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Journal of Physics Conference Series

Universal asymptotic correlation functions for point groups C 4v and C 6v : Numerical observation for square- and triangular-lattice Q -state Potts models

Hiromi Otsuka, Masafumi Fujimoto

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Abstract We focus on asymptotic correlation functions of off-critical models and propose a universal form possessing the C 4v symmetry and that possessing the C 6v symmetry. The former (the latter) is adaptable to classical spin systems on a square (triangular) lattice. Intriguingly, the C 4v form includes three free parameters—an amplitude A , the modulus k , and a deformation parameter b . Meanwhile, the C 6v form does the two of them, i.e., A and k . To proffer numerical evidence to our proposal, we perform large-scale Monte Carlo (MC) simulations of the Q -state Potts models on square and triangular lattices in the paramagnetic phase. We determine the free parameters to fit the forms to the MC data of correlation functions. Then, we find that both universal forms can describe asymptotic behaviors of correlation functions with high accuracy. Also, the C 4v and C 6v forms yield the equilibrium crystal shapes of the square-lattice and honeycomb-lattice models, respectively. These extracted curves of crystal shapes differ, representing the difference of lattice symmetries. Meanwhile, they are both algebraic curves of genus 1 and can become birationally equivalent if the values of k coincide. Our observation indicates that k provides a new measure for comparing the properties of different models on different lattices in two dimensions.

DOI
10.1088/1742-6596/3161/1/012021

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Abstract We focus on asymptotic correlation functions of off-critical models and propose a universal form possessing the C 4v symmetry and that possessing the C 6v symmetry. The former (the latter) is adaptable to classical spin systems on a square (triangular) lattice. Intriguingly, the C 4v form includes three free parameters—an amplitude A , the modulus k , and a deformation parameter b . Meanwhile, the C 6v form does the two of them, i.e., A and k . To proffer numerical evidence to our proposal, we perform large-scale Monte Carlo (MC) simulations of the Q -state Potts models on square and triangular lattices in the paramagnetic phase. We determine the free parameters to fit the forms to the MC data of correlation functions. Then, we find that both universal forms can describe asymptotic behaviors of correlation functions with high accuracy. Also, the C 4v and C 6v forms yield the equilibrium crystal shapes of the square-lattice and honeycomb-lattice models, respectively. These extracted curves of crystal shapes differ, representing the difference of lattice symmetries. Meanwhile, they are both algebraic curves of genus 1 and can become birationally equivalent if the values of k coincide. Our observation indicates that k provides a new measure for comparing the properties of different models on different lattices in two dimensions.
We investigate a universal curve in asymptotic correlation functions of off-critical systems that possess C_{6v} symmetry following the argument for C_{4v} symmetry in our previous paper [Phys. Rev. E 102, 032141 (2020)2470-004510.1103/PhysRevE.102.032141]. Unlike the C_{4v} case, a minimal asymptotic form exists, which contains only two free parameters: the normalization constant and the modulus of the universal curve. We perform large-scale Monte Carlo simulations of the triangular lattice Q-state Potts model above the transition temperature. For Q=1, 2, 3, and 4, we successfully obtain numerical evidence that the minimal form gives the leading asymptotic behavior. We also discuss the possibility that the corrections to the minimal form are expressed using this form as a building block. From the minimal form with optimized parameters, we derive the equilibrium crystal shape of the honeycomb lattice Potts model, which is given by an algebraic curve of genus 1 and is universal among models with C_{6v}. Although the curve differs from those obtained in the C_{4v} case, the latter curves also have genus 1. We indicate that the birational equivalence concept can play an important role in comparing asymptotic forms for different point group symmetries, for example, C_{6v} and C_{4v}.
Reexamining algebraic curves found in the eight-vertex model, we propose an asymptotic form of the correlation functions for off-critical systems possessing rotational and mirror symmetries of the square lattice, i.e., the C_{4v} symmetry. In comparison with the use of the Ornstein-Zernike form, it is efficient to investigate the correlation length with its directional dependence (or anisotropy). We investigate the Q-state Potts model on the square lattice. Monte Carlo (MC) simulations are performed using the infinite-size algorithm by Evertz and von der Linden. Fitting the MC data with the asymptotic form above the critical temperature, we reproduce the exact solution of the the anisotropic correlation length (ACL) of the Ising model (Q=2) within a five-digit accuracy. For Q=3 and 4, we obtain numerical evidence that the asymptotic form is applicable to their correlation functions and the ACLs. Furthermore, we successfully apply it to the bond percolation problem which corresponds to the Q→1 limit. From the calculated ACLs, the equilibrium crystal shapes (ECSs) are derived via duality and Wulff's construction. Regarding Q as a continuous variable, we find that the ECS of the Q-state Potts model is essentially the same as those of the Ising models on the Union Jack and 4-8 lattices, which are represented in terms of a simple algebraic curve of genus 1.
We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q -state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model ( Q = 2) at temperature T , as well as a fully packed O( n ) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q , n or 1/ T . By using a carefully chosen expansion parameter, q ≪ 1, all expansions turn out to be of the form , where the coefficients α k and β k are periodic functions of k . Thanks to this periodicity property, we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 ⩽ q < 1. We analyse in detail the q → 1 − limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases. Finally we work out the FLM formulae for the case where some of the system's boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen–Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r . These results are in turn compared with CFT predictions.
. Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out. 1. Introduction The systematic study of subdominant terms in scaling provides researchers with guidelines on how to extrapolate, e.g. finite-size results to the thermodynamic limit [1--5]. Also, it may in itself bring out connections to underlying physical properties. These latter may be universal, such as the relationship between critical free-energy finite-width correction and conforma...
We present a method which allows calculation of whole universal finite-size-scaling functions from Monte Carlo data. The crux of the new method is a technique of isolating the singular part from the total free energy. We apply this method to the 3-state Potts model on a square lattice and find the normalized scaling function Y(x) in the form 1+x+5.31x2{\mathit{x}}^{2}x2-1.2x3{\mathit{x}}^{3}x3-0.67x4{\mathit{x}}^{4}x4 . . . near the bulk critical point x=0, together with the normalizing universal amplitude E=1.053 and a nonuniversal metric factor D=-0.387. It also behaves as Y(x)\ensuremath{\approxeq}(\ifmmode\pm\else\textpm\fi{}x${)}^{\mathit{d}\ensuremath{\nu}}$${\mathit{Q}}_{\ifmmode\pm\else\textpm\fi{}}$ for large x in agreement with the hyperuniversality hypothesis with Q+{\mathit{Q}}_{+}Q+​/${\mathit{Q}}_{\mathrm{\ensuremath{-}}}$\ensuremath{\approxeq}1.
Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out.