AbstractIn this project finite-size size scaling above the upper critical dimension dc is investigated. Finite-size scaling there has long been poorly understood, especially its dependency on boundary conditions. The violation of the hyperscaling relation above dc has also been one of the most visible issues. The breakdown in standard scaling is due to the dangerous irrelevant variables presented in the self-interacting term in theφ4theory, which were considered dangerous to the free energy density and associated thermodynamic functions, but not to the correlation sector. Recently, a modified finite-size scaling scheme has been proposed, which considers that the correlation length actually plays a pivotal role and is affected by dangerous variables too. This new scheme, named QFSS, con-siders that the correlation length, instead of having standard scaling behaviourξ∼L,scales asξ∼Lϙ. This pseudocritical exponent is connected to the critical dimension and dangerous variables. Below dc this exponent takes the valueϙ= 1, but above the upper critical dimension it isϙ=d/dc.
QFSS succeeded in reconciling the mean-field exponents and FSS derived from the renormalisation-group for the models with short-range interactions above dc with periodic boundary conditions. Ifϙis an universal exponent, the validity of that theory should also hold for the free boundary conditions. Initial tests for such systems faced new problems. Whereas QFSS is valid at pseudocritical points where quantities such as the magnetic susceptibility experience a peak for finite systems, at critical points the standard FSS seemed to prevail, i.e., mean-field exponents withξ∼L. Here, we show that this last picture at critical point is not correct and instead the exponents that applied there actually arise from the Gaussian fixed-point FSS where the dangerous variables are suppressed. To achieve this aim, we study Ising models with long-range interaction, which can be tuned above dc, with periodic and free boundary conditions. We also include a study of the Fourier modes which can be used as an example of scaling quantities without dangerous variables.
|Date of Award||2016|
|Supervisor||Ralph Kenna (Supervisor), Bertrand Berche (Supervisor) & Martin Weigel (Supervisor)|