首页/论文/1909.00518

arXiv 1909.005182019-09-23math-phcond-mat.stat-mechmath.MP

Bulk, surface and corner free energies of the anisotropic triangular Ising model: series expansions and critical behaviour

R. J. Baxter

备注：: 25 pages, 2 figures, 2 additonal references
发表信息：: Proc. Roy. Soc. A 476: 20190713 (2020) 18 pages

英文摘要：

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman's spinor method to calculate low-temperature series expansions for the partition function to high order. From these we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk free energy. For the isotropic case, they also agree with Vernier and Jacobsen's conjecture for the $60^{\circ}$ corners, and with Cardy and Peschel's conformal invariance predictions for the dominant behaviour at criticality.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

We present an efficient algorithm for computing the partition function of the q-colouring problem (chromatic polynomial) on regular two-dimensional lattice strips. Our construction involves writing the transfer matrix as a product of sparse matrices, each of dimension ~ 3^m, where m is the number of lattice spacings across the strip. As a specific application, we obtain the large-q series of the bulk, surface and corner free energies of the chromatic polynomial. This extends the existing series for the square lattice by 32 terms, to order q^{-79}. On the triangular lattice, we verify Baxter's …

2010-06-11cond-mat.stat-mechmath-phmath.MP

The bulk, surface and corner free energies of the square lattice Ising model

We use Kaufman's spinor method to calculate the bulk, surface and corner free energies $f_b, f_s, f_s', f_c$ of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For $f_b, f_s, f'_s$ our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy $f_c$ depends only on the elliptic modulus $k$ that enters the working, and not on the argument $v$, which means that VJ's conjecture applies for the full anisotropi…

2016-07-18cond-mat.stat-mechmath-phmath.MP

Finite size corrections for the Ising model on higher genus triangular lattices

We study the topology dependence of finite size corrections to the Ising model partition function by considering the model on a triangular lattice embedded on a genus two surface. At criticality we observe a universal shape dependent correction, expressible in terms of Riemann theta functions, that reproduces the modular invariant partition function of the corresponding conformal field theory. The period matrix characterizing the moduli parameters of the limiting Riemann surface is obtained by a numerical study of the lattice continuum limit. The same results are reproduced using a discrete ho…

2003-09-25cond-mat.stat-mechhep-thmath-ph

The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems

The diagonal spin-spin correlations $ \langle σ_{0,0}σ_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $ N \in \mathb…

2015-04-27cond-mat.stat-mechmath-phmath.CA

磁场中二维伊辛模型杨-李奇点（Yang-Lee singularity）的角转移矩阵方法研究

Corner Transfer Matrix Approach to the Yang-Lee Singularity in the 2D Ising Model in a magnetic field

本文结合Baxter变分角转移矩阵方法与解析技术，数值计算了复磁场中二维伊辛模型杨-李边缘奇点附近的标度函数及奇点位置，其结果与伊辛场论及${\mathcal M}_{2/5}$共形场论预测高度一致。

2024-01-02cond-mat.stat-mechhep-thmath-ph

Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices

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 expa…

2011-10-10cond-mat.stat-mechmath-phmath.MP

更多相关论文收起更多论文

Gibbs measures of an Ising model with competing interactions on the triangular chandelier-lattice

In this paper, we consider an Ising model with three competing interactions on a triangular chandelier-lattice (TCL). We describe the existence, uniqueness, and non-uniqueness of translation-invariant Gibbs measures associated with the Ising model. We obtain an explicit formula for Gibbs measures with a memory of length 2 satisfying consistency conditions. It is rigorously proved that the model exhibits phase transitions only for given values of the coupling constants. As a consequence of our approach, the dichotomy between alternative solutions of Hamiltonian models on TCLs is solved. Finally…

2019-06-27math-phmath.DSmath.MP

临界温度以上三角型三维伊辛模型中两种符号的渗透

Percolation of both signs in a triangular-type 3D Ising model above $T_c$

研究在临界温度以上三角型三维伊辛模型中，两种符号的共存现象及外加场对聚集体形成的影响。

2025-12-16math-phmath.MPmath.PR

自由费米子模型与零场及虚场 $i(π/2){k_B}T$ 下的二维伊辛模型

Free-fermion models and the two-dimensional Ising models under the zero field and imaginary field $i(π/2){k_B}T$

本文提出蜂窝、三角和Kagomé晶格上二维伊辛模型在零场和虚场下的自由费米子表述，通过精确映射为八顶点模型并求解，首次给出Kagomé晶格在虚场下的精确解，并证明受挫模型仍具有非零剩余熵。

2025-10-15cond-mat.stat-mechmath-phmath.MP

Bell polynomials in the series expansions of the Ising model

Through applying Bell polynomials to the integral representation of the free energy of the Ising model for the triangular and hexagonal lattices we obtain the exact combinatorial formulas for the number of spin configurations at a given energy (i.e. low-temperature series expansion of the partition function or, alternatively, the number of states). We also generalize this approach to the wider class of the (chequered) Utiyama graphs. Apart from the presented exact formulas, our technique allows one to establish the correspondence between the perfect gas of clusters and the Ising model on the l…

2020-07-31cond-mat.stat-mechmath-phmath.MP

The high temperature Ising model on the triangular lattice is a critical percolation model

The Ising model at inverse temperature $β$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-β}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $β$, this generates a dependent (spin) percolation model with parameter $r$. We show…

2008-06-18math-phmath.MPmath.PR

The spectral gap of the 2-D stochastic Ising model with nearly single-spin boundary conditions

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N-by-N box, with boundary conditions which are ``plus'' except for small regions at the corners which are either free or ``minus.'' The spectral gap decreases exponentially in the size of the corner regions, when these regions are of size at least of order \log N. This means that removing as few as O(\log N) plus spins from the corners produces a spectral gap far smaller than the order N^{-2} gap believed to hold under the all-plus boundary condition. Our results are valid at all subcritical t…

2000-08-29math-phmath.MPmath.PR

Variational approach to the scaling function of the 2D Ising model in a magnetic field

The universal scaling function of the square lattice Ising model in a magnetic field is obtained numerically via Baxter's variational corner transfer matrix approach. The high precision numerical data is in perfect agreement with the remarkable field theory results obtained by Fonseca and Zamolodchikov, as well as with many previously known exact and numerical results for the 2D Ising model. This includes excellent agreement with analytic results for the magnetic susceptibility obtained by Orrick, Nickel, Guttmann and Perk. In general the high precision of the numerical results underlines the …

2008-11-28cond-mat.stat-mechhep-thmath-ph

三角晶格二聚体模型中的液滴观测

Observation of droplets in dimer model on a triangular lattice

Saint Petersburg State University（圣彼得堡国立大学）；Beijing Institute of Mathematical Sciences and Applications（北京数学与应用科学研究所）

本文以伊辛模型平衡液滴现象为动机，利用其与二聚体模型的关联，研究三角晶格二聚体模型中液滴形成，证明特定局部能量与低温可产生宏观液滴（由恒定与可移动二聚体重叠构成）。

2024-11-26cond-mat.stat-mechmath-phmath.MP