Riemann–Hilbert approach to the six-vertex model

Summary

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite n by Korepin and Izergin. The solution is based on the Yang–Baxter equations and it represents the free energy in terms of an n × n Hankel determinant. Paul Zinn-Justin observed that the Izergin–Korepin formula can be expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large n asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann–Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.

1. Six-vertex model

The six-vertex model, or the model of two-dimensional ice, is stated on a square lattice with arrows on edges. The arrows obey the rule that at every vertex there are two arrows pointing in and two arrows pointing out. This rule is sometimes called the ice-rule. There are only six possible configurations of arrows at each vertex, hence the name of the model; see Figure 1.

We will consider the domain wall boundary conditions (DWBC), in which the arrows on the upper and lower boundaries point into the square, and the ones on the left and right boundaries point out. One possible configuration with DWBC on the 4 × 4 lattice is shown on Figure 2.

The name of the square ice comes from the two-dimensional arrangement of water molecules, H2O, with oxygen atoms at the vertices of the lattice and one hydrogen atom between each pair of adjacent oxygen atoms. We place an arrow in the direction from a hydrogen atom toward an oxygen atom if there is a bond between them. Thus, as we already noticed before, there are two in-bound and two out-bound arrows at each vertex.

Figure 1. The six arrow configurations allowed at a vertex.

Figure 2. An example of a 4 x 4 configuration (left) and the corresponding ice crystal (right).

For each possible vertex state we assign a weight w_i, i =1, ...., 6, and define, as usual, the partition function, as a sum over all possible arrow configurations of the product of the vertex weights,

where V_n is the n × n set of vertices,σ (x)∈{1,...,6} is the vertex configuration of σ at vertex x according to Figure 1, and N_i (σ) is the number of vertices of type i in the configuration σ.