(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 $ε$-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 $d\to 1$ limit of the conformal bootstrap, showing that bounds can be - and indeed are - discontinuous in this limit. This discontinuity implies that for small $ε=d-1$ 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 $2\leq d<4$ Ising "kink" splits into two features which grow apart below $d=2$.

Figures (7)

• Figure 1: Bounds for $1<d<2$. In the terminology of ElShowk:2012ht these were done with $n_{max}=15$. They correspond to a truncation of the constraints to 136 components.
• Figure 2: Comparison with different theoretical approaches, taken from Holovatch1993. For small enough $d$ some of the predictions lie outside the unitarity bounds and we do not show them.
• Figure 3: Spectra of solutions to crossing symmetry along the edge of the allowed region. Shown are the low-lying spin-0 and spin-2 operators. Plots were made using truncations to 78 components ($n_{max}=11$).
• Figure 4: Spectrum along the bound curves for dimensions 1.8 and 1.5. Two spectra rearrangements can be clearly seen in the first case. For $d=1.5$ the first one seems to have disappeared or moved out towards the origin.
• Figure 5: Comparison of fractal Ising models on Sierpinski carpets with the $d=1.65$ bound.