Finite-size scaling of the high-dimensional Ising model in the loop representation

Besides its original spin representation, the Ising model is known to have the Fortuin-Kasteleyn (FK) bond and loop representations, of which the former was recently shown to exhibit two upper critical dimensions (d₂=4, d₏=6). Using a lifted worm algorithm, we determine the critical coupling as K₂=0. 0770. 16em{0ex}7080. 16em{0ex}91 (4) for d=7, which significantly improves over the previous results, and then study critical geometric properties of the loop Ising clusters on tori for spatial dimensions d=5 to 7. We show that as the spin representation, the loop Ising model has only one upper critical dimension at d₂=4. However, sophisticated finite-size scaling (FSS) behaviors, such as two length scales, two configuration sectors, and two scaling windows, still exist as the interplay effect of the Gaussian fixed point and complete-graph asymptotics. Moreover, using the loop-cluster algorithm, we provide an intuitive understanding of the emergence of the percolation-like upper critical dimension d₏=6 in the FK-Ising model. As a consequence, a unified physical picture is established for the FSS behaviors in all three representations of the Ising model above d₂=4.