arXiv 0809.23512008-09math-phhep-thmath.MP

手征Potts模型和单位根下的离散Sine-Gordon模型

Chiral Potts model and the discrete Sine-Gordon model at roots of unity

Vladimir V. Bazhanov

备注：: 26 pages, contribution to "Exploration of New Structures and Natural Constructions in Mathematical Physics", Nagoya, March 5-8, 2007. On the occasion of Professor Akihiro Tsuchiya's retirement. To be published in Advanced Studies in Pure Mathematics; v2: minor corrections

摘要：

单位根处的离散量子Sine-Gordon模型将经典的可积系统与可积的量子自旋系统结合起来，其参数服从经典的运动方程。我们表明，基本的R-矩阵的模型（满足差分性质杨巴克斯特方程）自然分裂成一个产品的奇异的“经典”的一部分和一个有限维的量子部分。$R$ -矩阵的经典部分本身满足量子杨-巴克斯特方程，因此可以被分解掉，然而，产生了量子部分的某种“扭曲”。我们证明了所得方程与N态手征Potts模型的星三角关系完全吻合。事实上，整个晶格上的相关自旋模型比手征波茨模型更一般，并且仅在最简单（恒定）的经典背景下才简化为后者。在一般情况下，该模型是非齐次的：它的玻尔兹曼权重由经典离散正弦-戈登模型的运动方程的非平凡背景解确定。

英文摘要：

The discrete quantum Sine-Gordon model at roots of unity remarkably combines a classical integrable system with an integrable quantum spin system, whose parameters obey classical equations of motion. We show that the fundamental R-matrix of the model (which satisfies a difference property Yang-Baxter equation) naturally splits into a product of a singular "classical" part and a finite dimensional quantum part. The classical part of the $R$-matrix itself satisfies the quantum Yang-Baxter equation, and therefore can be factored out producing, however, a certain "twist" of the quantum part. We show that the resulting equation exactly coincides with the star-triangle relation of the N-state chiral Potts model. The associated spin model on the whole lattice is, in fact, more general than the chiral Potts and reduces to the latter only for the simplest (constant) classical background. In a general case the model is inhomogeneous: its Boltzmann weights are determined by non-trivial background solutions of the equations of motion of the classical discrete sine-Gordon model.

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