No unitary 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 -> 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 <= d < 4 Ising kink splits into two features which grow apart below d = 2.