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

TL;DR

This paper constructs Baxter's Q operator for the XXZ model at roots of unity, providing explicit forms, proving key relations, and elucidating spectral degeneracies and FM strings, advancing understanding of the model's spectrum.

Contribution

It introduces a method to explicitly construct the Q operator at roots of unity and derives related spectral relations, addressing open issues in the XXZ model's spectrum.

Findings

Explicit Q operator at roots of unity

Proof of transfer matrix fusion and Wronskian relations

Conjecture for FM string creation and annihilation operators

Abstract

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 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.