Modular invariance in the gapped XYZ spin 1/2 chain

TL;DR

This paper demonstrates that the XYZ spin 1/2 chain's coupling constants can be extended across its entire phase diagram using modular properties of elliptic functions, revealing a deep symmetry akin to modular invariance in conformal field theories.

Contribution

The authors establish a modular invariance of the XYZ chain's partition function, extending the elliptic parametrization beyond its natural domain and relating parameter space rotations to the modular group.

Findings

Partition function invariant under PGL(2,Z) transformations

Elliptic parametrization extended to entire phase diagram

Reveals modular symmetry in integrable models

Abstract

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,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.