BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Six-vertex and Ashkin-Teller models: order/disorder phase transition Glazman, Alexander


Ashkin-Teller model is a classical four-component spin model introduced in 1943. It can be viewed as a pair of Ising models tau and tauâ with parameter J that are coupled by assigning parameter U for the interaction of the products tau*tauâ at every two neighbouring vertices. On the self-dual curve sinh 2J = e^{-2U}, the Ashkin-Teller model can be coupled with the six-vertex model with parameters a=b=1, c=coth 2J and is conjectured to be conformally invariant. The latter model has a height-function representation. We show that the height at a given face diverges logarithmically in the size of the domain when c=2 and remains uniformly bounded when c>2. In the latter case we obtain a complete description of translation-invariant Gibbs states and deduce that the Ashkin-Teller model on the self-dual line exhibits the following symmetry-breaking whenever J < U: correlations of spins tau and tauâ decay exponentially fast, while the product tau*tauâ is ferromagnetically ordered. The proof uses the Baxter-Kelland-Wu coupling between the six-vertex and the random-cluster models, as well as the recent results establishing the order of the phase transition in the latter model. However, in the talk, we will focus mostly on other parts of the proof: - description of the height-function Gibbs states via height-function mappings and T-circuits, - coupling between the Ashkin-Teller and the six-vertex models via an FK-Ising-type representation of these two models. (this is joint work with Ron Peled)

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