Field-theoretical calculations predict that, at the upper critical dimension ${d}_{c}=4$, 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 ${d}_{c}=4$ diverges logarithmically.
Field-theoretical calculations predict that, at the upper critical dimension dc=4, the finite-size scaling (FSS) behaviors of the Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents (y\ifmmode^\elseˆ\fit,y\ifmmode^\elseˆ\fih)=(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 C1\ensuremath∼L3(lnL)y\ifmmode^\elseˆ\fih, while the size of the second-largest cluster scales as C2\ensuremath∼L3(lnL)y\ifmmode^\elseˆ\fih2 with y\ifmmode^\elseˆ\fih2=\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 F1\ensuremath∼L2(lnL)y\ifmmode^\elseˆ\fit, and the specific heat scales as cE\ensuremath∼(lnL)2y\ifmmode^\elseˆ\fit. This clarifies the long-standing open question that whether the specific heat for the critical Ising model at dc=4 diverges logarithmically.
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.
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.
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). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size 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 $x\ensuremath{\equiv}{C}_{1}/{L}^{{d}_{\mathrm{F}}}$ (L is the linear size), where the fractal dimension 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) and 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) and dB(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents dmin and 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 study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds, Nc and Nb, respectively. These quantities include ${C}_{2c}=(⟨{N}_{c}^{2}⟩\ensuremath{-}{⟨{N}_{c}⟩}^{2})∕{L}^{d}$ and ${C}_{cb}=(⟨{N}_{c}{N}_{b}⟩\ensuremath{-}⟨{N}_{c}⟩⟨{N}_{b}⟩)∕{L}^{d}$, where L is the linear system size and d is the spatial dimensionality. In statistical models with thermal fluctuations, C2c and Ccb are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of C2c and Ccb is described by the thermal critical exponent ${y}_{t}\text{\ensuremath{-}}d$. We also measure quantity ${\ensuremath{\kappa}}_{b}=2⟨{N}_{b}{S}_{2}⟩∕⟨{S}_{2}⟩\ensuremath{-}⟨{N}_{b}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩∕⟨{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩\ensuremath{-}⟨{N}_{b}⟩$ and an analogous quantity ${\ensuremath{\kappa}}_{c}$ for Nc, where S2 and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}$ are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ diverge as Lyt for $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The analysis of the data of ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ yields yt=1.145(2) for the three-dimensional percolation, in good agreement with existing results.
Abstract Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (dc = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows r2−dc governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as L−dc/2(lnL)p^ with p^=1/2 a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.
The four-dimensional Ising model is simulated on Creutz cellular automatons using finite lattices with linear dimensions 4 ≤ L ≤ 8. The temperature variations and finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature for 7, 14, and 21 independent simulations. Approximate values for the critical temperature of the infinite lattice of Tc(∞) = 6.6965(35), 6.6961(30), 6.6960(12), 6.6800(3), 6.6801(2), 6.6802(1) and 6.6925(22) (without the logarithmic factor), 6.6921(22) (without the logarithmic factor), 6.6909(2) (without the logarithmic factor), 6.6822(13) (with the logarithmic factor), 6.6819(11) (with the logarithmic factor), and 6.6808(8) (with the logarithmic factor) are obtained from the intersection points of the specific heat curves, the Binder parameter curves, and straight line fits of specific heat maxima for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the results, 6.6802(1) and 6.6808(8), are in very good agreement with the results of a series expansion of Tc(∞), 6.6817(15) and 6.6802(2), the dynamic Monte Carlo value Tc(∞) = 6.6803(1), the cluster Monte Carlo value Tc(∞) = 6.680(1), and the Monte Carlo value using the Metropolis-Wolff cluster algorithm Tc(∞) = 6.6802632 ± 5 · 10−5. The average values calculated for the critical exponent of the specific heat are α =− 0.0402(15), − 0.0393(12), − 0.0391(11) with 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the result, α =− 0.0391(11), agrees with the series expansions result, α =− 0.12 ± 0.03 and the Monte Carlo result using the Metropolis-Wolff cluster algorithm, α ≥ 0 ± 0.04. However, α =− 0.0391(11) is inconsistent with the renormalization group prediction of α = 0.