  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.
