Adrien Poncelet, Philippe Ruelle
We consider the Abelian sandpile model on triangular and hexagonal lattices. We compute several height probabilities on the full plane and on half-planes, and discuss some properties of the universality of the model.
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In a recent paper S. Friedland and the author presented a formal expression for lambda_d(p) of the monomer-dimer problem on a d-dimensional rectangular lattice, which involved a power series in p. Herein, we find simlar expressions for lambda_tri(p) and lambda_hex(p), the entropy per site for triangular and hexagonal lattices.
Location of the Multicritical Point for the Ising Spin Glass on the Triangular and Hexagonal Lattices
对三角格子上+/- J伊辛模型相图中多临界点的确切位置提出了一个猜想。结果p_c=0.8358058与最近的一个数值估计很好地吻合。从这个值，它是可能的，推导出一个可比的猜想的准确位置的多临界点的六方晶格，p_c=0.9327041，再次在良好的协议与数值研究。该方法是对偶变换的一种变体，它将三角形晶格直接与其对偶三角形晶格联系起来，而不依赖于六边形晶格，并结合副本方法。
Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus. II. Triangular and hexagonal lattices
本文计算了包在环面上的三角晶格和六角晶格上的临界二维自旋1/2伊辛模型的自由能、内能和比热的有限尺寸修正。我们找到了这些量的有限尺寸修正的一般形式，以及每个展开式的第一系数的显式公式。我们分析了这些发现对模型的重正化群描述的影响。
The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome,...), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and…
In recent work the author presented a formal expansion for lambda_d associated to the dimer problem on a d-dimensional rectangular lattice. Expressed in terms of d it yielded a presumed asymptotic expansion for lambda_d in inverse powers of d. We also considered an expansion in powers of x, a formal variable ultimately set equal to 1. We believe this series has better asymptotic properties than the expansion in inverse powers of d. We discuss this, and apply the same method to the two-dimensional triangular and hexagonal lattices. Viewed as a test of the x-expansion the results on those two la…
Renormalization group approach to an Abelian sandpile model on planar lattices
重整化群（RG）方法求解格子沙堆模型的一个重要步骤是精确计算RG变换单元内沙堆动力学的所有可能的倾倒过程。在这里，我们提出了一个计算机算法来执行这样的精确计数的平面晶格的细胞在RG方法的Bak-Tang-Wiesenfeld沙堆模型[物理。{ \bf 59}，381（1987）]，并考虑由Pietronero，Vespignani和Zaplani（PVZ）提出的简化高RG方程[Phys. Rev. Lett. { \bf 72}，1690（1994）]和Ivashkevich [Phys. Rev. Lett. { \bf 76}，3368（1996）]。使用这种算法，我们能够更快地进行RG变换与大的细胞大小，例如$3 \times 3$细胞的正方形（sq）格在PVZ RG方程，这是目前最大的细胞大小，并发现一些错误，在以前的文件[物理评论E { \bf 51}，1711（1995）]。对于正方形格和平面三角形格，我们得到了每个格的唯一吸引不动点，并计算了雪崩指数$τ$和动力学指数$z$。我们的研究结果表明，在PVZ RG转化中细胞大小的增加不会导致更准确的结果。讨论了这一结果的意义。
Finite dipolar hexagonal columns on piled layers of triangular lattice
我们用蒙特卡罗模拟研究了磁性纳米粒子的自旋系统，这些自旋系统代表了磁性纳米粒子之间仅通过偶极-偶极相互作用相互作用的磁矩。在本文中，我们的目标是了解有限尺寸效应的磁性纳米粒子排列在六角形柱切割的紧密堆积结构或从那些单轴压缩。在具有真正的密堆积结构的柱中，我们观察到一个单一的涡旋状态，这也是以前在有限的二维系统中观察到的。另一方面，在层间距离为密堆积层距离的$1/\sqrt{2}$倍的体系中，我们发现了依赖于层数的基态。这种依赖性是由有限尺寸效应引起的，并且与相应的体系统中的取向转变有关。
Triangular Trimers on the Triangular Lattice: an Exact Solution
提出了一个由三角形格子上的三角形三聚体组成的模型。类似于二聚体问题，这些粒子完全覆盖晶格而没有重叠。该模型具有由刚性畴壁分隔的六边形单元的蜂窝结构。传递矩阵可以通过具有两种类型粒子的Bethe Answer对角化。这导致两个参数空间的二维子集上的熵的精确表达式。
The phase diagram of the hexagonal lattice quantum dimer model
讨论了量子二聚体模型在六方（蜂窝）晶格上的相图。除了在以前的工作中已经讨论过的柱状和交错价键固体，我们建立了一个plaquette价键固体的存在。在$v/t=-0.2 \pm 0.05$的晶元和柱状相之间的转变被认为是一阶的。我们注意到这个模型应该描述六角晶格上受挫海森堡模型的价键主导相，并讨论了它与J.B. Fouet等人对同一格点上的J_1-J_2模型进行了研究。我们的结果还揭示了三角晶格上的横向场Ising反铁磁体和堆叠三角晶格上的经典Ising反铁磁体的性质，这些性质与二聚体模型通过对偶性联系在一起。
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…
We study a system of particles in two dimensions interacting via a dipolar long-range potential $D/r^3$ and subject to a square-lattice substrate potential $V({\bf r})$ with amplitude $V$ and lattice constant $b$. The isotropic interaction favors a hexagonal arrangement of the particles with lattice constant $a$, which competes against the square symmetry of the substrate lattice. We determine the minimal-energy states at fixed external pressure $p$ generating the commensurate density $n = 1/b^2 = (4/3)^{1/2}/a^2$ in the absence of thermal and quantum fluctuations, using both analytical and nu…
