Geometric properties of the complete-graph Ising model in the loop representation

Abstract

The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random-cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation, and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the Loop-Cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales and two scaling windows.

arXiv comment10 pages, 10 figures

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• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
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• Universality of the complete-graph Potts model with $0<q\le 2$