Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro

doi:10.1007/s10955-014-1042-7

Title: Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

Journal Article · Thu Jun 26 00:00:00 EDT 2014 · Journal of Statistical Physics

DOI:https://doi.org/10.1007/s10955-014-1042-7· OSTI ID:1599339

El-Showk, Sheer ^[1]; Paulos, Miguel F. ^[2]; Poland, David ^[3]; Rychkov, Slava ^[4]; Simmons-Duffin, David ^[5]; Vichi, Alessandro ^[6]

European Organization for Nuclear Research (CERN), Geneva (Switzerland)
Brown Univ., Providence, RI (United States)
Yale Univ., New Haven, CT (United States)
European Organization for Nuclear Research (CERN), Geneva (Switzerland); École Normale Supérieure, Paris (France); Univ. Pierre et Marie Curie, Paris (France)
Institute for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)

Here, we use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge $c$ in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are even able to precisely reconstruct the first several $\mathbb {Z}_2$-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension $\Delta _\sigma = 0.518154(15)$, and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

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Research Organization:: Institute for Advanced Study, Princeton, NJ (United States); Univ. of California, Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF)

Grant/Contract Number:: SC0009988; PHY11-25915; 05-BLAN-NT09-573739; PITN-GA-2009-237920; FG02-11ER41742; AC02-05CH1123

OSTI ID:: 1599339

Journal Information:: Journal of Statistical Physics, Vol. 157, Issue 4-5; ISSN 0022-4715

