The Bethe ansatz for the six-vertex and XXZ models: an exposition

Understand

In this paper, we review a few known facts on the coordinate Bethe ansatz.

• We present a detailed construction of the Bethe ansatz vector $\psi$ and energy $\Lambda$, which satisfy $V \psi = \Lambda \psi$, where $V$ is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a= b=1$ and $c > 0$.
• We also show that the same vector $\psi$ satisfies $H \psi = E \psi$, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly.
• Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities.