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arXiv:2112.10162 · Physical review. E

Backbone and shortest-path exponents of the two-dimensional Q -state Potts model

Sheng Fang, Da Ke, Wei Zhong, Youjin Deng

Published Apr 15, 2022• 9 citations
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Abstract

We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional $Q$-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents ${d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)$ and 1.794(2) for $Q=2,3$, respectively. However, for large $Q$, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the $\mathrm{O}(n)$ loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with $Q={n}^{2}$. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops ${d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)$ for $Q=1,2,3,2+\sqrt{3},4$, respectively, and ${d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)$ for $Q=2,3,2+\sqrt{3},4$, respectively. Our estimates significantly improve over the existing results for both ${d}_{B}$ and ${d}_{\mathrm{min}}$. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of $n$ for the leading correction exponent.

DOI
10.1103/physreve.105.044122

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We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional QQQ-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents dB=1.7320(3){d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)dB​=1.7320(3) and 1.794(2) for Q=2,3Q=2,3Q=2,3, respectively. However, for large QQQ, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O(n)\mathrm{O}(n)O(n) loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with Q=n2Q={n}^{2}Q=n2. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops dB=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6){d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)dB​=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6) for Q=1,2,3,2+3,4Q=1,2,3,2+\sqrt{3},4Q=1,2,3,2+3​,4, respectively, and dmin=1.0945(2),1.0675(3),1.0475(3),1.0322(4){d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)dmin​=1.0945(2),1.0675(3),1.0475(3),1.0322(4) for Q=2,3,2+3,4Q=2,3,2+\sqrt{3},4Q=2,3,2+3​,4, respectively. Our estimates significantly improve over the existing results for both dB{d}_{B}dB​ and dmin{d}_{\mathrm{min}}dmin​. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of nnn for the leading correction exponent.
We study the three-point correlation function of the backbone in the two-dimensional Q-state Potts model using the Fortuin-Kasteleyn (FK) representation. The backbone is defined as the biconnected skeleton of an FK cluster after removing all dangling ends and bridges. To circumvent the severe critical slowing down in direct Potts simulations for large Q, we employ large-scale Monte Carlo simulations of the O(n) loop model on the hexagonal lattice, which is regarded to correspond to the Potts model with Q=n^{2}. Using a highly efficient cluster algorithm, we compute the universal three-point amplitude ratios for the backbone (R_{BB}) and FK clusters (R_{FK}). Our computed R_{FK} exhibits excellent agreement with exact conformal field theory predictions, validating the reliability of our numerical approach. In the critical regime, we find that R_{BB} is systematically larger than R_{FK}. Conversely, along the tricritical branch, R_{BB} and R_{FK} coincide within numerical accuracy, strongly suggesting that R_{BB}=R_{FK} holds throughout this regime. This finding mirrors the known equality of the backbone and FK cluster fractal dimensions at tricriticality, jointly indicating that both structures share the same geometric universality.
We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.
The two-dimensional Potts models were simulated with the use of a Monte Carlo method. This study is unique in that we simulated the weighted percolation clusters first described by Kasteleyn and Fortuin in 1969. We describe an auxiliary data structure which enables us to determine the connectedness of large clusters very efficiently. Our simulation of the percolation problem was found not to be affected by a critical slowing down. Critical exponents were found by studying the size and shape of large clusters. Our results agree with previous studies done on integer values of qqq, and with the conjecture of den Nijs.
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.
Abstract We have considered clusters of like spins in the Q -Potts model, the spin Potts clusters (S clusters). Using Monte Carlo simulations, we studied these clusters on a square lattice with periodic boundary conditions for values of Q ∈ [1, 4]. We continue the work initiated by Delfino et al (2013 J. Stat. Mech. P11011) by measuring the universal finite size corrections of the two-point connectivity. The numerical data are perfectly compatible with the conformal field theory (CFT) prediction, thus supporting the existence of a consistent CFT, still unknown, describing the connectivity Potts spin clusters. We provided in particular new insights on the energy field of such theory. For Q = 2, we found a good agreement with the prediction that the Ising spin clusters behave as the Fortuin–Kasteleyn ones at the tri-critical point of the dilute one-Potts model. We show that the structure constants are likely to be given by the imaginary Liouville structure constants, consistently with the results by Delfino et al (2013 J. Stat. Mech. P11011); Ang and Sun (2021 arXiv:2107.01788). For Q ≠ 2 instead, the structure constants we measure do not correspond to any known bootstrap solutions. The validity of our analysis is backed up by the measures of the spin Potts cluster wrapping probability for Q = 3. We evaluate the main critical exponents and the correction to the scaling. A new exact and compact expression for the torus one-point of the Q -Potts energy field is also given.
The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a-1 (a=e(betaJ)) and the coefficients of this polynomial, Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z(CP)=Z(Q,a(c)(Q)), where a(c)(Q)=1+square root[Q]. We find a set of zeros in the complex w=square root[Q] plane that map to (or close to) the Beraha numbers for real positive Q. We also identify Q(c)(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/square root[Q] plane lies on the unit circle. By finite-size scaling we find that 1/Q(c)(L)-->0 as L-->infinity. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q(c)(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter's conjecture Q(AF)(c)(a)=(1-a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q(FM)(c)(a)=(a-1)(2). We show that the edge singularity in the complex Q plane approaches Q(c) as Q(c)(L) approximately Q(c)+AL(-y(q)), and determine the scaling exponent y(q) for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y(t) as a function of Q in the range 2</=Q</=3. Using data for lattices of size 3</=L</=8 we find that y(t) is a smooth function of Q and is well fitted by y(t)=(1+Au+Bu2)/(C+Du) where u=-(2/pi)cos(-1)(squareroot[Q]/2). For Q=3 we find y(t) approximately 0.6; however if we include lattices up to L=12 we find y(t) approximately 0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].