A structural test for the conformal invariance of the critical 3d Ising model

Meneses, Simão; M., João, Simão; Rychkov, Slava; Lopes, J. M. Viana Parente; Yvernay, Pierre

doi:10.1007/jhep04(2019)115

Cited by 16 publications

(22 citation statements)

References 35 publications

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“…The version with general (m, n) will be important for the larger correlator system that puts Tµν on the same footing as σ, and χ 20. While these operators have never been accessed by the numerical bootstrap, results about Z2-even vectors in Monte Carlo simulations recently appeared in[90].…”

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confidence: 99%

Bootstrapping the long-range Ising model in three dimensions

Behan¹

2019

J. Phys. A: Math. Theor.

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The 3D Ising model and the generalized free scalar of dimension at least 0.75 belong to a continuous line of nonlocal fixed points, each referred to as a long-range Ising model. They can be distinguished by the dimension of the lightest spin-2 operator, which interpolates between 3 and 3.5 if we focus on the non-trivial part of the fixed line. A property common to all such theories is the presence of three relevant conformal primaries, two of which form a shadow pair. This pair is analogous to a superconformal multiplet in that it enforces relations between certain conformal blocks. By demanding that crossing symmetry and unitarity hold for a set of correlators involving the relevant operators, we compute numerical bounds on their scaling dimensions and OPE coefficients. Specifically, we raise the minimal spin-2 operator dimension to find successively smaller regions which eventually form a kink. Whenever a kink appears, its co-ordinates show good agreement with the epsilon expansion predictions for the critical exponents in the corresponding statistical model. As a byproduct, our results reveal an infinite tower of protected operators with odd spin.

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confidence: 99%

Bootstrapping the long-range Ising model in three dimensions

Behan¹

2019

J. Phys. A: Math. Theor.

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“…Non-unitary CFTs can arise for other values of N , such as N = 0, which describes self-avoiding random walks 5. The possibility that the critical theory is scale-, but not conformal, invariant has been recently excluded for the d = 3 φ 4 theory[16].…”

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confidence: 99%

Renormalization scheme dependence, RG flow, and Borel summability in ϕ4 theories in d<4

2019

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Renormalization group (RG) and resummation techniques have been used in Ncomponent φ 4 theories at fixed dimensions below four to determine the presence of non-trivial IR fixed points and to compute the associated critical properties. Since the coupling constant is relevant in d < 4 dimensions, the RG is entirely governed by renormalization scheme-dependent terms. We show that the known proofs of the Borel summability of observables depend on the renormalization scheme and apply only in "minimal" ones, equivalent in d = 2 to an operatorial normal ordering prescription, where the β-function is trivial to all orders in perturbation theory. The presence of a non-trivial fixed point can be unambiguously established by considering a physical observable, like the mass gap, with no need of RG techniques. Focusing on the N = 1, d = 2 φ 4 theory, we define a one-parameter family of renormalization schemes where Borel summability is guaranteed and study the accuracy on the determination of the critical exponent ν as the scheme is varied. While the critical coupling shows a significant sensitivity on the scheme, the accuracy in ν is essentially constant. As by-product of our analysis, we improve the determination of ν obtained with RG methods by computing three more orders in perturbation theory.

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“…(54)) is not valid anymore. This observation is at the heart of the criticism raised in in the recent preprint [30] where it was observed that, for N = 2, one can construct the conserved current J µ = ϕ 1 ∂ µ ϕ 2 − ϕ 2 ∂ µ ϕ 1 which is SO(2) invariant and has scaling dimension exactly equal to d − 1. At first sight this would be a counter-example of the present proof.…”

Section: Extension Of the Proof For Some O(n ) Modelsmentioning

confidence: 98%

“…Accordingly, in this important universality class, scale invariance implies conformal invariance in all dimensions. This proof has been criticized in [30] where it was argued that the assumptions made may not be fulfilled. Some elements of reply have already been presented in [31] but we discuss below in detail the issues raised in [30].…”

Section: Introductionmentioning

confidence: 99%

“…This proof has been criticized in [30] where it was argued that the assumptions made may not be fulfilled. Some elements of reply have already been presented in [31] but we discuss below in detail the issues raised in [30]. In [26] the scaling dimension of the most relevant vector operator was also calculated in d = 4 − ǫ in the Ising universality class obtaining D V = 3 + O(ǫ 2 ).…”

Section: Introductionmentioning

confidence: 99%

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Conformal Invariance and Vector Operators in the O(N) Model

2019

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It is widely expected that, for a large class of models, scale invariance implies conformal invariance. A sufficient condition for this to happen is that there exists no integrated vector operator, invariant under all internal symmetries of the model, with scaling dimension −1. In this article, we compute the scaling dimensions of vector operators with lowest dimensions in the O(N ) model. We use three different approximation schemes: ǫ expansion, large N limit and third order of the Derivative Expansion of Non-Perturbative Renormalization Group equations. We find that the scaling dimensions of all considered integrated vector operators are always much larger than −1. This strongly supports the existence of conformal invariance in this model. For the Ising model, an argument based on correlation functions inequalities was derived, which yields a lower bound for the scaling dimension of the vector perturbations. We generalize this proof to the case of the O(N ) model with N ∈ {2, 3, 4}.

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A structural test for the conformal invariance of the critical 3d Ising model

Cited by 16 publications

References 35 publications

Bootstrapping the long-range Ising model in three dimensions

Bootstrapping the long-range Ising model in three dimensions

Renormalization scheme dependence, RG flow, and Borel summability in ϕ4 theories in d<4

Conformal Invariance and Vector Operators in the O(N) Model

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