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

TL;DR

The paper advances a weakly coupled dual description of the 1d long-range Ising model near the crossover to short-range behavior, identifying a Kondo-like field theory whose U(1) singlet sector reproduces the LRI CFT at s=1 and remains tractable for s<1. By combining perturbative renormalization group analysis with analytic conformal bootstrap, the authors compute CFT data (scaling dimensions and OPE coefficients) near s=1 up to high order in the small parameter δ=1−s, and verify agreement between the two methods. They reveal a protected spectrum including σ and χ with exact dimensions under the crossover and derive numerous OPE relations that constrain the operator content, providing strong evidence for conformal invariance of the IR fixed point. The work also connects the 1d LRI to the AYK model and to a defect-CFT formulation, offering a coherent framework for systematic perturbative and bootstrap-based calculations of long-range critical behavior in one dimension.

Abstract

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.

Figures (3)

• Figure 1: $s=1$
• Figure 2: $s=0.9$
• Figure 4: A typical low-temperature configuration of kinks and antikinks in the LRI model, with $n=2$ and $a/L\sim 10^{-2}$.