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 365 works

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Conformal Bootstrap in the Regge Limit Li, Daliang; Meltzer, David; Poland, David arXiv https://doi.org/10.48550/arxiv.1705.03453	text	January 2017
Bootstrapping 3D Fermions with Global Symmetries Iliesiu, Luca; Kos, Filip; Poland, David arXiv https://doi.org/10.48550/arxiv.1705.03484	text	January 2017
Weight Shifting Operators and Conformal Blocks Karateev, Denis; Kravchuk, Petr; Simmons-Duffin, David arXiv https://doi.org/10.48550/arxiv.1706.07813	text	January 2017
The 3d Stress-Tensor Bootstrap Dymarsky, Anatoly; Kos, Filip; Kravchuk, Petr arXiv https://doi.org/10.48550/arxiv.1708.05718	text	January 2017
Dynamic scaling of topological ordering in classical systems Xu, Na; Castelnovo, Claudio; Melko, Roger G. arXiv https://doi.org/10.48550/arxiv.1711.03557	text	January 2017
Anomalous dimensions of spinning operators from conformal symmetry Gliozzi, Ferdinando arXiv https://doi.org/10.48550/arxiv.1711.05530	text	January 2017
The Conformal Bootstrap: Theory, Numerical Techniques, and Applications Poland, David; Rychkov, Slava; Vichi, Alessandro arXiv https://doi.org/10.48550/arxiv.1805.04405	text	January 2018
Bootstrapping the long-range Ising model in three dimensions Behan, Connor arXiv https://doi.org/10.48550/arxiv.1810.07199	text	January 2018
Bootstrapping Mixed Correlators in Three-Dimensional Cubic Theories Kousvos, Stefanos R.; Stergiou, Andreas arXiv https://doi.org/10.48550/arxiv.1810.10015	text	January 2018
Levy HBT results at NA61/SHINE Porfy, Barnabas arXiv https://doi.org/10.48550/arxiv.1906.06065	text	January 2019
Convergence of Non-Perturbative Approximations to the Renormalization Group Balog, Ivan; Chaté, Hugues; Delamotte, Bertrand arXiv https://doi.org/10.48550/arxiv.1907.01829	text	January 2019
Conformal invariance and vector operators in the $O(N)$ model De Polsi, Gonzalo; Tissier, Matthieu; Wschebor, Nicolás arXiv https://doi.org/10.48550/arxiv.1907.09981	text	January 2019
Superconformal blocks: general theory Burić, Ilija; Schomerus, Volker; Sobko, Evgeny Journal of High Energy Physics, Vol. 2020, Issue 1 https://doi.org/10.1007/jhep01(2020)159	journal	January 2020
Localizing gauge theories on S d Minahan, Joseph A. Journal of High Energy Physics, Vol. 2016, Issue 4 https://doi.org/10.1007/jhep04(2016)152	journal	April 2016
Perturbative 4D conformal field theories and representation theory of diagram algebras de Mello Koch, Robert; Ramgoolam, Sanjaye Journal of High Energy Physics, Vol. 2020, Issue 5 https://doi.org/10.1007/jhep05(2020)020	journal	May 2020
Scalar-fermion analytic bootstrap in 4D Elkhidir, Emtinan; Karateev, Denis Journal of High Energy Physics, Vol. 2019, Issue 6 https://doi.org/10.1007/jhep06(2019)026	journal	June 2019
On Mellin amplitudes in SCFTs with eight supercharges Zhou, Xinan Journal of High Energy Physics, Vol. 2018, Issue 7 https://doi.org/10.1007/jhep07(2018)147	journal	July 2018
Crossing invariant correlation functions at c = 1 from isomonodromic τ functions Gavrylenko, Pavlo; Santachiara, Raoul Journal of High Energy Physics, Vol. 2019, Issue 11 https://doi.org/10.1007/jhep11(2019)119	journal	November 2019
A generalized Nachtmann theorem in CFT Kundu, Sandipan Journal of High Energy Physics, Vol. 2020, Issue 11 https://doi.org/10.1007/jhep11(2020)138	journal	November 2020
Revisiting the dilatation operator of the Wilson–Fisher fixed point Liendo, Pedro Nuclear Physics B, Vol. 920 https://doi.org/10.1016/j.nuclphysb.2017.04.020	journal	July 2017
An AdS/EFT correspondence at large charge Loukas, Orestis; Orlando, Domenico; Reffert, Susanne Nuclear Physics B, Vol. 934 https://doi.org/10.1016/j.nuclphysb.2018.07.020	journal	September 2018
Remarks on the convergence properties of the conformal block expansion Rychkov, Slava; Yvernay, Pierre Physics Letters B, Vol. 753 https://doi.org/10.1016/j.physletb.2016.01.004	journal	February 2016
Describing phase transitions in field theory by self-similar approximants Yukalov, V. I.; Yukalova, E. P. EPJ Web of Conferences, Vol. 204 https://doi.org/10.1051/epjconf/201920402003	journal	January 2019
