Universality of spin correlations in the Ising model on isoradial graphs

Abstract

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i. e., as the mesh size δ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave aswhere the universal constant Cσ and the function Ξ(|u 1 − u 2 |, m) are independent of the lattice. The mass m is defined by the relation k − 1 ∼ 4mδ, where k is the Baxter elliptic parameter. This includes m of both signs as well as the critical case when Ξ(r, 0) = r −1/4 .These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S. C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice.As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel z −1/2 for m = 0 and of z −1/2 e ±2m|z| for m = 0. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

Conformal invariance in the quantum Ising model

“…This result generalizes the full-plane magnetization result below criticality proven for rectangular grids in [CHM19] and in general Z-invariant isoradial graphs with bounded angles in [CIM21]. It is known by [DCLM18] that above criticality, the spin-spin correlation decays exponentially fast.…”

“…They are expressed in terms of the space-time spin representation of the quantum Ising model. They also extend a series of results proved on isoradial graphs [HS13,Hon14,CHI15,CIM21] to the semi-discrete lattice, showing the existence and the conformal covariance of a scaling limit in simply connected domains. Such an extension has previously been made for interfaces in the FK-loop representation [Li19], but more work is required in this paper for general correlations, one of the reasons being that normalizing factors depending on the mesh size δ of the lattice are now required.…”

“…Proof. This is a consequence of Lemma 2.9 and Remark 2.13 in [CHI21], or [CIM21], figure 6. The introduction of Ω δ,a amounts to introducing a cut γ a = [u a v a ] in the double cover Ω [ua,va] which branches around u a and v a .…”

“…In [DCLM18], authors obtained finer estimations in this particular procedure from discrete to semi-discrete, thus extending phase transition results from discrete graphs described in Figure 2.3 to the semi-discrete lattice 2Z × R. This approach might be tempting knowing universality of spin-correlation on isoradial graphs proven in [CIM21], provided they keep the bounded angle-property. Still, in this case, additional difficulties rise due to local normalizing factor appearing before smashing the grid in the vertical direction.…”

“…in a connected set S of Λ(G) where the observable has no branchings (or vanishes near its branchings as described above), the maximum and the minimum of H is attained at the boundary of S. When integrating observables definded on a double cover with a finite number of branchings, this integration procedure kills the branching structure and provides a well-defined function on the planar domain. We now present a trick already mentioned in [CIM21], lemma 3.14 that takes advantages of the previous integration procedure to extact values of observables near their branchings.…”