Ground-state structure for the periodic Motzkin chain

Determine the exact structure of the zero-energy ground-state subspace of the periodic Motzkin spin-1 chain with N sites by proving that it has dimension 2N+1 with independent ground states v_{S^z} labeled by the eigenvalues S^z = 0, ±1, …, ±N of the total spin operator S^z, and showing that each v_{S^z} is given by the sum of all lattice paths from (0,0) to (N,S^z) with steps Δx = 1 and Δy ∈ {−1, 0, 1} without any restriction on y along the path.

Background

The Motzkin chain is a nearest-neighbor spin-1 model whose open-boundary ground state is the uniform superposition of Motzkin paths. This paper considers periodic boundary conditions and proposes a path-based description of the entire ground-state space together with its degeneracy.

Section 3 presents the formal conjecture specifying the 2N+1 degeneracy and a path characterization for each ground state labeled by S^z. The abstract states this as a conjecture and frames it as part of the paper’s main open results.