A Shortcut to the Q-Operator

TL;DR

This paper constructs the Baxter Q-operator for the spin-1/2 Heisenberg XXX chain using transfer matrices with a twist, filling a longstanding gap in integrable model analysis.

Contribution

It provides a novel method to explicitly construct Q-operators for the XXX chain via transfer matrices with a twist, comparing with previous approaches.

Findings

Constructed two linearly independent Q-operators as transfer matrices.

Demonstrated the role of a twist field in the construction.

Discussed implications for AdS/CFT Y-systems.

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.