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

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