No unitary bootstrap for the fractal Ising model

Golden, John; Paulos, Miguel F.

doi:10.1007/jhep03(2015)167

Cited by 23 publications

(21 citation statements)

References 67 publications

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“…Along these lines, it is worth pointing out a similarity between our study and that of [38], where Z 2 -invariant CFTs were examined in d < 3. For d ≥ 3, the upper bounds on the dimension ∆ σ of the lowest scalar that appears in the φ × φ OPE (where φ is the lowest dimension Z 2 -odd operator) exhibit a kink in the (∆ φ , ∆ σ ) plane corresponding to the Ising model.…”

Section: Discussionsupporting

confidence: 67%

“…While it is difficult to solve these constraints exactly in d > 2, the recent reformulation of the bootstrap uses unitarity to rephrase the constraint problem as a convex optimization problem, which can be numerically solved to get bounds on the CFT data in any number d of spacetime dimensions (see, for example, ). In several cases [18,29,35,38], these bounds have featured kinks that are believed to be located very close to known CFTs. The conformal bootstrap has therefore allowed these known CFTs to be studied nonperturbatively.…”

Section: Introductionmentioning

confidence: 85%

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BootstrappingO(N)vector models in4<d<6

2015

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We use the conformal bootstrap to study conformal field theories with O(N ) global symmetry in d = 5 and d = 5.95 spacetime dimensions that have a scalar operator φ i transforming as an O(N ) vector. The crossing symmetry of the four-point function of this O(N ) vector operator, along with unitarity assumptions, determine constraints on the scaling dimensions of conformal primary operators in the φ i × φ j OPE. Imposing a lower bound on the second smallest scaling dimension of such an O(N )-singlet conformal primary, and varying the scaling dimension of the lowest one, we obtain an allowed region that exhibits a kink located very close to the interacting O(N )-symmetric CFT conjectured to exist recently by Fei, Giombi, and Klebanov. Under reasonable assumptions on the dimension of the second lowest O(N ) singlet in the φ i × φ j OPE, we observe that this kink disappears in d = 5 for small enough N , suggesting that in this case an interacting O(N ) CFT may cease to exist for N below a certain critical value. *

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Section: Discussionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 85%

BootstrappingO(N)vector models in4<d<6

2015

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“…In that limit, the kinematics becomes the same as in d = 1 CFTs (conformal quantum mechanics). Then the four-point correlators are functions of the invariant u = t 12 t 34 /t 13 t 24 and using crossing symmetry and the OPE, it is possible to obtain sum rules analogous to the usual bootstrap-type equations as in CFTs (the conformal bootstrap in 0 + 1-dimensional CFT has been discussed in [68,69]). One possible obstacle to carrying out this program is that in the NRCFT case, the OPE Wilson coefficients are derivatives of an unknown function F (z = x 2 /t) appearing in the three-point function.…”

Section: Discussionmentioning

confidence: 99%

OPE convergence in non-relativistic conformal field theories

Goldberger

Khandker

Prabhu

2015

J. High Energ. Phys.

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Motivated by applications to the study of ultracold atomic gases near the unitarity limit, we investigate the structure of the operator product expansion (OPE) in non-relativistic conformal field theories (NRCFTs). The main tool used in our analysis is the representation theory of charged (i.e. non-zero particle number) operators in the NR-CFT, in particular the mapping between operators and states in a non-relativistic "radial quantization" Hilbert space. Our results include: a determination of the OPE coefficients of descendant operators in terms of those of the underlying primary state, a demonstration of convergence of the (imaginary time) OPE in certain kinematic limits, and an estimate of the decay rate of the OPE tail inside matrix elements which, as in relativistic CFTs, depends exponentially on operator dimensions. To illustrate our results we consider several examples, including a strongly interacting field theory of bosons tuned to the unitarity limit, as well as a class of holographic models. Given the similarity with known statements about the OPE in SO(2, d) invariant field theories, our results suggest the existence of a bootstrap approach to constraining NRCFTs, with applications to bound state spectra and interactions. We briefly comment on a possible implementation of this non-relativistic conformal bootstrap program.

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“…Applications have been found in diverse field theories ranging from supersymmetric conformal field theories [13][14][15][16][17] to the 3d-Ising model at criticality [18][19][20][21]. The lessons learnt using these methods transcend any underlying Lagrangian formulation and are hoped to be very general.…”

Section: Jhep11(2015)083mentioning

confidence: 99%

Analytic bootstrap at large spin

Kaviraj

Sen

Sinha

2015

J. High Energ. Phys.

106

137

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We use analytic conformal bootstrap methods to determine the anomalous dimensions and OPE coefficients for large spin operators in general conformal field theories in four dimensions containing a scalar operator of conformal dimension ∆ φ . It is known that such theories will contain an infinite sequence of large spin operators with twists approaching 2∆ φ + 2n for each integer n. By considering the case where such operators are separated by a twist gap from other operators at large spin, we analytically determine the n, ∆ φ dependence of the anomalous dimensions. We find that for all n, the anomalous dimensions are negative for ∆ φ satisfying the unitarity bound. We further compute the first subleading correction at large spin and show that it becomes universal for large twist. In the limit when n is large, we find exact agreement with the AdS/CFT prediction corresponding to the Eikonal limit of a 2-2 scattering with dominant graviton exchange.

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No unitary bootstrap for the fractal Ising model

Cited by 23 publications

References 67 publications

BootstrappingO(N)vector models in4<d<6

BootstrappingO(N)vector models in4<d<6

OPE convergence in non-relativistic conformal field theories

Analytic bootstrap at large spin

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