We consider deterministic walks on square, triangular and hexagonal two dimensional lattices. In each case, there is a scatterer at every site that can be in one of two states that force the walker to turn either to his/her immediate right or left. After the walker is scattered, the scatterer changes state. A lattice with an arrangement of scatterers is an environment. We show that there are only two environments for which the scattering rules are injective, mirrors or rotators, on the three lattices. On hexagonal lattices, B. Z. Webb and E. G. D. Cohen, proved that given an initial position a…
Ground state of the spin-1/2 Heisenberg antiferromagnet on a two-dimensional square-hexagonal-dodecagonal lattice
到目前为止，在最近邻自旋为1/2的反铁磁（AF）晶格系统中，已经从现有的11个二维均匀晶格中，对7个系统的磁性Neel长程序（NLRO）的存在性进行了研究。形成这些均匀（阿基米德）晶格（例如正方形，三角形，kagome）的小块是不同的正多边形。本文研究了由正方形、六边形和十二边形组成的第七类均匀（二分）晶格上AF自旋系统的基态非线性振荡。NLRO被证明发生在这个系统中。本文提出了一个关于二维阿基米德晶格上自旋为1/2的反铁磁系统的基态中存在非线性振荡的简单猜想。
We consider electrons on a honeycomb or triangular lattice doped to the saddle point of the bandstructure. We assume system parameters are such that spin density wave (SDW) order emerges below a temperature $T_N$ and investigate the nature of the SDW phase. We argue that at $T < T_N$ the system develops a uniaxial SDW phase whose ordering pattern breaks $O(3) \times Z_4$ symmetry and corresponds to an eight site unit cell with non-uniform spin moments on different sites. This state is a half-metal -- it preserves full original Fermi surface, but has gapless charged excitations in one spin bran…
A Comparative Study of the Magnetization Process of Two-Dimensional Antiferromagnets
本文研究了正方形、三角形和六角晶格自旋为1/2XXZ反铁磁体磁化曲线中的平台。人们发现一个零磁化高原（对应于自旋间隙）的正方形和六边形晶格与伊辛样各向异性，和高原的三分之一的饱和磁化强度的三角形晶格上幸存的少量易平面各向异性。在这里，我们开始与转移矩阵计算的伊辛极限，并继续在XXZ-各向异性的平台边界使用伊辛极限的基态系列。然后，主要重点是数值计算的磁化曲线与各向异性的各向同性的情况附近。最后，我们讨论了普遍性类相关的磁化曲线接近饱和的渐近行为，观察到的数值在两个和更高的维度。
Classical dimers on the triangular lattice
我们研究了三角晶格上的经典硬核二聚体模型。根据Kasteleyn关于平面图的基本定理，这个问题可以由Pfavians解决。这个模型是特别有趣的，不像二聚体问题的二分正方形和六边形晶格，它的相关性是短程的相关长度小于一个晶格常数。我们计算了二聚体-二聚体和单体-单体相关系数，发现模型是去精细化的：单体-单体相关系数指数下降到常数sin（pi/12）/sqrt（3）= .1494.仅略低于最近邻值1/6。我们还考虑了各向异性的三角形晶格模型，其中正方形晶格是由一个方向和小逸度的对角键扰动。我们表明，该模型立即成为非临界的，这种扰动相当于增加了一个质量项的两个马约拉纳费米子，是目前在长波长限制的正方形晶格问题。
Height representation, critical exponents, and ergodicity in the four-state triangular Potts antiferromagnet
研究了三角晶格上的四态反铁磁Potts模型。我们发现，该模型有六种类型的缺陷，扩散和湮灭，根据一定的守恒定律与他们有一个矢量值拓扑电荷。利用这些缺陷的性质，我们推导出一个2+2维的高度表示的模型，因此表明该模型是等效的三态Potts反铁磁体上的Kagome晶格和键着色模型上的三角形和六边形晶格。我们还计算了该模型的基态系综的临界指数。我们发现，由于路径长度的限制，自旋-自旋关联函数和自旋涨落的指数违反了Fisher标度律，这增加了自旋算符在高度晶格上的有效波长.我们证实了我们的预测广泛的蒙特卡罗模拟的模型，使用Wang-Swendsen-Kotecky聚类算法。虽然这个算法是不是遍历格与环形边界条件，我们证明了它是遍历格与自由边界条件，或更一般的格上拥有无限阶的非收缩回路。为了防止由于缺乏遍历性而引入的偏差，因此我们在环面和投影平面上进行模拟。
Using Monte Carlo simulations and finite-size scaling, we investigate the XY antiferromagnet on the triangular, Union Jack and bisected-hexagonal lattices, and in each case find both Ising and Kosterlitz-Thouless transitions. As is well-known, on the triangular lattice, as the temperature decreases the system develops chiral order for temperatures $T < \Tc$, and then quasi-long-range magnetic order on its sublattices when $T < \Ts$, with $\Ts < \Tc$. The behavior $\Ts<\Tc$ is predicted by theoretical arguments due to Korshunov, based on the unbinding of kink-antikink pairs. On the Union Jack a…
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