Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 025, 29 pages      arXiv:2010.10615      https://doi.org/10.3842/SIGMA.2021.025
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model

Vladimir V. Bazhanov a, Gleb A. Kotousov b, Sergii M. Koval a and Sergei L. Lukyanov cd
a) Department of Theoretical Physics, Research School of Physics, Australian National University, Canberra, ACT 2601, Australia
b) DESY, Theory Group, Notkestrasse 85, Hamburg, 22607, Germany
c) NHETC, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA
d) Kharkevich Institute for Information Transmission Problems, Moscow, 127994, Russia

Received October 30, 2020, in final form February 26, 2021; Published online March 16, 2021

Abstract
The inhomogeneous six-vertex model is a 2$D$ multiparametric integrable statistical system. In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, have remained unexplored. For general values of the parameters and twisted boundary conditions the model possesses ${\rm U}(1)$ invariance. In this paper we discuss the restrictions imposed on the parameters for which additional global symmetries arise that are consistent with the integrable structure. These include the lattice counterparts of ${\mathcal C}$, ${\mathcal P}$ and ${\mathcal T}$ as well as translational invariance. The special properties of the lattice system that possesses an additional ${\mathcal Z}_r$ invariance are considered. We also describe the Hermitian structures, which are consistent with the integrable one. The analysis lays the groundwork for studying the scaling limit of the inhomogeneous six-vertex model.

Key words: solvable lattice models; Bethe ansatz; Yang-Baxter equation; discrete symmetries; Hermitian structures.

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