Abstract
We study the conformal bootstrap constraints for 3D conformal field theories with a ℤ2 or parity symmetry, assuming a single relevant scalar operator ϵ that is invariant under the symmetry. When there is additionally a single relevant odd scalar σ, we map out the allowed space of dimensions and three-point couplings of such “Ising-like” CFTs. If we allow a second relevant odd scalar σ′, we identify a feature in the allowed space compatible with 3D \( \mathcal{N} \) = 1 superconformal symmetry and conjecture that it corresponds to the minimal \( \mathcal{N} \) = 1 supersymmetric extension of the Ising CFT. This model has appeared in previous numerical bootstrap studies, as well as in proposals for emergent supersymmetry on the boundaries of topological phases of matter. Adding further constraints from 3D \( \mathcal{N} \) =1 superconformal symmetry, we isolate this theory and use the numerical bootstrap to compute the leading scaling dimensions Δσ = Δϵ − 1 = .58444(22) and three-point couplings λσσϵ = 1.0721(2) and λϵϵϵ = 1.67(1). We additionally place bounds on the central charge and use the extremal functional method to estimate the dimensions of the next several operators in the spectrum. Based on our results we observe the possible exact relation λϵϵϵ/λσσϵ = tan(1).
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References
S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
S. El-Showk et al., Solving the 3d Ising model with the conformal bootstrap II. c-minimization and precise critical exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
D. Poland andD. Simmons-Duffin, The conformal bootstrap, Nature Phys. 12 (2016) 535.
D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, arXiv:1805.04405 [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions with global symmetries, JHEP 01 (2018) 036 [arXiv:1705.03484] [INSPIRE].
T. Grover, D.N. Sheng and A. Vishwanath, Emergent space-time supersymmetry at the boundary of a topological phase, Science 344 (2014) 280 [arXiv:1301.7449] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Yukawa CFTs and emergent supersymmetry, PTEP 2016 (2016) 12C105 [arXiv:1607.05316] [INSPIRE].
N. Zerf et al., Four-loop critical exponents for the Gross-Neveu-Yukawa models, Phys. Rev. D 96 (2017) 096010 [arXiv:1709.05057] [INSPIRE].
B. Ihrig, L.N. Mihaila and M.M. Scherer, Critical behavior of Dirac fermions from perturbative renormalization, Phys. Rev. B 98 (2018) 125109 [arXiv:1806.04977] [INSPIRE].
J.-H. Park, Superconformal symmetry in three-dimensions, J. Math. Phys. 41 (2000) 7129 [hep-th/9910199] [INSPIRE].
J. Rong and N. Su, Bootstrapping minimal \( \mathcal{N} \) = 1 superconformal field theory in three dimensions, arXiv:1807.04434 [INSPIRE].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
T. Ohtsuki, CBoot: a Sage module to create (convolved) conformal block table, https://github.com/tohtsky/cboot.
D. Bashkirov, Bootstrapping the \( \mathcal{N} \) = 1 SCFT in three dimensions, arXiv:1310.8255 [INSPIRE].
Z. Li and N. Su, 3D CFT archipelago from single correlator bootstrap, arXiv:1706.06960 [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrapping conformal field theories with the extremal functional method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
L. Iliesiu et al., Fermion-scalar conformal blocks, JHEP 04 (2016) 074 [arXiv:1511.01497] [INSPIRE].
A. Dymarsky et al., The 3d stress-tensor bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
L.F. Alday and A. Zhiboedov, Conformal bootstrap with slightly broken higher spin symmetry, JHEP 06 (2016) 091 [arXiv:1506.04659] [INSPIRE].
L.F. Alday and A. Zhiboedov, An algebraic approach to the analytic bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
S. El-Showk et al., Conformal field theories in fractional dimensions, Phys. Rev. Lett. 112 (2014) 141601 [arXiv:1309.5089] [INSPIRE].
A.A. Nizami, T. Sharma and V. Umesh, Superspace formulation and correlation functions of 3d superconformal field theories, JHEP 07 (2014) 022 [arXiv:1308.4778] [INSPIRE].
E.I. Buchbinder, S.M. Kuzenko and I.B. Samsonov, Superconformal field theory in three dimensions: correlation functions of conserved currents, JHEP 06 (2015) 138 [arXiv:1503.04961] [INSPIRE].
A. Atanasov, A. Hillman and D. Poland, https://github.com/ABAtanasov/IsingBootstrap/.
D. Simmons-Duffin, https://gitlab.com/bootstrapcollaboration/spectrum-extraction.
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Atanasov, A., Hillman, A. & Poland, D. Bootstrapping the minimal 3D SCFT. J. High Energ. Phys. 2018, 140 (2018). https://doi.org/10.1007/JHEP11(2018)140
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DOI: https://doi.org/10.1007/JHEP11(2018)140