We show that the elliptic parametrization of the coupling constants of the quantum $XYZ$ spin chain can be analytically extended outside of their natural domain, to cover the whole phase diagram of the model, which is composed of 12 adjacent regions, related to one another by a spin rotation. This extension is based on the modular properties of the elliptic functions and we show how rotations in parameter space correspond to the double covering $PGL(2,\mathbb{Z})$ of the modular group, implying that the partition function of the $XYZ$ chain is invariant under this group in parameter space, in the same way as a conformal field theory partition function is invariant under the modular group acting in real space. The encoding of the symmetries of the model into the modular properties of the partition function could shed light on the general structure of integrable models.

• Received 9 June 2013

DOI:https://doi.org/10.1103/PhysRevB.88.104418