On the transition between the disordered and antiferroelectric phases of the 6-vertex model

Abstract

The symmetric six-vertex model with parameters $a,b,c>0$ is expected to exhibit different behavior in the regimes $a+b<c$ (antiferroelectric), $|a-b|<c\le a+b$ (disordered) and $|a-b|>c$ (ferroelectric). In this work, we study the way in which the transition between the regimes $a+b=c$ and $a+b<c$ manifests. When $a+b<c$, we show that the associated height function is localized and its extremal periodic Gibbs states can be parametrized by the integers in such a way that, in the $n$-th state, the heights $n$ and $n+1$ percolate while the connected components of their complement have diameters with exponentially decaying tails. When $a+b=c$, the height function is delocalized. The proofs rely on the Baxter-Kelland-Wu coupling between the six-vertex and the random-cluster models and on recent results for the latter. An interpolation between free and wired boundary conditions is introduced by modifying cluster weights. Using triangular lattice contours ($\mathbb{T}$-circuits), we describe another coupling for height functions that in particular leads to a novel proof of the delocalization at $a=b=c$. Finally, we highlight a spin representation of the six-vertex model and obtain a coupling of it to the Ashkin-Teller model on ${\mathbb{Z}}^2$ at its self-dual line $\sinh 2J=e^{-2U}$. When $J<U$, we show that each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered.

v2: Improvements to the exposition, minor enhancements to the main results. Added a short proof for the delocalization of the height function of square-ice via triangular circuits. v3: Minor edits, published version

References in corpus (3)

• On delocalization in the six-vertex model
• Structure of Gibbs measures for planar FK-percolation and Potts models
• Finitary codings for gradient models and a new graphical representation for the six-vertex model

Cited by in corpus (13)

• Height function delocalisation on cubic planar graphs
• Delocalization of the height function of the six-vertex model
• Stability and Loop Models from Decohering Non-Abelian Topological Order
• Nonexistence and Uniqueness for Pure States of Ferroelectric Six-Vertex Models
• Finitary codings for gradient models and a new graphical representation for the six-vertex model
• Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps
• The Discrete Gaussian model, I. Renormalisation group flow at high temperature
• The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature
• Conformal invariance and Renormalization Group
• On antiferromagnetic regimes in the Ashkin-Teller model
• Phase transitions and crtical exponents in the six-vertex model on kagome lattices
• Macroscopic behavior of Lipschitz random surfaces
• On the duality between height functions and continuous spin models