1d Ising model with $1/r^{1.99}$ interaction

TL;DR

This work identifies a dual impurity-type description for the critical one-dimensional long-range Ising model with 1/r^{1+s} interactions, yielding a weakly coupled fixed point near the crossover s→1. The authors formulate a defect CFT Zs that couples a negative-dimension Gaussian free field to a qubit, reproduce the AYK Coulomb-gas physics, and obtain a controlled IR fixed point with g⋆ ∼ √δ and b⋆^2 ∼ 1−δ/4. They show that at s=1 the theory reduces to a conformal boundary condition for a 2d free scalar, enabling an exact spectrum and OPE data; perturbative CFT data around s=1 are then derived via both RG and analytic conformal bootstrap and shown to be in perfect agreement. The results provide a concrete, solvable framework for the 1d LRI crossover, deliver precise predictions for operator dimensions and OPE coefficients, and offer a pathway to extend these methods to related long-range and defect CFT problems.

Abstract

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 (CFTs) whose data depends nontrivially on $s$ in the range $1/2\leq s\leq 1$. 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 which 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 CFT 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.