Field-theoretical calculations predict that, at the upper critical dimension
dc=4, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents $({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h})=(1/6,1/4)$. Using high-efficient cluster algorithms and the lifted worm algorithm, we present a systematic study to the FSS of the four-dimensional Ising model at criticality in the Fortuin-Kasteleyn (FK) bond and loop representations. In the FK representation, the size of the largest cluster is observed to scale as ${C}_{1}\ensuremath{\sim}{L}^{3}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h}}$, while the size of the second-largest cluster scales as ${C}_{2}\ensuremath{\sim}{L}^{3}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h2}}$ with ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h2}=\ensuremath{-}1/4$ a new correction exponent not yet predicted from field theory. In the loop representation, we observe that the size of the largest loop cluster scales as ${F}_{1}\ensuremath{\sim}{L}^{2}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t}}$, and the specific heat scales as ${c}_{\mathrm{E}}\ensuremath{\sim}{(lnL)}^{2{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t}}$. This clarifies the long-standing open question that whether the specific heat for the critical Ising model at
dc=4 diverges logarithmically.
The authors review exact studies on finite-sized 2 dimensional Ising models and show that the point for an infinite-sized model at the critical temperature is a point of nonuniform approach in the temperature-size plane. They also illuminate some strong effects of finite-size on quantities which do not diverge at the critical point. They then review Monte Carlo studies for 3 dimensional Ising models of various sizes (L = 2--100) at various temperatures. From these results they find that the data for the renormalized coupling constant collapses nicely when plotted against the correlation length, determined in a system of edge length L, divided by L. They also find that {zeta}{sub L}/L {ge} 0.26 is definitely too large for reliable studies of the critical value, g*, of the renormalized coupling constant. They have reasonable evidence that {zeta}{sub L}/L {approx} 0.1 is adequate for results that are within one percent of those for the infinite system size. On this basis, they have conducted a series of Monte Carlo calculations with this condition imposed. These calculations were made practical by the development of improved estimators for use in the Swendsen-Wang cluster method. The authors found from these results, coupled with a reversed limit computation (size increases with the temperature fixed at the critical temperature), that g* > 0, although there may well be a sharp downward drop in g as the critical temperature is approached in accord with the predictions of series analysis. The results support the validity of hyperscaling in the 3 dimensional Ising model.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.