Algebraic arctic curves in the domain-wall six-vertex model

TL;DR

This paper derives explicit algebraic equations for the arctic curve in the six-vertex model with domain wall boundary conditions at root-of-unity weights, revealing new algebraic structures and bounds.

Contribution

It provides explicit algebraic descriptions of the arctic curve at root-of-unity weights, including degree bounds and detailed examples.

Findings

Arctic curve is described by algebraic equations at root-of-unity weights.

Explicit parametric solutions for the arctic curve are obtained.

Upper bounds on the degree of the algebraic equations are established.

Abstract

The arctic curve, i.e. the spatial curve separating ordered (or `frozen') and disordered (or `temperate) regions, of the six-vertex model with domain wall boundary conditions is discussed for the root-of-unity vertex weights. In these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve. Some interesting examples are discussed in detail. The upper bound on the maximal degree of the equation in a generic root-of-unity case is obtained.