标题： 周期Fredkin链的对称性 显示英文标题

标题： Symmetries of the periodic Fredkin chain

摘要： 弗雷德金链是一种自旋-$1/2$模型，具有三个最近邻的相互作用。 在周期性边界条件下，基态是简并的，可以用迪克路径的等价类来描述。 我们引入两个与哈密顿量对易的算符，在作用于基态时，它们分别起到降低和升高算符的作用。 这些算符根据格点数$N$是奇数还是偶数，分别生成$B$或$C$型李代数，其秩为$n=\lceil N/2\rceil$。 总自旋算符的第三分量可以表示为卡坦子代数元素和某些中心元素的和。 在$C$型李代数的情况下（格点数为偶数），这种表示与之前在周期性莫茨金链背景下针对自旋-$1$算符提出的类似公式一致。 显示英文摘要

摘要： Fredkin chain is a spin-$1/2$ model with interaction of three nearest neighbors. In the case of periodic boundary conditions the ground state is degenerated and can be described in terms of equivalence classes of the Dyck paths. We introduce two operators commuting with the Hamiltonian, which play the roles of lowering and raising operators when acting at the ground states. These operators generate the $B$- or $C$-type Lie algebras, depending on whether the number of sites $N$ is odd or even, respectively, with rank $n=\lceil N/2\rceil$. The third component of total spin operator can be represented as a sum of the Cartan subalgebra elements and some central element. In the $C$-type Lie algebra case (even number of sites) this representation coincides with a similar formula previously conjectured for spin-$1$ operators, in the context of periodic Motzkin chain.