Conformally covariant scaling limits of energy and spin correlations in non-integrable planar Ising models

A 2021 review of the constructive renormalization group (RG) approach to the two-dimensional Ising model (A. Giuliani, arXiv:2012.01084) formulates two linked open problems whose resolution would establish rigorous universality of the Ising universality class beyond integrability: (1) extend the constructive-RG construction of the scaling limit of energy correlations from the cylinder to simply connected planar domains of arbitrary shape, for the boundary conditions open, $+$ and $-$ (handling these three suffices for this item), which requires explicit, sharp quantitative bounds on the remainder $g_a^\Omega - g_0^\Omega$ between the finite-lattice fermionic propagator in a generic domain and its continuum limit — bounds strong enough for the multiscale RG machinery to start; and (2) construct the scaling limit of bulk spin-spin correlations for non-integrable finite-range perturbations, where the spin correlator is represented as a fermionic propagator on a discretized Riemann surface with a monodromy cut between the spin insertions, so that the non-translation-invariant RG must handle a one-dimensional defect. Both core questions remain open: the boundary/half-plane cases are settled, but the arbitrary-domain energy sector and the bulk spin sector are not.

Background

The two-dimensional Ising model at its critical point is the paradigmatic lattice system whose large-scale behavior is described by a conformal field theory (CFT), i.e. a field theory invariant under angle-preserving transformations of the plane. For the exactly solvable ('integrable') nearest-neighbor coupling this is a theorem: the Kac–Ward/Grassmann method (a representation of the partition function and correlations as determinants of an explicit free-fermion kernel over anticommuting Grassmann variables) makes the model solvable, and the discrete holomorphicity of fermionic observables (exact lattice analogues of holomorphic functions, built from the order variables, i.e. the spins themselves, and the disorder variables, which measure the frustration accumulated along a loop) implies that, as the lattice mesh $a\to0$, the energy density and all spin correlation functions converge to conformally covariant limits in arbitrary simply connected planar domains, with the CFT scaling dimensions (1 for the energy, 1/8 for the spin) — Hongler–Smirnov for the energy density, Chelkak–Hongler–Izyurov for spin correlations, building on Chelkak–Smirnov's universality of fermionic observables. The central rigorous universality question for this class is whether the same conformally covariant limits persist when the interaction is perturbed by finite-range terms that destroy solvability (couplings moved off the Kac–Ward/Grassmann-solvable manifold, the 'non-integrable' regime, with perturbation strength $\lambda\neq0$). The constructive renormalization group (RG) answers this on translation-invariant geometries: integrating out lattice fermions scale by scale through Grassmann Gaussian integrals, Giuliani–Greenblatt–Mastropietro (2012) proved that in the full plane the energy correlations of the model at sufficiently small $\lambda\neq0$ converge to the same limit as the integrable model up to an explicit non-universal amplitude analytic in $\lambda$, Antinucci–Giuliani–Greenblatt (2020) established the cylinder scaling limit, up to the same non-universal multiplicative amplitude (independent of the shape and size of the domain), and Antinucci–Giuliani–Greenblatt (2021) developed the corresponding Grassmann representation in cylindrical geometry. The 2021 review by Giuliani poses, as the program's natural next targets: (i) energy correlations in domains of arbitrary shape, for different boundary conditions (say open, $+$ or $-$), which hinges on explicit bounds on the remainder $g_a^\Omega - g_0^\Omega$ between the finite-lattice fermionic propagator (the inverse of the lattice Dirac/Kac–Ward operator in the domain $\Omega$) and its continuum counterpart, sharp enough for the multiscale RG to start — the bounds implicit in the integrable-side convergence proofs are too weak; and (ii) bulk spin correlations, whose representation as a fermionic propagator on a discretized Riemann surface with a monodromy cut between the spin insertions (the two-sheeted cover of the lattice on which the fermions are single-valued; the cut is the one-dimensional line of discontinuity forced by the semilocality of the lattice fermions, which pick up a sign when a spin insertion is encircled) introduces a one-dimensional defect that the non-translation-invariant RG must handle. The two sectors target one objective: a constructive proof that conformal invariance, not merely criticality, survives the loss of integrability.

Problem Statement

Consider the two-dimensional Ising model at criticality, discretized with mesh $a$ inside a planar domain $\Omega$ of arbitrary shape, where the interaction is a finite-range perturbation of the critical nearest-neighbor couplings with sufficiently small strength $\lambda \neq 0$ — the non-integrable regime in which Kac–Ward/Grassmann solvability of the interaction no longer applies. Establish the constructive scaling-limit program of Giuliani's review in this setting, in its two linked observable sectors:

1. (Energy sector, arbitrary simply connected domains) Prove that the critical energy-density correlations admit a scaling limit in any simply connected domain $\Omega \subset \mathbb{R}^2$ of arbitrary shape, for the boundary conditions open, $+$ and $-$ (handling these three suffices for this item), and prove conformal covariance of the limit; the limit should coincide — up to the non-universal amplitude $Z_{\mathrm{energy}}(\lambda)$ constructed in the cylinder theory — with the limit of the integrable model. The source identifies the key required ingredient: explicit quantitative bounds on the remainder $g_a^\Omega - g_0^\Omega$ between the finite-lattice fermionic propagator in $\Omega$ and its continuum counterpart, sharp enough for the multiscale renormalization group to run in $\Omega$ (the bounds available from the integrable-side convergence proofs are explicitly too weak).

2. (Spin sector, bulk) Prove that the critical bulk spin-spin correlation functions at sufficiently small $\lambda \neq 0$ converge, as $a \to 0$ — the full-plane case alone resolves this sector, while convergence in bounded domains is a further stage rather than a prerequisite — to conformally covariant limits coinciding, up to a non-universal amplitude $Z_{\mathrm{spin}}(\lambda)$ constructed together with the limit, with the conformal-field-theory predictions (the limiting functions of Chelkak–Hongler–Izyurov, with anomalous exponent 1/8 per spin insertion). The source's proposed route is the representation of the spin correlator as a fermionic propagator on a discretized Riemann surface with a monodromy cut between the spin insertions, the cut acting as a one-dimensional defect to be handled by the non-translation-invariant multiscale RG.

The two sectors are the source's two open problems: both ask for the analogue, at $\lambda \neq 0$, of the integrable conformal-invariance theorems, for the two natural observable sectors of the same model class, and their resolution would deliver conformal invariance of the Ising universality class beyond integrability in domains of arbitrary shape. The source stresses that already the two-point spin correlation in the full-plane limit is highly non-trivial, and its understanding would represent a breakthrough in the field. Results for symmetric domains (torus, cylinder, half-plane) and for boundary spin observables are already established in the cited literature.

References

1. A. Giuliani, "Conformal invariance and Renormalization Group", arXiv:2012.01084 (2021), DOI 10.48550/arXiv.2012.01084, https://arxiv.org/abs/2012.01084
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