A shortcut to the<i>Q</i>-operator

TLDR

Baxter's Q‑operator is a powerful tool for exact diagonalization of integrable models, yet it has not been properly constructed for the simplest system, the compact spin‑1/2 Heisenberg‑Bethe XXX spin chain. The authors aim to fill this gap by constructing two linearly independent operatorial solutions to Baxter's TQ equation as commuting transfer matrices in the presence of a twist field, and they speculate on the relevance of Q‑operators to recent Y‑system proposals for AdS/CFT. These solutions are obtained by tracing over infinitely many oscillator states in the auxiliary channel of an associated monodromy matrix, and the authors compare and contrast their method with earlier approaches to constructing the Q‑operator for the XXX chain.

Abstract

Baxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1/2 Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matrix. We furthermore compare and differentiate our approach to earlier articles addressing the problem of the construction of the Q-operator for the XXX chain. Finally we speculate on the importance of Q-operators for the physical interpretation of recent proposals for the Y-system of AdS/CFT.