Elliptic pfaffians and solvable lattice models

Hjalmar Rosengren

The Izergin-Korepin formula expresses the partition function of the domain wall six-vertex model as a simple determinant. It has proved very useful for investigating relation to combinatorics as well as thermodynamic properties of various physical quantities. In spite of recent progress, there is no very pleasing extension of the Izergin-Korepin formula to elliptic lattice models. In our talk, we will discuss expressions involving pfaffians rather than determinants. We introduce twelve pfaffians with elliptic function entries, which are all related by modular transformations. The domain wall partition function for the 8VSOS model (at the combinatorial line) is expressed as a linear combination of two such pfaffians. Similar expressions can be given for certain eigenvalues of Q-operators on inhomogeneous XYZ spin chains. In the homogeneous limit, we obtain new Hankel determinant formulas for the corresponding quantities.

http://scgp.stonybrook.edu/video_portal/video.php?id=2487

Simons- Workshop: Six-vertex models, dimers, shapes, and all that