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

In a recent paper (cond-mat/0009279), Fabricius and McCoy studied the spectrum of the spin 1/2 XXZ model at roots of unity, i.e. ∆ = (q + q−1)/2 with q2N = 1 for integer N ≥ 2. They found a certain pattern of degeneracies and linked it to the sl2-loop symmetry present in the commensurable spin sector Sz ≡ 0 mod N . We show that the degeneracies are due to an unusual type of zero-energy “transparent” excitation, the cyclic bound state. The cyclic bound states exist both in the commensurable and in the incommensurable sectors indicating a symmetry group present, of which sl2-loop algebra is a partial manifestation. Our approach treats both sectors on even footing and allows us to obtain analytically an explicit expression for the degeneracies in the case N = 3. Typeset using REVTEX e-mail: [email protected] email: [email protected]