1d Long-range Ising Model Near the Short-range Crossover

Seed Seminar of Mathematics and Physics
One of the features of the critical behavior of the long-range Ising model (LRI) with interactions decaying as 1/rs+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 φ4 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 . 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.
 
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