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arXiv:1406.2913 · Journal of Statistical Mechanics Theory and Experiment

Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit

István A Kovács, Jean-Christian Anglès d’Auriac, Ferenc Iglói

Published Sep 16, 2014• 1 citation
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Abstract

We consider the random-bond Potts model in the large- Q limit and calculate the excess entropy, S Γ , of a contour, Γ , which is given by the mean number of Fortuin–Kasteleyn clusters which are crossed by Γ . In two dimensions, S Γ is proportional to the length of Γ , to which—at the critical point—there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess entropy and independently from the finite-size correction of the free-energy as: lim Q → ∞ c ( Q )/ln Q = 0.74(2), close to previous estimates calculated at finite values of Q .

DOI
10.1088/1742-5468/2014/09/p09019

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We consider the random-bond Potts model in the large- Q limit and calculate the excess entropy, S Γ , of a contour, Γ , which is given by the mean number of Fortuin–Kasteleyn clusters which are crossed by Γ . In two dimensions, S Γ is proportional to the length of Γ , to which—at the critical point—there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess entropy and independently from the finite-size correction of the free-energy as: lim Q → ∞ c ( Q )/ln Q = 0.74(2), close to previous estimates calculated at finite values of Q .
We consider the random-bond Potts model in the large-Q limit and calculate the excess entropy, SΓ, of a contour, Γ, which is given by the mean number of Fortuin-Kasteleyn clusters which are crossed by Γ. In two dimensions SΓ is proportional to the length of Γ, to which - at the critical point - there are universal logarithmic corrections due to corners. These are calculated by applying tech- niques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess entropy and independently from the finite-size correction of the free-energy as: limQ→∞ c(Q)/ ln Q = 0.74(2), close to previous estimates calculated at finite values of Q.
For the two-dimensional QQQ-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number ${N}_{\ensuremath{\Gamma}}$ of clusters that intersect a given contour $\ensuremath{\Gamma}$. To leading order, ${N}_{\ensuremath{\Gamma}}$ is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of $\ensuremath{\Gamma}$. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition, we find agreement with these predictions from large-scale numerical simulations. For the spin clusters, on the other hand, the cluster numbers are not found to be consistent with the values obtained by analytic continuation, as conventionally assumed.
The entanglement entropy for a quantum critical system across a boundary with a corner exhibits a subleading logarithmic scaling term with a scale-invariant coefficient. Using a Numerical Linked Cluster Expansion, we calculate this universal quantity for a square-lattice bilayer Heisenberg model at its quantum critical point. We find, for this 2 + 1 dimensional O (3) universality class, that it is thrice the value calculated previously for the Ising universality class. This relation gives substantial evidence that this coefficient provides a measure of the number of degrees of freedom of the theory, analogous to the central charge in a 1 + 1 dimensional conformal field theory.
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 study the number of clusters in two-dimensional (2d2d2d) critical percolation, ${N}_{\ensuremath{\Gamma}}$, which intersect a given subset of bonds, $\ensuremath{\Gamma}$. In the simplest case, when $\ensuremath{\Gamma}$ is a simple closed curve, ${N}_{\ensuremath{\Gamma}}$ is related to the entanglement entropy of the critical diluted quantum Ising model, in which $\ensuremath{\Gamma}$ represents the boundary between the subsystem and the environment. Due to corners in $\ensuremath{\Gamma}$ there are universal logarithmic corrections to ${N}_{\ensuremath{\Gamma}}$, which are calculated in the continuum limit through conformal invariance, making use of the Cardy-Peschel formula. The exact formulas are confirmed by large-scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity.
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.