theory against Monte Carlo results for the 5D Ising model, highlighting its improved accuracy over previous single-variable approaches.

Findings

The two-variable scaling theory better explains Monte Carlo data than the lowest-mode theory.

Discrepancies remain in susceptibility predictions, indicating limitations of the current theory.

The theory's corrections improve understanding of finite-size effects near criticality.

Abstract

Monte Carlo results for the moments <M^k> of the magnetization distribution of the nearest-neighbor Ising ferromagnet in a L^d 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 (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_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 <M^2>). Reasons for these problems are briefly discussed.