Corner contribution to cluster numbers in the Potts model

Abstract

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

Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit

“…In the following section we show, that in the random cluster representation S Γ is simply given by the mean number of clusters in the optimal sets which are crossed by Γ. This type of problem has already been considered by two of us in the case of the non-random Potts model both for Q = 1, representing percolation 28,29 and for general values of Q ≤ 4 30 . Repeating the reasoning applied in these papers we show that the dominant term of S Γ represents the area law to which there are logarithmic corrections at the critical point due to corners and these are calculated by conformal techniques.…”

“…2. To subtract the corner contribution from the data we have used the so called geometric approach 28,30,35 : for each sample N Γ is calculated in two different geometries, which have the same boundary term, but different corner ones. Thus the corner contribution is obtained from their difference.…”

“…We have also studied subsystems comprised of n = 1, 2, 3 or 4 crosses (see in Fig. 2 of Ref [30]), in which case we have obtained finite-size estimates for c ′ . For 1 and 3 crosses these are presented in Fig.…”

Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics

“…The study of statistical systems and their field theory representation in the presence of corners has been covered extensively, e.g. Ising and Potts model, loop model and percolation [73,137,100,99], using various theoretical and numerical machinery. In two dimensions, as pointed out by Cardy and Peschel [25], the universal contribution to the free energy of a critical system in a domain with a corner with angle θ has been determined using the complex transformation…”

Corner contribution to percolation cluster numbers in three dimensions

“…This prefactor has been calculated analytically through the application of the Cardy-Peschel formula 8 and the results have been checked trough Monte-Carlo simulations. Very recently these investigations have been extended to the Fortuin-Kasteleyn and spin clusters in the Q ≤ 4 state Potts model 9 (percolation being recovered in the limit Q → 1).…”