Bootstrapping the O(N ) vector models

By Filip Kos, David Poland, David Simmons-Duffin. 2013

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A bstractWe study the conformal bootstrap for 3D CFTs with O(N ) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N ) singlet and symmetric tensor operators appearing in the ϕi × ϕj OPE, where ϕi is a fundamental of O(N ). Comparing these bounds to previous determinations of critical exponents in the O(N ) vector models, we find strong numerical evidence that the O(N ) vector models saturate the bootstrap constraints at all values of N . We also compute general lower bounds on the central charge, giving numerical predictions for the values realized in the O(N ) vector models. We compare our predictions to previous computations in the 1/N expansion, finding precise agreement at large values of N .

Subfields: Computational Mathematics, Computational Mechanics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/4103375a6a81af842eef6c6380b3e5c080985057

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Citations in the corpus

Analyticity in spin in conformal theories (2017, background): “eq. (3.7). These poles are all below the unitarity bound = J+d 2, but are nonetheless to the right of the contour = d=2, and so they should be included. Poles of conformal blocks were studied e.g. in [31,32]. Their possible location is heavily constrained by the Casimir equations, which imply that t”
Bootstrapping mixed correlators in the 3D Ising model (2014, methodology): “Representations of this type were first developed for 2D (Virasoro) conformal blocks by Alyosha Zamolodchikov [37, 38], and a generalization for global conformal blocks with identical external scalars in general spacetime dimension D was developed in [16].”
Weight shifting operators and conformal blocks (2017, neutral): “where r is the radial coordinate of [141, 142] and h∞,ρ is some relatively easily computable function, then we can write the conformal block as a sum over residues [14, 105].”
Bootstrapping the O(N) archipelago (2015, neutral): “Each region is roughly triangular, with an upper-left vertex that corresponds to the kinks in previous bounds [15].”
EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions (2016, methodology): “David Simmons-Duffin work [42] provides a modern introduction to the “shadow formalism”.”
The N=2$$ \mathcal{N}=2 $$ superconformal bootstrap (2014, methodology): “The semidefinite programming approach to the numerical bootstrap depends on our ability to approximate conformal blocks of fixed spin ` and varying conformal dimension ∆ by polynomials in ∆ [39, 44].”
A semidefinite program solver for the conformal bootstrap (2015, background): “Matching polynomial coefficients on each side, as done in [7, 9, 17], leads to higher-rank matrices Ap.”
An algebraic approach to the analytic bootstrap (2015, methodology): “We hope that we can check this statement using the results obtained by other methods in the near future [16, 17].”
A scaling theory for the long-range to short-range crossover and an infrared duality (2017, methodology): “In three dimensions, the conformal blocks are not known exactly; however, they can be computed efficiently as a series in r and η = cos θ, where ρ = re, using a recursion relation [24, 25].”
Deconfined quantum critical points: symmetries and dualities (2017, neutral): “However, subsequent investigations [25,26] of SO(5)-symmetric CFTs using the conformal bootstrap [127,128] did not find a sufficiently stable [143] CFT in the expected region of parameter space.”
Bootstrapping 3D fermions (2015, methodology): “This method achieved an impressive degree of success, for example, by enabling accurate determinations of the dimensions of low-lying operators in particular CFTs, such as the 3D Ising [5–8] and critical O(N) vector [9, 10] models.”
Witten diagrams revisited: the AdS geometry of conformal blocks (2015, background): “The blocks can also be expressed as infinite sums over poles in ∆ associated with null states of SO(d, 2), in analogy with Zamolodchikov’s recursion relations in d = 2 [37–39]; these provide excellent rational approximations to the blocks that are used in numerical work.”
Looking for a bulk point (2015, background): “Solving this equation in the large ∆ limit gives [68]”
The ϵ ?> -expansion from conformal field theory (2015, background): “Many other conformal theories, such as the O N ( ) model [21, 22], models with supersymmetry in various dimensions [23–32], as well as other less familiar or conjectural conformal theories [33–39] are being studied.”