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/r¹⁺ˢ. In the critical regime, for 1/2 ≤ s ≤ 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 → 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.