Xiao T, Li Z, Zhou Z, Fang S, Deng Y. Finite-size scaling of the high-dimensional Ising model in the loop representation.
Phys Rev E 2024;
109:034125. [PMID:
38632761 DOI:
10.1103/physreve.109.034125]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2023] [Accepted: 02/22/2024] [Indexed: 04/19/2024]
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, 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_{p}=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 d_{c}=4.
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