Existence and explicit form of raising/lowering symmetry operators

Construct operators Σ^+ and Σ^− on (C^3)^{⊗N} that commute with the periodic Motzkin Hamiltonian H^Periodic and act on the ground-state basis {v_{S^z}} as Σ^± v_{S^z} = c_±(S^z) v_{S^z±1} for S^z ≠ ±N and Σ^± v_{±N} = 0, and establish that Σ^± are given by the explicit global-spin formulas Σ^± = Σ_{r_1,…,r_N∈{−2,−1,0,1,2}, r_1+…+r_N=±1} s_1^{r_1}…s_N^{r_N} or equivalently by the residue expression Σ^± = Res_{λ=0} ∏_{i=1}^N (λ^{−2}(s_i^{±})^2 + λ^{−1}s_i^{±} + I + λ s_i^{∓} + λ^2 (s_i^{∓})^2).

Background

The degeneracy of the proposed ground-state space suggests nontrivial symmetries. Section 3 formulates a conjecture giving explicit nonlocal formulas for global raising and lowering operators and asserting their commutation with the Hamiltonian and their action on ground states.

The introduction explicitly states the conjectural existence of such operators with explicit formulas and their interpretation as raising/lowering operators on the ground-state space.