(No) Bootstrap for the Fractal Ising Model

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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 ε-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 ε=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.