One-Dimensional Ising Model with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>1.99</mml:mn></mml:msup></mml:math> Interaction

We study the 1D Ising model with long-range interactions decaying as 1/r^{1+s}. The critical model corresponds to a family of 1D conformal field theories whose data depend nontrivially on s in the range 1/2≤s≤1. The model is known to be described by a generalized free field with quartic interaction, which is weakly coupled near s=1/2 but strongly coupled near the short-range crossover at s=1. We propose a dual description that becomes weakly coupled at s=1. At s=1, our model becomes an exactly solvable conformal boundary condition for the 2D free scalar. We perform a number of consistency checks of our proposal and calculate the perturbative conformal field theory data around s=1 analytically using both (1) our proposed field theory and (2) the analytic conformal bootstrap. Our results show complete agreement between the two methods.