Arctic curve of the free-fermion six-vertex model with reflecting end boundary condition

Abstract

We consider the six-vertex model with reflecting end boundary condition. We study the asymptotic behavior of the boundary correlations. This asymptotic behavior is used as an input into the Tangent Method in order to derive analytically the arctic curve at the free fermion point. The obtained curve is a semicircle, which is in agreement with previous Monte Carlo simulations. *

Arctic curves of the reflecting boundary six vertex and of the twenty vertex models

“…After revisiting the case of the 6V model for pedagogical purposes in section 3, we will apply the tangent method in section 4 to the case of the 6V model on the (2n − 1) × n rectangular grid (a simplified version of the U-turn 6V model), in section 5 to the case of the 20V model with DWBC3 on the quadrangle Q n , and finally in section 6 to the domino tilings of the Aztec triangle T n . Note that the tangent method was previously applied in [PR19a] to a particular 'free fermion' case of the U-turn 6V model, where the arctic curve is a half-circle: the results of section 3 extend this to arbitrary values of the parameters.…”

“…In figure 13 (left) we represent arctic curves for η = π 4 and the isotropic value v = − π 2 with u ranging from 0 + to π 4 − . The u = 0 arctic curve is given by (x, y) = (cos(2ξ) − 1, sin(2ξ) + 1): it is the half-circle (x + 1) 2 + (y − 1) 2 = 1 with x −1, first obtained in [PR19a]. The case u = π 4 is singular.…”

Arctic curves of the 6V model with partial DWBC and double Aztec rectangles

“…The EFP technique yields an arctic curve whose shape is the solution of F(x, y, z) = 0 and ∂ z F(x, y, z) = 0, which exactly describes the envelope of a family of curves parametrized by z, which moreover turned out to be straight lines. This observation was then elevated to a universal geometric principle by the development of the tangent method [13], which provides an alternative derivation of the arctic curve and which has been applied to a large class of models [2,3,[14][15][16][17][18][19][20][21][22][23].…”

“…There exist other boundary conditions (all resembling DWBC in some way) for which the 6V model exhibits an arctic curve [4]. The reflecting boundary conditions have been studied analytically in [20][21][22], but the so-called partial domain-wall boundary conditions (pDWBC) have not been considered to this day. This paper is organised as follows.…”