Bootstrapping the O(N ) vector models

Kos, Filip; Poland, David; Simmons-Duffin, David

doi:10.1007/jhep06(2014)091

Cited by 326 publications

(639 citation statements)

References 65 publications

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“…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: 82%

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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: Introductionmentioning

confidence: 82%

“…A recent notable application of these techniques has been to CFTs with global O(N ) symmetry in d = 3 [18]. This class of CFTs are of interest because they include the critical O(N ) vector model, which has several physical applications for d = 3 and N ≤ 3.…”

Section: Introductionmentioning

confidence: 99%

BootstrappingO(N)vector models in4<d<6

2015

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“…The seminal work [8] proposed an efficient numerical procedure for extracting information about the space of all conformal field theories, and has been followed by many other works in a variety of contexts [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Similarly, the studies [24][25][26][27] have shown that it is even possible to analytically derive completely generic constraints on the spectrum of CFTs.…”

Section: Introductionmentioning

confidence: 99%

No unitary bootstrap for the fractal Ising model

Golden

Paulos

2015

J. High Energ. Phys.

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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.

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“…Recent numerical studies [49][50][51][52][53] have shed more light on the non perturbative regime of the O(N ) models where they have showed that it is possible to obtain results of the dimensions of certain operators for finite N case which resembles realistic models e.g the Ising model. Meanwhile, on the analytical side, the authors of [54] have shown that it is…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

“…The details can be found in [45][46][47][48][49][50][51][52][53]61]. We focus on theories containing a scalar field φ i in the fundamental representation of O(N ) in d = 4.…”

Section: O(n ) Fundamentalsmentioning

confidence: 99%

More on analytic bootstrap for O(N) models

Dey

Kaviraj

Sen

2016

J. High Energ. Phys.

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This note is an extension of a recent work on the analytical bootstrapping of O(N ) models. An additonal feature of the O(N ) model is that the OPE contains trace and antisymmetric operators apart from the symmetric-traceless objects appearing in the OPE of the singlet sector. This in addition to the stress tensor (T µν ) and the φ i φ i scalar, we also have other minimal twist operators as the spin-1 current J µ and the symmetric-traceless scalar in the case of O(N ). We determine the effect of these additional objects on the anomalous dimensions of the corresponding trace, symmetric-traceless and antisymmetric operators in the large spin sector of the O(N ) model, in the limit when the spin is much larger than the twist. As an observation, we also verified that the leading order results for the large spin sector from the −expansion are an exact match with our n = 0 case. A plausible holographic setup for the special case when N = 2 is also mentioned which mimics the calculation in the CFT.

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Bootstrapping the O(N ) vector models

Cited by 326 publications

References 65 publications

BootstrappingO(N)vector models in4<d<6

BootstrappingO(N)vector models in4<d<6

No unitary bootstrap for the fractal Ising model

More on analytic bootstrap for O(N) models

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