正方形格上的16顶点模型及其奇偶8顶点子情形
The 16-vertex model and its even and odd 8-vertex subcases on the square lattice
摘要：
本文对16-顶点模型的已知映射进行了综述和扩充，重点讨论了一般模型的8-顶点偶子情形和8-顶点奇子情形之间的映射，并给出了这些模型之间的新映射，这些映射在有限环形格上是有效的。特别是，我们发现新的模型之间的映射，通过使用它们的代数不变量的$SL(2)\times SL(2)$对称性的16顶点模型，我们还发现了一个更大的弱图变换。我们展示了许多负权重模型的例子，这些模型映射到只有正权重的模型。利用偶、奇八顶点模型的代数不变关系，我们找到了场中正方形格子伊辛模型复场平面上映射到偶、奇八顶点模型的点的完备集，这些点也对应于模型的自由费米子点集。我们没有发现任何新的可积点，但我们发现一个新的映射之间的奇数8顶点模型和正方形晶格伊辛模型在磁场$H= iπ/(2β)$，有效的有限环形晶格。我们还直接通过各种例子表明，通过代数不变量的映射不完全用尽可能的映射模型可能与另一个模型。我们构造了一个新的解决方案的奇数8顶点自由费米子模型，这是有效的有限格点，因为以前已知的解决方案，从一个有效的映射只在热力学极限。最后，我们首次详细讨论了列交错自由费米子八顶点模型的相变，并证明了它可以映射到二分交错自由费米子模型。
英文摘要：
We survey and enlarge the known mappings of the 16-vertex model, with emphasis on mappings between the even and odd 8-vertex subcases of the general model, also giving new mappings between these models, valid on finite toroidal lattices. In particular, we find new mappings between the models by using their algebraic invariants with respect to the $SL(2)\times SL(2)$ symmetry of the 16-vertex model; we also find a larger set of weak-graph transformations. We show many examples of models with negative weights which map to models with only positive weights. Using the algebraic invariant relations of the even and odd 8-vertex models, we find the complete set of points in the complex field plane of the square lattice Ising model in a field which map to the even or odd 8-vertex models; these points also correspond to the set of free-fermionic points of the model. We do not find any new integrable points, but we find a new mapping between the odd 8-vertex model and the square lattice Ising model at magnetic field $H= iπ/(2β)$, valid on finite toroidal lattices. We also show directly through various examples that mappings via algebraic invariants do not fully exhaust the possible mappings a model may have with another model. We construct a new solution to the odd 8-vertex free-fermion model which is valid on the finite lattice, since the previous known solution resulted from a mapping valid only in the thermodynamic limit. Finally, we detail for the first time the phase transitions of the column staggered free-fermion 8-vertex model, and show that it can be mapped to the bi-partite staggered free-fermion model.
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