Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911833632276480 |
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| author | Xiao, Tianning Li, Zhiyi Zhou, Zongzheng Fang, Sheng Deng, Youjin |
| author_facet | Xiao, Tianning Li, Zhiyi Zhou, Zongzheng Fang, Sheng Deng, Youjin |
| contents | 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 $d_c=4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_11712 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation Xiao, Tianning Li, Zhiyi Zhou, Zongzheng Fang, Sheng Deng, Youjin Statistical Mechanics 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 $d_c=4$. |
| title | Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2310.11712 |