Critical exponents of the 3d Ising and related models from conformal bootstrap

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 -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 = 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.

Section: Discussionmentioning

confidence: 99%

No unitary bootstrap for the fractal Ising model

Golden

Paulos

2015

“…While our results could be compared to the leading results of perturbation theory, the most interesting question that remains open is on how to generalize our use of the CFT constraints to successfully reproduce higher orders of the -expansion. It is possible that the correct path is to follow the conformal bootstrap program [35,36]: possibly using the Mellin space representation [37,38] and ensuring that the non-unitarity of some theories poses no obstacle [39,40], or perhaps exploiting the idea of large spin perturbation theory [41,42], which may prove useful in this direction.…”

Section: Jhep04(2017)127mentioning

confidence: 99%

Leading CFT constraints on multi-critical models in d > 2

Codello

Safari

Vacca

et al. 2017

Abstract:We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial φ m below their upper critical dimensions d c = 2m m−2 , and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m ≥ 4 these theories coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d = 2, while for odd m the theories are non-unitary and start at m = 3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators φ k and of some families of structure constants in either the coupling's or the -expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling's expansion.

“…This framework for computing matrix elements would allow us to start from any CFT where OPE coefficients are known explicitly or could be found through the numerical bootstrap [48][49][50] (for CFTs without a Lagrangian description).…”

Section: Jhep07(2016)140mentioning

confidence: 99%

A conformal truncation framework for infinite-volume dynamics

Katz

Khandker

Walters

2016

107

We present a new framework for studying conformal field theories deformed by one or more relevant operators. The original CFT is described in infinite volume using a basis of states with definite momentum, P , and conformal Casimir, C. The relevant deformation is then considered using lightcone quantization, with the resulting Hamiltonian expressed in terms of this CFT basis. Truncating to states with C ≤ C max , one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method requires the introduction of an appropriate regulator, which can be chosen to preserve the conformal structure of the basis. We check this framework in three dimensions for various perturbative deformations of a free scalar CFT, and for the case of a free O(N ) CFT deformed by a mass term and a non-perturbative quartic interaction at large-N . In all cases, the truncation scheme correctly reproduces known analytic results. We also discuss a general procedure for generating a basis of Casimir eigenstates for a free CFT in any number of dimensions.