Upper bound on the mass anomalous dimension in many-flavor gauge theories: a conformal bootstrap approach Iha, Hisashi; Makino, Hiroki; Suzuki, Hiroshi Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 5 https://doi.org/10.1093/ptep/ptw046	journal	May 2016
The ϵ-expansion of the codimension two twist defect from conformal field theory Yamaguchi, Satoshi Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 9 https://doi.org/10.1093/ptep/ptw115	journal	September 2016
Dimensional reduction by conformal bootstrap Hikami, Shinobu Progress of Theoretical and Experimental Physics, Vol. 2019, Issue 8 https://doi.org/10.1093/ptep/ptz081	journal	August 2019
Operator mixing in the ε -expansion: Scheme and evanescent-operator independence Di Pietro, Lorenzo; Stamou, Emmanuel Physical Review D, Vol. 97, Issue 6 https://doi.org/10.1103/physrevd.97.065007	journal	March 2018
Unitarity violation in noninteger dimensional Gross-Neveu-Yukawa model Ji, Yao; Kelly, Michael Physical Review D, Vol. 97, Issue 10 https://doi.org/10.1103/physrevd.97.105004	journal	May 2018
Scaling solutions in the derivative expansion Defenu, N.; Codello, A. Physical Review D, Vol. 98, Issue 1 https://doi.org/10.1103/physrevd.98.016013	journal	July 2018
Lévy Femtoscopy with PHENIX at RHIC Csanad, Mate Preprints.org https://doi.org/10.20944/preprints201712.0019.v1	journal	December 2017
Harmonic Analysis in d-dimensional Superconformal Field Theory Buric, Ilija Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2021-01105	text	January 2021
Emergent black hole physics from conformal field theory thermodynamics Vos, Gideon University of Groningen https://doi.org/10.33612/diss.98791310	dissertation	October 2019
Bounds on OPE Coefficients in 4D Conformal Field Theories Caracciolo, Francesco; Echeverri, Alejandro Castedo; von Harling, Benedict arXiv https://doi.org/10.48550/arxiv.1406.7845	text	January 2014
Conformal Bootstrap Approach to O(N) Fixed Points in Five Dimensions Bae, Jin-Beom; Rey, Soo-Jong arXiv https://doi.org/10.48550/arxiv.1412.6549	preprint	January 2014
The ${\mathcal N}=2$ superconformal bootstrap Beem, Christopher; Lemos, Madalena; Liendo, Pedro arXiv https://doi.org/10.48550/arxiv.1412.7541	text	January 2014
Boundary and Interface CFTs from the Conformal Bootstrap Gliozzi, F.; Liendo, P.; Meineri, M. arXiv https://doi.org/10.48550/arxiv.1502.07217	text	January 2015
Bootstrapping the O(N) Archipelago Kos, Filip; Poland, David; Simmons-Duffin, David arXiv https://doi.org/10.48550/arxiv.1504.07997	text	January 2015
The $(2,0)$ superconformal bootstrap Beem, Christopher; Lemos, Madalena; Rastelli, Leonardo arXiv https://doi.org/10.48550/arxiv.1507.05637	text	January 2015
Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories Qualls, Joshua D. arXiv https://doi.org/10.48550/arxiv.1508.00548	preprint	January 2015
Conformal Invariance in the Long-Range Ising Model Paulos, Miguel F.; Rychkov, Slava; van Rees, Balt C. arXiv https://doi.org/10.48550/arxiv.1509.00008	text	January 2015
Stress-tensor OPE in N=2 Superconformal Theories Liendo, Pedro; Ramirez, Israel; Seo, Jihye arXiv https://doi.org/10.48550/arxiv.1509.00033	text	January 2015
Axiomatic conformal theory in dimensions >2 and AdS/CT correspondence Schwarz, Albert arXiv https://doi.org/10.48550/arxiv.1509.08064	text	January 2015
Functional renormalization group approach to non-collinear magnets Delamotte, B.; Dudka, M.; Mouhanna, D. arXiv https://doi.org/10.48550/arxiv.1510.00169	text	January 2015
Conformal Collider Physics from the Lightcone Bootstrap Li, Daliang; Meltzer, David; Poland, David arXiv https://doi.org/10.48550/arxiv.1511.08025	text	January 2015
3d $\mathcal{N}=2$ minimal SCFTs from Wrapped M5-branes Bae, Jin-Beom; Gang, Dongmin; Lee, Jaehoon arXiv https://doi.org/10.48550/arxiv.1610.09259	text	January 2016
Generalized Wilson-Fisher critical points from the conformal OPE Gliozzi, Ferdinando; Guerrieri, Andrea; Petkou, Anastasios C. arXiv https://doi.org/10.48550/arxiv.1611.10344	text	January 2016
The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT Simmons-Duffin, David arXiv https://doi.org/10.48550/arxiv.1612.08471	text	January 2016
Inverse Bootstrapping Conformal Field Theories Li, Wenliang arXiv https://doi.org/10.48550/arxiv.1706.04054	text	January 2017
Conformal Bootstrap Analysis for Yang-Lee Edge Singularity Hikami, S. arXiv https://doi.org/10.48550/arxiv.1707.04813	text	January 2017
