论文标题
旋转链可以将组合学与数字理论联系起来吗？
Can a spin chain relate combinatorics to number theory?
论文作者
论文摘要
Motzkin自旋链是Shor＆Movassagh引入的旋转型$ 1 $的无挫折型号。基态是通过在方格的上半部分绘制随机步行来构建的，以旋转配置。它具有异常大的纠缠熵[量子波动]。 Motzkin链的基态可以通过Motzkin路径进行分析描述。没有对激发态的分析描述。该模型无法解决。我们通过删除Motzkin路径的局部等效移动之一来简化模型。系统变得可集成[类似于XXX自旋链]。我们称其为免费的Motzkin链。从量子集成性的角度来看，该模型是特殊的，因为其$ r $ -Matrix没有交叉单位性。我们通过概括功能性bethe ansatz方法来解决周期性的自由motzkin链。我们构建了一个具有附加参数的$ T-Q $关系，以制定能量谱。该新参数与统一根有关，可以用数字理论中的Möbius函数来描述。我们观察到数字理论的进一步模式。
The Motzkin spin chain is a spin-$1$ frustration-free model introduced by Shor & Movassagh. The ground state is constructed by mapping random walks on the upper half of the square lattice to spin configurations. It has unusually large entanglement entropy [quantum fluctuations]. The ground state of the Motzkin chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states. The model is not solvable. We simplify the model by removing one of the local equivalence moves of the Motzkin paths. The system becomes integrable [similar to the XXX spin chain]. We call it free Motzkin chain. From the point of view of quantum integrability, the model is special since its $R$-matrix does not have crossing unitarity. We solve the periodic free Motzkin chain by generalizing the functional Bethe Ansatz method. We construct a $T-Q$ relation with an additional parameter to formulate the energy spectrum. This new parameter is related to the roots of unity and can be described by the Möbius function in number theory. We observe further patterns of number theory.
