Abstract
We develop the numerical bootstrap technique to study the 2 + 1 dimensional \( \mathcal{N} \) = 1 superconformal field theories (SCFTs). When applied to the minimal \( \mathcal{N} \) = 1 SCFT, it allows us to determine its critical exponents to high precision. This model was argued in [1] to describe a quantum critical point (QCP) at the boundary of a 3 + 1D topological superconductor. More interestingly, this QCP can be reached by tuning a single parameter, where supersymmetry (SUSY) is realized as an emergent symmetry. We show that the emergent SUSY condition also plays an essential role in bootstrapping this SCFT. But performing a “two-sided” Padé re-summation of the large N expansion series, we calculate the critical exponents for Gross-Neveu-Yukawa models at N =4 and N =8.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
A. M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
S. Ferrara, A. F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [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].
S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, 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, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) Models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
S. M. Chester et al., Carving out OPE space and precise O(2) model critical exponents, JHEP 06 (2020) 142 [arXiv:1912.03324] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
J.-L. Gervais and B. Sakita, Field Theory Interpretation of Supergauges in Dual Models, Nucl. Phys. B 34 (1971) 632 [INSPIRE].
P. Ramond, Dual Theory for Free Fermions, Phys. Rev. D 3 (1971) 2415 [INSPIRE].
J. Wess and B. Zumino, Supergauge Transformations in Four-Dimensions, Nucl. Phys. B 70 (1974) 39 [INSPIRE].
A. Rahmani, X. Zhu, M. Franz and I. Affleck, Emergent supersymmetry from strongly interacting majorana zero modes, Phys. Rev. Lett. 115 (2015) 166401 [Erratum ibid. 116 (2016) 109901] [arXiv:1504.05192] [INSPIRE].
S.-K. Jian, Y.-F. Jiang and H. Yao, Emergent spacetime supersymmetry in 3d weyl semimetals and 2d dirac semimetals, Phys. Rev. Lett. 114 (2015) 237001 [arXiv:1407.4497] [INSPIRE].
T. H. Hsieh, G. B. Halász and T. Grover, All majorana models with translation symmetry are supersymmetric, Phys. Rev. Lett. 117 (2016) 166802 [arXiv:1604.08591] [INSPIRE].
S.-K. Jian, C.-H. Lin, J. Maciejko and H. Yao, Emergence of supersymmetric quantum electrodynamics, Phys. Rev. Lett. 118 (2017) 166802 [arXiv:1609.02146] [INSPIRE].
Z.-X. Li, Y.-F. Jiang and H. Yao, Edge quantum criticality and emergent supersymmetry in topological phases, Phys. Rev. Lett. 119 (2017) 107202 [arXiv:1610.04616] [INSPIRE].
Z.-X. Li, A. Vaezi, C. B. Mendl and H. Yao, Observation of Emergent Spacetime Supersymmetry at Superconducting Quantum Criticality, arXiv:1711.04772 [INSPIRE].
L. Huijse, B. Bauer and E. Berg, Emergent Supersymmetry at the Ising-Berezinskii-Kosterlitz-Thouless Multicritical Point, Phys. Rev. Lett. 114 (2015) 090404 [arXiv:1403.5565] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long Multiplet Bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].
D. J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
A. Hasenfratz, P. Hasenfratz, K. Jansen, J. Kuti and Y. Shen, The Equivalence of the top quark condensate and the elementary Higgs field, Nucl. Phys. B 365 (1991) 79 [INSPIRE].
J. Zinn-Justin, Four fermion interaction near four-dimensions, Nucl. Phys. B 367 (1991) 105 [INSPIRE].
W. Wetzel, Two Loop β-function for the Gross-Neveu Model, Phys. Lett. B 153 (1985) 297 [INSPIRE].
L. Karkkainen, R. Lacaze, P. Lacock and B. Petersson, Critical behavior of the three-dimensional Gross-Neveu and Higgs-Yukawa models, Nucl. Phys. B 415 (1994) 781 [Erratum ibid. 438 (1995) 650] [hep-lat/9310020] [INSPIRE].
N. A. Kivel, A. S. Stepanenko and A. N. Vasiliev, On calculation of (2 + ϵ) RG functions in the Gross-Neveu model from large N expansions of critical exponents, Nucl. Phys. B 424 (1994) 619 [hep-th/9308073] [INSPIRE].
J. A. Gracey, Four loop MS-bar mass anomalous dimension in the Gross-Neveu model, Nucl. Phys. B 802 (2008) 330 [arXiv:0804.1241] [INSPIRE].
