The Z-invariant Ising model via dimers

Abstract

The Z-invariant Ising model [3] is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k. When k = 0 the model is critical, and as k varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers [7,8] to the full Z-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of [8]: it involves a local function and the massive discrete exponential function introduced in [10]. This shows in particular that Z-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the Z-invariant spanning forests of [10], and deduce that the two models have the same second order phase transition in k. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this Z-invariant Ising model. √ 1−k 2 on the dual isoradial graph G * yield the same probability measure on polygon configurations of the graph G. The elliptic parameters k and k * can be interpreted as parametrizing dual temperatures, see Section 4.2 and also [11,47].where Z Ising (G, J) is the normalizing constant known as the Ising partition function.A polygon configuration of G is a subset of edges such that every vertex has even degree; let P(G) denote the set of polygon configurations of G. Then, the high temperature expansion [36,37] of the Ising model partition function gives the following identity: Z Ising (G, J) = 2 |V| e∈E cosh J e P∈P(G) e∈P tanh J e .

Planar Ising Model at Criticality: State-of-the-Art and Perspectives

“…More generally, one can consider an arbitrary infinite tiling ♦ of the complex plane by rhombi with angles uniformly bounded from below, split the vertices of the bipartite graph Λ formed by the vertices of these rhombi into two isoradial graphs Γ • and Γ • , and define the Ising model weights on Γ • by setting x e := tan 1 2 θ e , where θ e is the half-angle of the corresponding rhombus at vertices from Γ • . This model is called a self-dual Z-invariant Ising model on isoradial graphs and it can be viewed as a particular point in a family of the so-called Z-invariant Ising models studied by Baxter and parameterized by an elliptic parameter k. Recently, it was shown by Boutillier, de Tilière and Raschel that, similarly to the case of regular lattices, the Z-invariant Ising model on a given isoradial graph exhibits a second order phase transition at its self-dual point k = 0, see [5] and references therein for more details.…”

The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z-Invariance

“…3), and the goal is to define Boltzmann weights in terms of θ so as to transform these relations into the Yang-Baxter equations. Several Z -invariant models have been studied on lozenge graphs, including the bipartite dimer model [45], Ising model [13,16], Laplacian (or spanning forest model) [14,45], random cluster model [28]. The results of [49] also imply that we do not lose anything by considering the Yang-Baxter equations on a lozenge graph rather than on a pseudoline arrangement, like that of [5].…”

“…We prove this in Proposition 31. When (1 − k 2 )(1 − l 2 ) = 1 (or k * = l in the notations of [16]) the weights no longer depend on the bipartite coloring of Q (i.e if a face f has a half-angle θ and g has a half-angle π 2 − θ , then A( f ) = B(g), etc. ), and we recover Baxter's solution in the free-fermion case.…”

“…), and we recover Baxter's solution in the free-fermion case. When k = l we get a Z -invariant 6V model whose corresponding dimer model can be found in [16]. At this point it is unknown if this new parametrization of checkerboard Yang-Baxter equations could be extended outside of the free-fermion manifold.…”

Circle Patterns and Critical Ising Models

“…The proof of Chelkak and Smirnov uses the fact that differential operators admit well behaved discretizations on isoradial graphs [16,28]. One should also mention that dimer models on isoradial graphs related to the Ising model were studied in [7][8][9].…”