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arXiv:2310.11712 · Physical review. E

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

Tianning Xiao, Zhiyi Li, Zongzheng Zhou, Sheng Fang, Youjin Deng

Published Mar 21, 2024• 3 citations
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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.

DOI
10.1103/physreve.109.034125

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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.
Field-theoretical calculations predict that, at the upper critical dimension dc=4{d}_{c}=4dc​=4, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents $({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h})=(1/6,1/4)$. Using high-efficient cluster algorithms and the lifted worm algorithm, we present a systematic study to the FSS of the four-dimensional Ising model at criticality in the Fortuin-Kasteleyn (FK) bond and loop representations. In the FK representation, the size of the largest cluster is observed to scale as ${C}_{1}\ensuremath{\sim}{L}^{3}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h}}$, while the size of the second-largest cluster scales as ${C}_{2}\ensuremath{\sim}{L}^{3}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h2}}$ with ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{h2}=\ensuremath{-}1/4$ a new correction exponent not yet predicted from field theory. In the loop representation, we observe that the size of the largest loop cluster scales as ${F}_{1}\ensuremath{\sim}{L}^{2}{(lnL)}^{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t}}$, and the specific heat scales as ${c}_{\mathrm{E}}\ensuremath{\sim}{(lnL)}^{2{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}}_{t}}$. This clarifies the long-standing open question that whether the specific heat for the critical Ising model at dc=4{d}_{c}=4dc​=4 diverges logarithmically.
Recently, we argued [Chin. Phys. Lett. 39, 080502 (2022)0256-307X10.1088/0256-307X/39/8/080502] that the Ising model simultaneously exhibits two upper critical dimensions (d_{c}=4,d_{p}=6) in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions d from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for 4<d<6 and d≥6, and thus strongly support the argument that 6 is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.
The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the loop-cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales, and two scaling windows.
The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.
We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as Kc(3D)=0.221654631(8){K}_{c}(3\mathrm{D})=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}631(8)Kc​(3D)=0.221654631(8). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C1{C}_{1}C1​ follows a single-variable function as $P({C}_{1},L)\phantom{\rule{0.16em}{0ex}}d{C}_{1}=\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}(x)\phantom{\rule{0.16em}{0ex}}dx$ with $x\ensuremath{\equiv}{C}_{1}/{L}^{{d}_{\mathrm{F}}}$ (LLL is the linear size), where the fractal dimension dF{d}_{\mathrm{F}}dF​ is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}(x)$ in three dimensions, and attributed to the different approaching behaviors for $K\ensuremath{\rightarrow}{K}_{c}+{0}^{\ifmmode\pm\else\textpm\fi{}}$. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as dmin(3D)=1.25936(12){d}_{\mathrm{min}}(3\mathrm{D})=1.259\phantom{\rule{0.16em}{0ex}}36(12)dmin​(3D)=1.25936(12) and dmin(2D)=1.0940(2){d}_{\mathrm{min}}(2\mathrm{D})=1.094\phantom{\rule{0.16em}{0ex}}0(2)dmin​(2D)=1.0940(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as dB(3D)=2.1673(15){d}_{\mathrm{B}}(3\mathrm{D})=2.167\phantom{\rule{0.16em}{0ex}}3(15)dB​(3D)=2.1673(15) and dB(2D)=1.7321(4){d}_{\mathrm{B}}(2\mathrm{D})=1.732\phantom{\rule{0.16em}{0ex}}1(4)dB​(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents dmin{d}_{\mathrm{min}}dmin​ and dB{d}_{\mathrm{B}}dB​ either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.
We present an extensive Markov chain Monte Carlo study of the finite-size scaling behavior of the Fortuin-Kasteleyn Ising model on five-dimensional hypercubic lattices with periodic boundary conditions. We observe that physical quantities, which include the contribution of the largest cluster, exhibit complete graph asymptotics. However, for quantities where the contribution of the largest cluster is removed, we observe that the scaling behavior is mainly controlled by the Gaussian fixed point. Our results therefore suggest that both scaling predictions, i.e., the complete graph and the Gaussian fixed point asymptotics, are needed to provide a complete description for the five-dimensional finite-size scaling behavior on the torus.