标题： kagome晶格Potts模型的临界流形 显示英文标题

标题： Critical manifold of the kagome-lattice Potts model

摘要： 任何二维无限正则格子G都可以通过用G的一个有限子图B来铺满平面得到；我们称B为G的基。我们引入了一个依赖于B及其在G中的嵌入的两参数图多项式P_B(q,v)。代数曲线P_B(q,v)=0被证明可以作为q状态Potts模型的临界流形的近似，该模型在G上定义，耦合v=exp(K)-1。该曲线预测了铁磁(v>0)和反铁磁(v<0)区域的相图。 对于更大的基B，近似值变得越来越准确，我们猜想当B无限大时，P_B(q,v)=0提供了精确的临界流形。 此外，对于某些格子G，或者对于任何G上的Ising模型(q=2)，无论选择何种B，P_B(q,v)都会分解：递归因子的零点则提供了精确的临界流形。 从这个意义上说，计算P_B(q,v)可以用来检测G上Potts模型的精确可解性。 我们以正方格子为例说明该方法，该模型已经被精确求解，以及Kagome格子，该模型尚未被精确求解。 对于正方格子，我们正确地再现了已知的相图，包括反铁磁相变和Berker-Kadanoff相中的奇点。 对于Kagome格子，使用包含六条边的最小基，我们恢复了一个著名的（但现已被推翻）F.Y. Wu的猜想。 更大的基对该公式进行了逐步改进，给出了Wu方法的自然扩展。 多项式预测与数值计算结果高度一致。 对于v>0，6边基的预测临界耦合v_c的精度约为10^{-4}或10^{-5}，而对于研究的最大基（36边）则提高到10^{-6}或10^{-7}。 显示英文摘要

摘要： Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B of G; we call B a basis of G. We introduce a two-parameter graph polynomial P_B(q,v) that depends on B and its embedding in G. The algebraic curve P_B(q,v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = exp(K)-1, defined on G. This curve predicts the phase diagram both in the ferromagnetic (v>0) and antiferromagnetic (v<0) regions. For larger bases B the approximations become increasingly accurate, and we conjecture that P_B(q,v) = 0 provides the exact critical manifold in the limit of infinite B. Furthermore, for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises for any choice of B: the zero set of the recurrent factor then provides the exact critical manifold. In this sense, the computation of P_B(q,v) can be used to detect exact solvability of the Potts model on G. We illustrate the method for the square lattice, where the Potts model has been exactly solved, and the kagome lattice, where it has not. For the square lattice we correctly reproduce the known phase diagram, including the antiferromagnetic transition and the singularities in the Berker-Kadanoff phase. For the kagome lattice, taking the smallest basis with six edges we recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases provide successive improvements on this formula, giving a natural extension of Wu's approach. The polynomial predictions are in excellent agreement with numerical computations. For v>0 the accuracy of the predicted critical coupling v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to 10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).