Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation

Abstract

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 $K_c=0.07770891(4)$ 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 $d_c=4$.

12 pages, 12 figures

References in corpus (7)

• A Guide to Stochastic Loewner Evolution and its Applications
• Worm Monte Carlo study of the honeycomb-lattice loop model
• The discontinuity of the specific heat for the 5D Ising model
• Geometric Upper Critical Dimensions of the Ising Model
• Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
• Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph
• Geometric properties of the complete-graph Ising model in the loop representation

Cited by in corpus (1)

• Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model