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

TL;DR

This paper investigates a one-dimensional long-range Ising model with interactions decaying as 1/r^{1+s}, exploring its conformal field theory descriptions, dualities, and exact solutions near the critical point at s=1.

Contribution

It introduces a dual description of the 1d long-range Ising model that simplifies analysis at s=1 and verifies the model's CFT data through both field theory and bootstrap methods.

Findings

The model is described by a family of 1d CFTs with data depending on s.

A dual description becomes weakly coupled at s=1, enabling exact solutions.

Analytic calculations agree with bootstrap results near s=1.

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.