Abstract
There has been a long running debate on the finite size scaling for the Ising model with free boundary conditions above the upper critical dimension, where the standard picture gives an L2 scaling for the susceptibility and an alternative theory has promoted an L5/2 scaling, as would be the case for cyclic boundary. In this paper we present results from simulation of the far largest systems used so far, up to side L=160 and find that this data clearly supports the standard scaling. Further we present a discussion of why rigorous results for the random-cluster model provide both supports for the standard scaling picture and a clear explanation of why the scalings for free and cyclic boundary should be different.
Highlights
IntroductionAs can be seen from the references in those two papers there has been some debate on whether the standard scaling picture, saying that e.g. the susceptibility scales as L2 for free boundary, holds or whether an alternative theory proposing that it scales as L5/2 is correct
The upper critical dimension d = 4, for the Ising model with nearest-neighbour interaction, the critical exponents assume their mean field values [1,2]; α = 0, β = 1/2, γ = 1, ν = 1/2, and the so-called hyperscaling law dν = 2−α fails for d > 4
As can be seen from the references in those two papers there has been some debate on whether the standard scaling picture, saying that e.g. the susceptibility scales as L2 for free boundary, holds or whether an alternative theory proposing that it scales as L5/2 is correct
Summary
As can be seen from the references in those two papers there has been some debate on whether the standard scaling picture, saying that e.g. the susceptibility scales as L2 for free boundary, holds or whether an alternative theory proposing that it scales as L5/2 is correct. In a reply [11] it was again suggested that the alternative picture is correct and that the results of [10] were due to too small systems, dominated by finite size effects stemming from their large boundaries. We compare how well the standard scaling and the alternative theory fit our new large system data, and discuss why, based on mathematical results on the random cluster model, there are good reasons for expecting the standard picture to be the correct one, as the data suggests
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