Fredkin spin chain

The most famous example of a spin chain is the XXX Heisenberg model. The density of the Hamiltonian can be expressed as a permutation operator [swap gate]. Faddeev represented the eigenfunctions in terms of the algebraic Bethe Ansatz. It corresponds to a matrix product state representation. We introduce a new model of interacting spin 1/2. It describes the interactions of three nearest neighbors. The density of the Hamiltonian is a controlled permutation operator [Fredkin gate]. Our construction generalizes the Shor-Movassagh chain to half- integer spins. The Fredkin chain can be solved using combinatorics related to Catalan numbers, where we consider random walks in the upper half plane of a square lattice [Dyck walks]. Each Dyck path can be mapped to a wave function of spins. The ground state is an equally weighted superposition of Dyck walks. We can also express it as a matrix product state. We further construct a model of interacting spins 3/2 and greater half-integer spins. The models with higher spins correspond to colorings of the Dyck walks. We construct a SU(k) symmetric model [where k is the number of colors]. The Fredkin chain with spins 3/2 and higher has stronger quantum fluctuations than XXX.