L. N. Mihaila, N. Zerf, B. Ihrig, I. F. Herbut and M. M. Scherer, Gross-Neveu-Yukawa model at three loops and Ising critical behavior of Dirac systems, Phys. Rev. B 96 (2017) 165133 [arXiv:1703.08801] [INSPIRE].
N. Zerf, L. N. Mihaila, P. Marquard, I. F. Herbut and M. M. Scherer, Four-loop critical exponents for the Gross-Neveu-Yukawa models, Phys. Rev. D 96 (2017) 096010 [arXiv:1709.05057] [INSPIRE].
J. A. Gracey, Three loop calculations in the O(N) Gross-Neveu model, Nucl. Phys. B 341 (1990) 403 [INSPIRE].
J. A. Gracey, Calculation of exponent η to O(1/N2) in the O(N) Gross-Neveu model, Int. J. Mod. Phys. A 6 (1991) 395 [Erratum ibid. 6 (1991) 2755] [INSPIRE].
J. A. Gracey, Anomalous mass dimension at O(1/N2) in the O(N) Gross-Neveu model, Phys. Lett. B 297 (1992) 293 [INSPIRE].
J. A. Gracey, Computation of β (g) at O(1/N2) in the O(N) Gross-Neveu model in arbitrary dimensions, Int. J. Mod. Phys. A 9 (1994) 567 [hep-th/9306106] [INSPIRE].
S. Chandrasekharan and A. Li, Quantum critical behavior in three dimensional lattice Gross-Neveu models, Phys. Rev. D 88 (2013) 021701 [arXiv:1304.7761] [INSPIRE].
Z.-X. Li, Y.-F. Jiang and H. Yao, Fermion-sign-free Majarana-quantum-Monte-Carlo studies of quantum critical phenomena of Dirac fermions in two dimensions, New J. Phys. 17 (2015) 085003 [arXiv:1411.7383] [INSPIRE].
S. Hesselmann and S. Wessel, Thermal Ising transitions in the vicinity of two-dimensional quantum critical points, Phys. Rev. B 93 (2016) 155157 [arXiv:1602.02096] [INSPIRE].
E. Huffman and S. Chandrasekharan, Fermion bag approach to Hamiltonian lattice field theories in continuous time, Phys. Rev. D 96 (2017) 114502 [arXiv:1709.03578] [INSPIRE].
I. F. Herbut, Interactions and phase transitions on graphene’s honeycomb lattice, Phys. Rev. Lett. 97 (2006) 146401 [cond-mat/0606195] [INSPIRE].
C. Cordova, T. T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP 03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
D. Bashkirov, Bootstrapping the \( \mathcal{N} \) = 1 SCFT in three dimensions, arXiv:1310.8255 [INSPIRE].
L. Iliesiu, F. Kos, D. Poland, S. S. Pufu, D. Simmons-Duffin and R. Yacoby, Bootstrapping 3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
J.-H. Park, Superconformal symmetry in three-dimensions, J. Math. Phys. 41 (2000) 7129 [hep-th/9910199] [INSPIRE].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
J. Rong and N. Su, Bootstrapping minimal \( \mathcal{N} \) = 1 superconformal field theory in three dimensions, arXiv:1807.04434 [INSPIRE].
A. Atanasov, A. Hillman and D. Poland, Bootstrapping the Minimal 3D SCFT, JHEP 11 (2018) 140 [arXiv:1807.05702] [INSPIRE].
L. Fei, S. Giombi, I. R. Klebanov and G. Tarnopolsky, Yukawa CFTs and Emergent Supersymmetry, PTEP 2016 (2016) 12C105 [arXiv:1607.05316] [INSPIRE].
J. A. Gracey, Computation of critical exponent η at O(1/N3) in the four Fermi model in arbitrary dimensions, Int. J. Mod. Phys. A 9 (1994) 727 [hep-th/9306107] [INSPIRE].
L. Iliesiu, F. Kos, D. Poland, S. S. Pufu and D. Simmons-Duffin, Bootstrapping 3D Fermions with Global Symmetries, JHEP 01 (2018) 036 [arXiv:1705.03484] [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M. F. Paulos, Bootstrapping the Three-Dimensional Supersymmetric Ising Model, Phys. Rev. Lett. 115 (2015) 051601 [arXiv:1502.04124] [INSPIRE].
S. M. Chester, J. Lee, S. S. Pufu and R. Yacoby, The \( \mathcal{N} \) = 8 superconformal bootstrap in three dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE].
M. F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1807.04434
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Rong, J., Su, N. Bootstrapping the minimal \( \mathcal{N} \) = 1 superconformal field theory in three dimensions. J. High Energ. Phys. 2021, 154 (2021). https://doi.org/10.1007/JHEP06(2021)154
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2021)154