Combinatorial and topological approach to the 3D Ising model
Combinatorial and topological approach to the 3D Ising model
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DOI：
10.1088/0305-4470/33/4/308
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发表时间：
1999-09
期刊：
影响因子：
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通讯作者：
T. Regge;R. Zecchina
中科院分区：
文献类型：
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作者：
T. Regge;R. Zecchina
我们将平面的Pfaffian形式推广到具有高拓扑格g的格上，详细讨论了格上的三维Ising模型，其中g与点的数目成正比。配分函数的展开式是用晶格的取向同调环，即自旋结构分类的2^{2g} Pfaffians给出的。通过一个简单的双线性公式，一个依赖于有向环的拓扑交的签名项保证了正确计数。讨论了规范对称在上述膨胀中所起的作用。同样的形式可以应用于一般格上完美匹配的计数问题，并提供了0-1矩阵的恒量的行列式展开式。
We extend the planar Pfaffian formalism for the evaluation of the Ising partition function to lattices of high topological genus g. The 3D Ising model on a cubic lattice, where g is proportional to the number of sites, is discussed in detail. The expansion of the partition function is given in terms of 2^{2 g} Pfaffians classified by the oriented homology cycles of the lattice, i.e. by its spin-structures. Correct counting is guaranteed by a signature term which depends on the topological intersection of the oriented cycles through a simple bilinear formula. The role of a gauge symmetry arising in the above expansion is discussed. The same formalism can be applied to the counting problem of perfect matchings over general lattices and provides a determinant expansion of the permanent of 0-1 matrices.