We give an intuitive geometric explanation for the apparent breakdown of standard finite-size scaling in systems with periodic boundaries above the upper critical dimension. The Ising model and self-avoiding walk are simulated on five-dimensional hypercubic lattices with free and periodic boundary conditions, by using geometric representations and recently introduced Markov-chain Monte Carlo algorithms. We show that previously observed anomalous behaviour for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings.Finite-size Scaling (FSS) is a fundamental physical theory within statistical mechanics, describing the asymptotic approach to the thermodynamic limit of finite systems in the neighbourhood of a critical phase transi-It is well-known [3] that models of critical phenomena typically possess an upper critical dimension, d c , such that in dimensions d ≥ d c , their thermodynamic behaviour is governed by critical exponents taking simple mean-field values [4]. In contrast to the simplicity of the thermodynamic behaviour, however, the theory of FSS in dimensions above d c is surprisingly subtle, and remains the subject of ongoing debate [5][6][7][8][9][10][11][12]. We will show here that such subtleties can be explained in a simple way, by taking an appropriate geometric perspective.Perhaps the most important class of models in equilibrium statistical mechanics are the n-vector models [13], describing systems of pairwise-interacting unit-vector spins in R n [14]. The cases n = 1, 2, 3 respectively correspond to the Ising, XY and Heisenberg models of ferromagnetism, while the limiting case n = 0 corresponds to the Self-avoiding Walk (SAW) model of polymers [3].The n-vector model has wide-ranging applications in condensed matter physics, particularly in the theory of superfluidity/superconductivity and quantum magnetism. In addition, the case n = 2 is related to the Bose-Hubbard model [15] which is actively studied in the field of ultra-cold atom physics. In such quantum applications, the quantum system in d spatial dimensions is related to the classical model in d + 1 dimensions. Since [3] d c = 4 for the nearest-neighbour n-vector model, this shows that understanding its FSS when d ≥ d c is of importance not only to the theory of FSS itself, but also in the field of condensed matter physics more generally. We also note that the value of d c can be reduced by the introduction of long-range interactions.

The simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down. A number of cluster algorithms, usually based on the Fortuin-Kasteleyn representation of the Potts model, and suitable generalizations for continuousspin models have been used to increase simulation efficiency. The first algorithm making use of this representation, suggested by Sweeny in 1983, has not found widespread adoption due to problems in its implementation. However, it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well-known algorithm due to Swendsen and Wang. Here, we present an efficient implementation of Sweeny's approach for the random-cluster model using recent algorithmic advances in dynamic connectivity algorithms.

Original citation:Kovacs, Istvan A., Elci, Eren Metin., Weigel, Martin., and Igloi, Ferenc Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. For the two-dimensional Q-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number N of clusters that intersect a given contour . To leading order, N is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of . These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition, we find agreement with these predictions from large-scale numerical simulations. For the spin clusters, on the other hand, the cluster numbers are not found to be consistent with the values obtained by analytic continuation, as conventionally assumed. CURVE is the Institutional Repository for Coventry University