(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

References in corpus (5)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Bounds on 4D Conformal and Superconformal Field Theories
• Bounds in 4D Conformal Field Theories with Global Symmetry
• Central Charge Bounds in 4D Conformal Field Theory
• Critical Behavior of an Ising System on the Sierpinski Carpet: A Short-Time Dynamics Study

Cited by in corpus (2)

• Bootstrapping $O(N)$ Vector Models in $4<d<6$
• The Epsilon-Expansion from Conformal Field Theory