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} 1 / r 1 + s . In the critical regime, for 1/2 ≤ s ≤ 1 1 / 2 ≤ s ≤ 1 , this system realizes a family of nontrivial one-dimensional conformal field theories (CFTs), whose data vary continuously with s s . For s;gt1 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 s = 1 / 2 but strongly coupled in the vicinity of the short-range crossover at s=1 s = 1 . We introduce a dual formulation that becomes weakly coupled as s \to 1 s → 1 . Precisely at s=1 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 s = 1 , up to next-to-next-to-leading order in 1-s 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.