Conformal Bootstrap Analysis for Single and Branched Polymers Hikami, S. arXiv https://doi.org/10.48550/arxiv.1708.03072	text	January 2017
An appetizer to modern developments on the Kardar-Parisi-Zhang universality class Takeuchi, Kazumasa A. arXiv https://doi.org/10.48550/arxiv.1708.06060	text	January 2017
Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey Chang, Chi-Ming; Fluder, Martin; Lin, Ying-Hsuan arXiv https://doi.org/10.48550/arxiv.1710.08418	text	January 2017
New method for the conformal bootstrap with OPE truncations Li, Wenliang arXiv https://doi.org/10.48550/arxiv.1711.09075	preprint	January 2017
Correlations in low-dimensional quantum gases Lang, Guillaume arXiv https://doi.org/10.48550/arxiv.1712.05424	preprint	January 2017
The unitary subsector of generalized minimal models Behan, Connor arXiv https://doi.org/10.48550/arxiv.1712.06622	text	January 2017
OPE inversion in Mellin space Cardona, Carlos arXiv https://doi.org/10.48550/arxiv.1803.05086	preprint	January 2018
Observables in Inhomogeneous Ground States at Large Global Charge Hellerman, Simeon; Kobayashi, Nozomu; Maeda, Shunsuke arXiv https://doi.org/10.48550/arxiv.1804.06495	text	January 2018
Pushing the Limits of Monte Carlo Simulations for the 3d Ising Model Ferrenberg, Alan M.; Xu, Jiahao; Landau, David P. arXiv https://doi.org/10.48550/arxiv.1806.03558	text	January 2018
Conformal Bootstrap Signatures of the Tricritical Ising Universality Class Gowdigere, Chethan N.; Santara, Jagannath arXiv https://doi.org/10.48550/arxiv.1811.11442	text	January 2018
Closed-form expression for cross-channel conformal blocks near the lightcone Li, Wenliang arXiv https://doi.org/10.48550/arxiv.1906.00707	text	January 2019
Introduction to a renormalisation group method Bauerschmidt, Roland; Brydges, David C.; Slade, Gordon arXiv https://doi.org/10.48550/arxiv.1907.05474	text	January 2019
Scaling the semidefinite program solver SDPB Landry, Walter; Simmons-Duffin, David arXiv https://doi.org/10.48550/arxiv.1909.09745	preprint	January 2019
Boundary conformal field theory at the extraordinary transition: The layer susceptibility to $O(\varepsilon)$ Shpot, M. A. arXiv https://doi.org/10.48550/arxiv.1912.03021	text	January 2019
Precise Critical Exponents of the O(N)-Symmetric Quantum field Model using Hypergeometric-Meijer Resummation Shalaby, Abouzeid M. arXiv https://doi.org/10.48550/arxiv.2004.08711	text	January 2020
Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions Qiao, Jiaxin arXiv https://doi.org/10.48550/arxiv.2005.09105	text	January 2020
Bootstrapping Coulomb and Higgs branch operators Gimenez-Grau, Aleix; Liendo, Pedro arXiv https://doi.org/10.48550/arxiv.2006.01847	text	January 2020
An Automated Generation of Bootstrap Equations for Numerical Study of Critical Phenomena Go, Mocho arXiv https://doi.org/10.48550/arxiv.2006.04173	preprint	January 2020
Long-range multi-scalar models at three loops Benedetti, Dario; Gurau, Razvan; Harribey, Sabine arXiv https://doi.org/10.48550/arxiv.2007.04603	text	January 2020
Energy trapped Ising model Amoretti, Andrea; Costagliola, Gianluca; Magnoli, Nicodemo arXiv https://doi.org/10.48550/arxiv.2007.07150	text	January 2020
Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory Elvang, Henriette arXiv https://doi.org/10.48550/arxiv.2007.08436	text	January 2020
Recursive construction of the operator product expansion in curved space Fröb, Markus B. arXiv https://doi.org/10.48550/arxiv.2007.15668	text	January 2020
Bootstrapping boundary-localized interactions Behan, Connor; Di Pietro, Lorenzo; Lauria, Edoardo arXiv https://doi.org/10.48550/arxiv.2009.03336	text	January 2020
Universality in Non-Equilibrium Quantum Systems Berdanier, William arXiv https://doi.org/10.48550/arxiv.2009.05706	preprint	January 2020
blocks_3d: Software for general 3d conformal blocks Erramilli, Rajeev S.; Iliesiu, Luca V.; Kravchuk, Petr arXiv https://doi.org/10.48550/arxiv.2011.01959	text	January 2020
A CFT Distance Conjecture Perlmutter, Eric; Rastelli, Leonardo; Vafa, Cumrun arXiv https://doi.org/10.48550/arxiv.2011.10040	text	January 2020
Factorized lightcone expansion of conformal blocks Li, Wenliang arXiv https://doi.org/10.48550/arxiv.2012.09710	text	January 2020

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