Boundary one-point function of the rational six-vertex model with partial domain wall boundary conditions: explicit formulas and scaling properties

TL;DR

This paper derives explicit formulas and analyzes the scaling properties of the boundary one-point function in the rational six-vertex model with partial domain wall boundary conditions, revealing step-wise behavior and error function scaling near the steps.

Contribution

It provides explicit determinant formulas for finite lattices and analyzes the asymptotic behavior of the boundary one-point function in the large lattice limit.

Findings

One-point function exhibits step-wise behavior as lattice size grows.

Near the steps, the one-point function scales as the error function.

Asymptotic expansion derived from a second-order differential equation.

Abstract

We consider the six-vertex model with the rational weights on an $s\times N$ square lattice, $s\leq N$, with partial domain wall boundary conditions. We study the one-point function at the boundary where the free boundary conditions are imposed. For a finite lattice, it can be computed by the quantum inverse scattering method in terms of determinants. In the large $N$ limit, the result boils down to an explicit terminating series in the parameter of the weights. Using the saddle-point method for an equivalent integral representation, we show that as $s$ next tends to infinity, the one-point function demonstrates a step-wise behavior; at the vicinity of the step it scales as the error function. We also show that the asymptotic expansion of the one-point function can be computed from a second-order ordinary differential equation.