# No unitary bootstrap for the fractal Ising model

Golden, John (Department of Physics, Brown University, Box 1843, Providence, RI, 02912-1843, U.S.A.) ; Paulos, Miguel (CERN, Theory Division, Geneva, Switzerland)

01 April 2015

Abstract: 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.

Published in: JHEP 1503 (2015) 167