Boundary One-Point Function of the Rational Six-Vertex Model with Partial Domain Wall Boundary Conditions: Explicit Formulas and Scaling Properties

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2021
• Main Authors: Minin, Mikhail D., Pronko, Andrei G.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2021
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211416
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Boundary One-Point Function of the Rational Six-Vertex Model with Partial Domain Wall Boundary Conditions: Explicit Formulas and Scaling Properties. Mikhail D. Minin and Andrei G. Pronko. SIGMA 17 (2021), 111, 29 pages

We consider the six-vertex model with the rational weights on an 𝑠 × 𝑁 square lattice, 𝑠 ≤ 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; in 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.