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

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.