On the Spectrum of the XXZ-chain at roots of unity

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In a recent paper (cond-mat/0009279), Fabricius and McCoy studied the spectrum of the spin 1/2 XXZ-model at Delta = (q+q⁻¹)/2 and q²ⁿ=1 for integer N >1. They found a certain pattern of degeneracies and linked it to the sl(2)-loop symmetry present in the commensurable spin sector (N divides Sᶻ). We show that the degeneracies are due to zero-energy, transparent excitations, the cyclic bound states. These exist both in commensurable and incommensurable sectors, indicating a symmetry, of which sl(2)-loop is a partial manifestation. Our approach treats both sectors on even footing and yields an analytical expression for the degeneracies in the case N = 3.