Symmetries of the periodic Fredkin chain

Abstract: 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.