Symmetries of the periodic Fredkin chain

TL;DR

This paper explores the symmetry structure of the periodic Fredkin chain, revealing that its ground states form representations of Lie algebras depending on the parity of the number of sites, with implications for understanding its algebraic and physical properties.

Contribution

It introduces operators that generate Lie algebras acting on the ground states of the periodic Fredkin chain, connecting the model's symmetries to well-known algebraic structures.

Findings

Ground states form representations of B- or C-type Lie algebras

Operators acting on ground states generate these Lie algebras

Representation of total spin aligns with previous conjectures for related models

Abstract

The 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 degenerate and can be described in terms of equivalence classes of Dyck paths. We introduce two operators commuting with the Hamiltonian which play the roles of lowering and raising operators when acting on 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 the 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 the periodic Motzkin chain.