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

Glazman, Alexander; Peled, Ron

doi:10.1214/23-ejp980

Cited by 2 publications

(19 citation statements)

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“…In this section, we describe the connection between the self-dual Ashkin–Teller model on a domain in and the spin representation of the six-vertex model on the corresponding even domain in . This relation has first been noticed in [ Fan72a ] comparing their critical properties, it was made explicit in [ Fan72b , Weg72 ] (see also [ HDJS13 ]), and was upgraded to a coupling in [ GP23 , Lis22 ] (we note that [ Lis22 ] treats a more general case of two interacting Potts models). We consider two types of boundary conditions that will play an important role in proving Proposition 1.1 .…”

Section: Models Couplings and Required Inputmentioning

confidence: 84%

“…We also state couplings of this model with the ATRC model and FK-percolation that will be crucial to our arguments. A combination of these two couplings has been made explicit recently in the work of Peled and the third author [ GP23 ] and we summarize the results of that work that we will rely on.…”

Section: Models Couplings and Required Inputmentioning

confidence: 97%

“…We refer to [ DG04 ] for a survey on the physics literature on the critical behaviour of the AT model, as well as predictions on critical exponents using the quantum field theory. Recently, Peled and the third author have proven that spins (or ) and the products exhibit qualitatively different behavior at the self-dual line when [ GP23 ]: products are ordered, while (and ) exhibits exponential decay of correlations. We derive Theorem 1 by extending this statement to an open neighborhood of the self-dual line when .…”

Section: Introductionmentioning

confidence: 99%

“…Compared to [ GP23 ], the exponential decay is proven in finite volume and under the largest boundary conditions. This is crucial for applying the so-called “finite-size criterion” (or ) argument [ Sim80 , Lie80 , DT16a , DT16b ], since the Simon–Lieb inequality is not available (the Simon inequality is valid for [ Lis21 ]).…”

Section: Introductionmentioning

confidence: 99%

“…These two couplings were composed for the first time in the work of Peled and the third author [ GP23 ]. We also use -circuits introduced in [ GP23 ] to apply the non-coexistence theorem [ She05 , DRT19 ].…”

Section: Introductionmentioning

confidence: 99%

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Phase Diagram of the Ashkin–Teller Model

Aoun,

Dober,

Glazman

2024

Commun. Math. Phys.

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The Ashkin–Teller model is a pair of interacting Ising models and has two parameters: J is a coupling constant in the Ising models and U describes the strength of the interaction between them. In the ferromagnetic case $$J,U>0$$ J , U > 0 on the square lattice, we establish a complete phase diagram conjectured in physics in 1970s (by Kadanoff and Wegner, Wu and Lin, Baxter and others): when $$J<U$$ J < U , the transitions for the Ising spins and their products occur at two distinct curves that are dual to each other; when $$J\ge U$$ J ≥ U , both transitions occur at the self-dual curve. All transitions are shown to be sharp using the OSSS inequality. We use a finite-size criterion argument and continuity to extend the result of Glazman and Peled (Electron J Probab 28:1-53, 2023) from a self-dual point to its neighborhood. Our proofs go through the random-cluster representation of the Ashkin–Teller model introduced by Chayes–Machta and Pfister–Velenik and we rely on couplings to FK-percolation.

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