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No unitary bootstrap for the fractal Ising model

  • John Golden
  • Miguel F. Paulos
Open Access
Regular Article - Theoretical Physics

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.

Keywords

Field Theories in Lower Dimensions Nonperturbative Effects 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of PhysicsBrown UniversityProvidenceU.S.A.
  2. 2.CERN, Theory DivisionGenevaSwitzerland

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