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 ≤ 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-nextto- 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.