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

Abstract

The spin-\frac{1}{2}12 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 a 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 auxiliary representation we give a conjecture for creation and annihilation operators of FM strings for all roots of unity. We connect our findings with the `string-charge duality’ in the thermodynamic limit, leading to a conjecture for the imaginary part of the FM string centres with potential applications to out-of-equilibrium physics.

Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models

“…(21). As shown in [30], commensurate value of the twist φ leads to the descendant tower structure and Onsager algebra due to the existence of eigenstates associated with exact (Fabricius-McCoy) strings.…”

“…β = 0, the 2 -dimensional semi-cyclic representation becomes 2 -dimensional highest weight representation, cf. Appendix A in [30].…”

“…Due to the underlying quantum group structure at root of unity [23], the spectra have exponentially many degeneracies. This phenomenon has been studied in [24][25][26][27][28][29][30]. More recently, the author and collaborators have constructed the Baxter's Q operator for XXZ model at root of unity and studied the spectra in terms of descendant towers [30], which elucidated the origin of the exponentially many degeneracies due to the existence of eigenstates associated with exact (Fabricius-McCoy) strings.…”

“…This phenomenon has been studied in [24][25][26][27][28][29][30]. More recently, the author and collaborators have constructed the Baxter's Q operator for XXZ model at root of unity and studied the spectra in terms of descendant towers [30], which elucidated the origin of the exponentially many degeneracies due to the existence of eigenstates associated with exact (Fabricius-McCoy) strings. In particular, XX model, a special case of XXZ model at root of unity, possesses the Onsager algebra symmetry [16] additionally, cf.…”

“…Sec. 4.1, and its spectrum has similar descendant tower structure as the spectra at other roots of unity [30]. It is thus natural to consider the question whether XXZ models at arbitrary roots of unity value of the anisotropy would possess similar Onsager algebra symmetries.…”