Degenerate ground-state manifold and path-sum description for the periodic Motzkin chain

Determine whether the periodic Motzkin spin-1 chain with N sites and Hamiltonian H^Periodic = ∑_{i=1}^{N-1} Π_{i,i+1} + Π_{N,1} possesses a (2N+1)-fold degenerate zero-energy ground-state manifold with eigenstates v_{S^z} labeled by the eigenvalue S^z ∈ {−N, …, N} of the total spin operator S^z, and whether each v_{S^z} equals the uniform sum over all length-N lattice paths with steps Δx = 1 and Δy ∈ {−1, 0, 1} connecting (0, 0) to (N, S^z) without any nonnegativity restriction on y.

Background

In the open boundary case, the Motzkin chain has a unique ground state given by the uniform superposition of Motzkin paths and is exactly known. For periodic boundary conditions, the structure of the ground-state space has not been established.

The paper proposes that, under periodic boundary conditions, the ground state becomes highly degenerate with 2N+1 independent states distinguished by the eigenvalue of the total-spin operator S^z. Moreover, it posits an explicit path-sum description for each S^z sector, analogous to Motzkin paths but without the nonnegativity (y ≥ 0) constraint.