Thesl₂loop algebra symmetry of the twisted transfer matrix of the six-vertex model at roots of unity

Abstract

We discuss a family of operators which commute or anti-commute with the twisted transfer matrix of the six-vertex model at q being roots of unity: q 2N = 1. The operators commute with the Hamiltonian of the XXZ spin chain under the twisted boundary conditions, and they are valid also for the inhomogeneous case. For the case of the anti-periodic boundary conditions, we show explicitly that the operators generate the sl 2 loop algebra in the sector of the total spin operator S Z ≡ N/2 (mod N ). The infinite-dimensional symmetry leads to exponentially-large spectral degeneracies, as shown for the periodic boundary conditions [12]. Furthermore, we derive explicitly the sl 2 loop algebra symmetry for the periodic XXZ spin chain with an odd number of sites in the sector S Z ≡ N/2 (mod N ) when q is a primitive N th root of unity with N odd. Interestingly, in the case of N = 3, various conjectures of combinatorial formulas for the XXZ spin chain with odd-sites have been given by Stroganov and others. We also note a connection to the spectral degeneracies of the eight-vertex model. *

Continuum limit and symmetries of the periodic spin chain

“…Then, we only need to check the higher-order Serre relations Since we have seen that the e n and f n commute with the Hamiltonian, we have thus found a loop algebra symmetry of the Hamiltonian H in the gℓ(1|1) spin chain. This is much like the symmetry uncovered in [35,39], but a more careful comparison shows that the sectors we are considering are different: while in [35], the loop algebra is observed for periodic (antiperiodic) XX spin chain and even (odd) spin, ours is obtained in the opposite case, corresponding to periodicity for the gℓ(1|1) fermions. We stress that in contrast with the main focus of this paper, the loop algebra is only a symmetry of the Hamiltonian, and does not extend to the full JT L N algebra.…”

The twisted XXZ chain at roots of unity revisited

“…Here z denotes the (multiplicative) spectral parameter and q is the deformation parameter appearing in the spin-chain Hamiltonians (1) and (2). Central to our discussion will be the properties of the (inhomogeneous) six-vertex monodromy matrix which one usually decomposes over the two-dimensional auxiliary space,…”

“…Similar as for the periodic case the sl 2 symmetry algebra for λ = −1 has been restricted to certain commensurate spin-sectors. The discussion in [2] has also been extended to include the case of the inhomogeneous chain.…”

Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

“…This has been first established by Deguchi, Fabricius and McCoy in [3], where additional results for N = 3, 4 outside the commensurate sectors can be found. When λ = 1 the symmetry algebra will in general reduce to the upper or lower Borel subalgebra [4,5]. A special case is obtained when the boundary conditions are tuned to λ = q ±S z .…”