A strong-weak duality for the 1d long-range Ising model

We investigate the one-dimensional Ising model with long-range interactions decaying as $1/r^{1+s}$. In the critical regime, for $1/2 \leq s \leq 1$, this system realizes a family of nontrivial one-dimensional conformal field theories (CFTs), whose data vary continuously with $s$. For $s>1$ the model has instead no phase transition at finite temperature, as in the short-range case. In the standard field-theoretic description, involving a generalized free field with quartic interactions, the critical model is weakly coupled near $s=1/2$ but strongly coupled in the vicinity of the short-range crossover at $s=1$. We introduce a dual formulation that becomes weakly coupled as $s \to 1$. Precisely at $s=1$, the dual description becomes an exactly solvable conformal boundary condition of the two-dimensional free scalar. We present a detailed study of the dual model and demonstrate its effectiveness by computing perturbatively the CFT data near $s=1$, up to next-to-next-to-leading order in $1-s$, by two independent approaches: (i) standard renormalization of our dual field-theoretic description and (ii) the analytic conformal bootstrap. The two methods yield complete agreement.

💡 Research Summary

The paper investigates the one‑dimensional Ising model with power‑law decaying interactions, J ∑_{i≠j}σ_iσ_j/|i−j|^{1+s}, focusing on the critical regime ½ ≤ s ≤ 1 where a continuous phase transition occurs. In this interval the model is described in the continuum by a generalized free field (GFF) φ of scaling dimension Δ_φ = (1−s)/2 perturbed by λ₂ φ² and λ₄ φ⁴ interactions. Near s ≈ ½ the quartic coupling is small and the theory is weakly coupled, reproducing mean‑field exponents with small corrections. As s approaches 1, Δ_φ→0 and the quartic coupling becomes strong, rendering the standard φ⁴ description unsuitable.

To overcome this difficulty the authors construct a dual formulation inspired by the Anderson‑Yuval‑Kosterlitz (AYK) Coulomb‑gas picture of domain walls. The dual theory treats the domain walls as elementary excitations and is formulated as a one‑dimensional impurity (Kondo‑type) model. It contains two fields: a topological spin operator σ with Δ_σ = 0 (the Z₂ order parameter in the short‑range limit) and a GFF χ with Δ_χ = (1+s)/2. The interaction is linear, g ∫σ χ. At s = 1 the χ field coincides with the boundary value of a two‑dimensional free scalar, so the dual model becomes an exactly solvable conformal boundary condition of the 2d free boson. Consequently the dual description is weakly coupled for s→1, providing a complementary weak‑coupling expansion to the φ⁴ picture which is weak near s ≈ ½.

The authors perform two independent calculations of the conformal data of the resulting infrared fixed point. First, a perturbative renormalization‑group (RG) analysis is carried out in the small parameter δ = 1−s. They compute the β‑functions up to two loops, locate the non‑trivial fixed point, and determine anomalous dimensions of the primary operators σ, χ, the composite operator \hatσ₃, and others. They also obtain OPE coefficients for four‑point functions ⟨σσσσ⟩, ⟨σσχχ⟩, ⟨σχσχ⟩, etc., expanding them to order δ, √δ and δ². Logarithmic corrections at the crossover are discussed and the consistency of the operator spectrum with conformal invariance is verified.

Second, the authors apply the analytic conformal bootstrap for one‑dimensional CFTs. Using the exact data at s = 1 (the boundary CFT of the 2d free scalar) as seed, they assume that the CFT data admit an expansion in non‑negative powers of √δ. By constructing appropriate analytic functionals they solve the crossing equations for the light operators (σ, χ, and their composites) order by order in √δ. The bootstrap reproduces precisely the anomalous dimensions and OPE coefficients obtained from the RG calculation, and even predicts higher‑order terms (e.g., δ^{3/2}) that were not explicitly computed in the RG section. The agreement between the two methods provides strong evidence that the dual impurity model indeed flows to a genuine conformal fixed point for all s ≤ 1.

The paper’s main contributions are: (i) identification of the exact duality between the 1d long‑range Ising model at s = 1 and a conformal boundary condition of the 2d free scalar; (ii) construction of a weakly coupled field‑theoretic description valid near the short‑range crossover (s → 1); (iii) systematic perturbative determination of CFT data up to next‑to‑next‑to‑leading order in 1−s; (iv) independent verification of these results via analytic bootstrap, establishing the consistency of the proposed duality and the conformal nature of the infrared fixed point. The work opens avenues for studying other one‑dimensional long‑range models (e.g., Potts or O(N) generalizations) and for exploring higher‑dimensional analogues of the strong‑weak duality presented here.