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 $(d_c=4,d_p=6)$. Using a lifted worm algorithm, we determine the critical coupling as $K_c = 0.077\,708\,91(4)$ 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 $d_c=4$. However, sophisticated finite-size scaling (FSS) behaviors, like 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 $d_p=6$ in the FK-Ising model. As a consequence, a unified physical picture is established for the FSS behaviors in all the three representations of the Ising model above $d_c=4$.