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

SIGMA 17 (2021), 111, 29 pages      arXiv:2108.06190      https://doi.org/10.3842/SIGMA.2021.111
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

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
Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia

Received August 16, 2021, in final form December 18, 2021; Published online December 25, 2021

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.

Key words: lattice models; domain wall boundary conditions; phase separation; correlation functions; Yang-Baxter algebra.

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