On the Q operator and the spectrum of the XXZ model at root of unity

Yuan Miao, Jules Lamers, Vincent Pasquier

Published 2020-12-18Version 1

The spin-1/2 Heisenberg XXZ chain is a paradigmatic quantum integrable model. Although it can be solved exactly via Bethe ansatz techniques, there are still open issues regarding the spectrum at root of unity values of the anisotropy. We construct Baxter's Q operator at arbitrary anisotropy from a two-parameter transfer matrix associated to a complex-spin auxiliary space. A decomposition of this transfer matrix provides an simple proof of the transfer matrix fusion and Wronskian relations. At root of unity a truncation allows us to construct the Q operator explicitly in terms of finite-dimensional matrices. From its decomposition we derive truncated fusion and Wronskian relations as well as an interpolation-type formula that has been conjectured previously. We elucidate the Fabricius-McCoy (FM) strings and exponential degeneracies in the spectrum of the six-vertex transfer matrix at root of unity. Using a semicyclic representation, we give a conjecture for creation and annihilation operators of FM strings for most roots of unity. We connect our findings with the 'string-charge duality' in the thermodynamic limit, leading towards a conjecture on FM string centres and potential applications to out-of-equilibrium physics.