Numerical study of the F model with domain-wall boundaries

Abstract

We perform a numerical study of the F model with domain-wall boundary conditions. Various exact results are known for this particular case of the six-vertex model, including closed expressions for the partition function for any system size as well as its asymptotics and leading finite-size corrections. To complement this picture we use a full lattice multicluster algorithm to study equilibrium properties of this model for systems of moderate size, up to L = 512. We compare the energy to its exactly known large-L asymptotics. We investigate the model's infinite-order phase transition by means of finite-size scaling for an observable derived from the staggered polarization in order to test the method put forward in our recent joint work with Duine and Barkema. In addition we analyze local properties of the model. Our data are perfectly consistent with analytical expressions for the arctic curves. We investigate the structure inside the temperate region of the lattice, confirming the oscillations in vertex densities that were first observed by Syljuåsen and Zvonarev and recently studied by Lyberg et al. We point out "(anti)ferroelectric" oscillations close to the corresponding frozen regions as well as "higher-order" oscillations forming an intricate pattern with saddle-point-like features.

Phase separation in the six-vertex model with a variety of boundary conditions

“…4c, where the vertices (x, y) are now coloured in black if δρ c ≤ 0 and white otherwise. As noticed in [16] for the case of DWBC, we observe also here the presence of saddle-point-like features emerging in the average density.…”

“…Another density, which will be used in this paper, is the difference of the density vertices of type c 1 and vertices of type c 2 . This density [16]…”

“…Density profiles. The observables we use to study the behaviour of the model are the density profiles; see [16] and [15] for a detailed study of various density profiles in the case of DWBC. One density is the density of the curves which separate the level sets; we call this density ρ [15].…”

Arctic boundaries of the ice model on three-bundle domains

“…This phenomenon was soon observed to be ubiquitous within the context of highly correlated statistical mechanical systems; see, for instance, [1,2,5,6,7,10,12,13,15,16,17,18,19,20,25,28,29,30,32,33,34,35,42,43,44,45,51,54,57,61]. In particular, Cohn-Kenyon-Propp developed a variational principle [12] that prescribes a law of large numbers for random domino tilings on almost arbitrary domains, which was used effectively by to explicitly determine the arctic boundaries of uniformly random lozenge tilings on polygonal domains.…”

“…Still, extensive simulations [34,61,2,25,43,51,42] over the past two decades have provided strong numerical evidence indicating the existence of an arctic boundary for the domain-wall sixvertex model. Based on earlier free energy predictions due to Lieb [50] and Sutherland-Yang-Yang [60], variational principles have also been conjectured for the six-vertex model with general boundary conditions [65,53,58].…”