Arctic Curves of the Twenty-Vertex Model with Domain Wall Boundaries

Abstract

We use the tangent method to compute the arctic curve of the Twenty-Vertex (20V) model with particular domain wall boundary conditions for a wide set of integrable weights. To this end, we extend to the finite geometry of domain wall boundary conditions the standard connection between the bulk 20V and 6V models via the Kagome lattice ice model. This allows to express refined partition functions of the 20V model in terms of their 6V counterparts, leading to explicit parametric expressions for the various portions of its arctic curve. The latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases. A number of numerical simulations are also presented, which highlight the arctic curve phenomenon and corroborate perfectly the analytic predictions of the tangent method. We finally compute the arctic curve of the Quarter-turn symmetric Holey Aztec Domino Tiling (QTHADT) model, a problem closely related to the 20V model and whose asymptotics may be analyzed via a similar tangent method approach. Again results for the QTHADT model are found to be in perfect agreement with our numerical simulations. CONTENTS

Exact time evolution formulae in the XXZ spin chain with domain wall initial state

“…The general procedure is known as the tangent method [61]; it has been shown to apply to a wide class of -interacting or not-particle systems, see e.g. [14,[62][63][64]. As we shall see below, the only ingredient needed will be the asymptotic behavior of h(τ |z), combined with the free propagation result (110,111).…”

Arctic boundaries of the ice model on three-bundle domains

“…The tangent method was implemented in [20] to predict the arctic boundary of the domainwall six-vertex model; the result matched earlier predictions from [15]. It was also used later to heuristically derive the arctic boundaries for the six-vertex model on other domains [20,19] by Colomo-Sportiello and Colomo-Pronko-Sportiello; for vertically symmetric alternating sign matrices [32] by Di Francesco-Lapa; various classes of non-intersecting path models [27,28,29,30,31,32] by Debin-Granet-Ruelle, Debin-Ruelle, Di Francesco-Guitter, and Di Francisco-Lapa; for twentyvertex models by Debin-Di Francesco-Guitter [26]; and for random lecture hall tableaux by Corteel-Keating-Nicoletti [22].…”

“…This phenomenon was soon observed to be ubiquitous within the context of highly correlated statistical mechanical systems; see, for instance, [1, 2, 5, 6, 7, 10, 12, 13, 15, 16, 17, 18, 19, 20, 25, 28, by Debin-Granet-Ruelle, Debin-Ruelle, Di Francesco-Guitter, and Di Francisco-Lapa; for twentyvertex models by Debin-Di Francesco-Guitter [26]; and for random lecture hall tableaux by Corteel-Keating-Nicoletti [22].…”

Fluctuation of the phase boundary in the six-vertex model with domain wall boundary conditions: a Monte Carlo study

“…In the disorder phase, analogous MC algorithms are nowadays commonly adopted in numerical experiments with the six vertex model [46][47][48][49][50][51]. For earlier numerical approaches to ice models, see [52,53].…”