(No) Bootstrap for the Fractal Ising Model

We consider the conformal bootstrap for spacetime dimension $1<d<2$. We determine bounds on operator dimensions and compare our results with various theoretical and numerical models, in particular with resummed $\epsilon$-expansion and Monte Carlo simulations of the Ising model on fractal lattices. The bounds clearly rule out that these models correspond to unitary conformal field theories. We also clarify the $d\to 1$ limit of the conformal bootstrap, showing that bounds can be - and indeed are - discontinuous in this limit. This discontinuity implies that for small $\epsilon=d-1$ the expected critical exponents for the Ising model are disallowed, and in particular those of the $d-1$ expansion. Altogether these results strongly suggest that the Ising model universality class cannot be described by a unitary CFT below $d=2$. We argue this also from a bootstrap perspective, by showing that the $2\leq d<4$ Ising "kink" splits into two features which grow apart below $d=2$.