(No) Bootstrap for the Fractal Ising Model

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 $\epsilon$-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 $2\le d<4$ Ising "kink" splits into two features which grow apart below $d=2$.

v2: minor updates

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Cited by in corpus (9)

• Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions
• Bootstrapping SCFTs with Four Supercharges
• Bootstrapping $O(N)$ Vector Models in $4<d<6$
• PyCFTBoot: A flexible interface for the conformal bootstrap
• Non-Abelian Binding Energies from the Lightcone Bootstrap
• Random Field Ising Model and Parisi-Sourlas Supersymmetry. Part II. Renormalization Group
• The Epsilon-Expansion from Conformal Field Theory
• On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions
• Analytic Studies of Fermions in the Conformal Bootstrap