Ongoing trimester: Spin systems and phases of matter
1d Long-range Ising model near the short-range crossover
byPhiline van Vliet
Abstract:One of the features of the critical behavior of the long-range Ising model (LRI) with interactions decaying as \(1/r^{s + d}\) is that it has a line of interacting fixed points, and hence corresponds to an family of interacting d-dimensional CFT, for \(d/2 < s < s_{*}\). Both ends of this range of s admit a weakly-coupled description. For small \(s\), the theory is described as a generalized free field with a \(\varphi^4\) interaction. For large \(s\), the theory can be described by the \(d\)-dimensional short-range Ising model (SRI) coupled to a generalized free field. While these descriptions have been used to study the LRI in dimensions \(d>1\), in \(d = 1\) it leads to a fascinating puzzle. The \(1d\) LRI still has a line of interacting fixed points for \(1/2 < s < 1\). However, it is well-known that the SRI does not have a second-order phase transition in \(1d\) and leaves the question what happens when \(s\) approaches \(1\).
In this talk I will discuss our model which provides a weakly-coupled description for the \(1d\) LRI around \(s = 1\). 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 CFT data around \(s = 1\) using perturbation theory and the analytic bootstrap. This is based on https://arxiv.org/pdf/2412.12243.