Order-disorder phase transition in 2D lattice models




The project P 34713 has been funded by FWF as one of the Stand-Alone Projects at the decision board no. 83 of May 10th, 2021 with a budget of 399.577,50 Euro for the duration of 4 years.


  1. We prove the existence of macroscopic loops in the loop \(O(2)\) model with \(\frac12\leq x^2\leq 1\). This implies a logarithmic delocalisation of an integer-valued Lipschitz function on the triangular lattice and settles one side of the conjecture of Fan, Domany, and Nienhuis: they predicted in the 1970s-80s that \(x^2=\frac12\) is the critical point for the localisation-delocalisation transition.
    We also prove delocalisation in the six-vertex model with \(0<a,b\leq c\leq a+b\). This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions, for \(1\leq q\leq 4\). We rely neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory.
    Our approach goes through a novel FKG property that enables the use of the non-coexistence theorem of Zhang and Sheffield to prove delocalisation all the way up to the critical point. In the six-vertex model, we additionally use the \(\mathbb T\)-circuit argument. Finally, in the regimes \(\frac12\leq x^2\leq 1\) and \(a=b\leq c\leq a+b\), we use additional symmetries to extend existing renormalisation inequalities and establish logarithmic fluctuations. This is consistent with the conjecture that the scaling limit is the Gaussian free field.

  2. Phase diagram of the Ashkin-Teller model, Preprint, with Yacine Aoun and Moritz Dober

    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 \(J,U>0\) 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 \(J<U\), the transitions for the Ising spins and their products occur at two distinct curves that are dual to each other; when \(J\geq U\), both transitions occur at the self-dual curve. All transitions are shown to be sharp using the OSSS inequality.
    We use a finite-criterion argument and continuity to extend the result of Peled and the third author 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.

  3. Heisenberg models and Schur--Weyl duality, AAM, with Jakob Björnberg, Hjalmar Rosengren, Kieran Ryan

    Advances in Applied Mathematics (AAM), Vol. 151 (2023)
    We present a detailed analysis of certain quantum spin systems with inhomogeneous (non-random) mean-field interactions. Examples include, but are not limited to, the interchange- and spin singlet projection interactions on complete bipartite graphs. Using two instances of the representation theoretic framework of Schur--Weyl duality, we can explicitly compute the free energy and other thermodynamic limits in the models we consider. This allows us to describe the phase-transition, the ground-state phase diagram, and the expected structure of extremal states.

  4. Probability and Mathematical Physics (PMP), Vol. 4 (2023), no. 2, 209–256
    We prove that all Gibbs measures of the \(q\)-state Potts model on \(\mathbb{Z}^2\) are linear combinations of the extremal measures obtained as thermodynamic limits under free or monochromatic boundary conditions. In particular all Gibbs measures are invariant under translations. This statement is new at points of first-order phase transition, that is at \(T=T_{c}(q)\) when \(q>4\). In this case the structure of Gibbs measures is the most complex in the sense that there exist \(q+1\) distinct extremal measures.
    Most of the work is devoted to the FK-percolation model on \(\mathbb{Z}^{2}\) with \(q\geq 1\), where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman and Higuchi, which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. The result for the Potts model is then derived using the Edwards-Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwards-Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation.
    Finally, the proof is generic enough to adapt to the FK-percolation and Potts models on the triangular and hexagonal lattices and to the loop \(O(n)\) model in the range of parameters for which its spin representation is positively associated.
  5. Macroscopic loops in the loop O(n) model via the XOR trick, Preprint, with Nicholas Crawford, Matan Harel, and Ron Peled 

    The loop \(O(n)\) model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by \((n,x)\), where \(n\) is a loop weight and \(x\) is an edge weight. Nienhuis predicts that, for \(0 \leq n \leq 2\), the model exhibits two regimes: one with short loops when \(x < x_c(n)\), and another with macroscopic loops when \(x \geq x_c(n)\), where \(x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}\).
    In this paper, we prove three results regarding the existence of long loops in the loop \(O(n)\) model. Specifically, we show that, for some \(\delta >0\) and any \((n,x) \in [1,1+\delta) \times (1- \delta, 1]\), there are arbitrarily long loops surrounding typical faces in a finite domain. If \(n \in [1,1+\delta)\) and \(x \in (1-\delta,1/\sqrt{n}]\), we can conclude the loops are macroscopic. Next, we prove the existence of loops whose diameter is comparable to that of a finite domain whenever \(n=1, x \in (1,\sqrt{3}]\); this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. Finally, we show the existence of non-contractible loops on a torus when \(n \in [1,2], x=1\).
    The main ingredients of the proof are: (i) the `XOR trick': if \(\omega\) is a collection of short loops and \(\Gamma\) is a long loop, then the symmetric difference of \(\omega\) and \(\Gamma\) necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary graph, built using the Chayes--Machta and Edwards--Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini--Schramm limits of planar graphs.

  6. Electronic Journal of Probability (EJP), 2023
    The symmetric six-vertex model with parameters~\(a,b,c>0\) is expected to exhibit different behavior in the regimes \(a+b<c\) (antiferroelectric), \(|a-b|<c\leq a+b\) (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 in the thermodynamic limit.
    It is shown that the height function of the six-vertex model delocalizes with logarithmic variance when \(a+b=c\) while remaining localized when \(a+b<c\). In the latter regime, the extremal translation-invariant Gibbs states of the height function are described. Qualitative differences between the two regimes are further exhibited for the Gibbs states of the six-vertex model itself and for the Gibbs states of its associated spin representation.
    Via a coupling, our results further allow to study the self-dual Ashkin--Teller model on \(\mathbb{Z}^2\). It is proved that on the portion of the self-dual curve \(\sinh 2J = e^{-2U}\) where \(J<U\) each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered. This is in contrast to the case \(J=U\) (the critical 4-state Potts model) where it is known that there is power-law decay for both Ising configurations and for their product.
    The proofs rely on the recently established order of the phase transition in the random-cluster model, which relates to the six-vertex model via the Baxter--Kelland--Wu coupling. Additional ingredients include the introduction of a random-cluster model with modified weight for boundary clusters, analysis of a spin (mixed Ashkin--Teller model) and bond representation (of the FK-Ising type) for the six-vertex model, and the introduction of triangular lattice contours and associated bijection for the analysis of the Gibbs states of the height function.


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