One-Dimensional Ising Model with 1/r1.99 Interaction

We study the 1D Ising model with long-range interactions decaying as 1/r1+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.