A semidefinite program solver for the conformal bootstrap

  • David Simmons-DuffinEmail author
Open Access
Regular Article - Theoretical Physics


We introduce SDPB: an open-source, parallelized, arbitrary-precision semidefinite program solver, designed for the conformal bootstrap. SDPB significantly outperforms less specialized solvers and should enable many new computations. As an example application, we compute a new rigorous high-precision bound on operator dimensions in the 3d Ising CFT, Δ σ = 0.518151(6), Δ ϵ = 1.41264(6).


Conformal and W Symmetry Field Theories in Lower Dimensions 


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.


  1. [1]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    S. Rychkov, Conformal bootstrap in three dimensions?, arXiv:1111.2115 [INSPIRE].
  4. [4]
    S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].ADSGoogle Scholar
  5. [5]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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].ADSCrossRefGoogle Scholar
  7. [7]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    C. Beem, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=4 \) superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    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].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].Google Scholar
  12. [12]
    F. Caracciolo and V.S. Rychkov, Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].ADSGoogle Scholar
  13. [13]
    D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R. Rattazzi, S. Rychkov and A. Vichi, Central charge bounds in 4D conformal field theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].ADSzbMATHGoogle Scholar
  15. [15]
    R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    A. Vichi, Improved bounds for CFTs with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    S. El-Showk et al., Conformal field theories in fractional dimensions, Phys. Rev. Lett. 112 (2014) 141601 [arXiv:1309.5089] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    L.F. Alday and A. Bissi, The superconformal bootstrap for structure constants, JHEP 09 (2014) 144 [arXiv:1310.3757] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    D. Bashkirov, Bootstrapping the \( \mathcal{N}=1 \) SCFT in three dimensions, arXiv:1310.8255 [INSPIRE].
  23. [23]
    M. Berkooz, R. Yacoby and A. Zait, Bounds on \( \mathcal{N}=1 \) superconformal theories with global symmetries, JHEP 08 (2014) 008 [arXiv:1402.6068] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Y. Nakayama and T. Ohtsuki, Approaching the conformal window of O(n) × O(m) symmetric Landau-Ginzburg models using the conformal bootstrap, Phys. Rev. D 89 (2014) 126009 [arXiv:1404.0489] [INSPIRE].ADSGoogle Scholar
  25. [25]
    Y. Nakayama and T. Ohtsuki, Five dimensional O(N)-symmetric CFTs from conformal bootstrap, Phys. Lett. B 734 (2014) 193 [arXiv:1404.5201] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    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].ADSCrossRefGoogle Scholar
  27. [27]
    F. Caracciolo, A.C. Echeverri, B. von Harling and M. Serone, Bounds on OPE coefficients in 4D conformal field theories, JHEP 10 (2014) 020 [arXiv:1406.7845] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    L.F. Alday and A. Bissi, Generalized bootstrap equations for \( \mathcal{N}=4 \) SCFT, JHEP 02 (2015) 101 [arXiv:1404.5864] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    Y. Nakayama and T. Ohtsuki, Bootstrapping phase transitions in QCD and frustrated spin systems, Phys. Rev. D 91 (2015) 021901 [arXiv:1407.6195] [INSPIRE].ADSGoogle Scholar
  30. [30]
    S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, Exact correlators of BPS operators from the 3d superconformal bootstrap, JHEP 03 (2015) 130 [arXiv:1412.0334] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  31. [31]
    J.-B. Bae and S.-J. Rey, Conformal bootstrap approach to O(N) fixed points in five dimensions, arXiv:1412.6549 [INSPIRE].
  32. [32]
    C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=2 \) superconformal bootstrap, arXiv:1412.7541 [INSPIRE].
  33. [33]
    S.M. Chester, S.S. Pufu and R. Yacoby, Bootstrapping O(N) vector models in 4 < d < 6, Phys. Rev. D 91 (2015) 086014 [arXiv:1412.7746] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Dymarsky, On the four-point function of the stress-energy tensors in a CFT, arXiv:1311.4546 [INSPIRE].
  38. [38]
    C. Beem et al., Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    A.L. Fitzpatrick et al., Covariant approaches to superconformal blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, \( \mathcal{N}=1 \) superconformal blocks for general scalar operators, JHEP 08 (2014) 049 [arXiv:1404.5300] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    Wolfram Research, Mathematica, Wolfram Research Inc., Champaign U.S.A. (2014).Google Scholar
  42. [42]
    GNU Linear Programming Kit,
  43. [43]
  44. [44]
    M. Yamashita, K. Fujisawa, M. Fukuda, K. Nakata and M. Nakata, A high-performance software package for semidefinite programs: SDPA 7, Research Report B-463, Dept. of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo Japan (2010).Google Scholar
  45. [45]
    M. Yamashita, K. Fujisawa and M. Kojima, Implementation and evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0), Optim. Method. Softw. 18 (2003) 491.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    M. Nakata, A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD, IEEE Int. Symp. CACSD (2010) 29.Google Scholar
  47. [47]
    Y. Nesterov and A. Nemirovskii, Interior-point polynomial algorithms in convex programming, Society for Industrial and Applied Mathematics (1994) [doi: 10.1137/1.9781611970791].
  48. [48]
    F. Alizadeh, Combinatorial optimization with interior point methods and semidefinite matrices, Ph.D. Thesis, University of Minnesota, Minneapolis U.S.A. (1991).Google Scholar
  49. [49]
    C. Helmberg, F. Rendl, R.J. Vanderbei and H. Wolkowicz, An interior-point method for semidefinite programming, SIAM J. Optim. 6 (1996) 342.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    L. Vandenberghe and S. Boyd, A primal-dual potential reduction method for problems involving matrix inequalities, Math. Program. 69 (1995) 205.MathSciNetzbMATHGoogle Scholar
  51. [51]
    C. Hanselka and M. Schweighofer, Matrix polynomials positive semidefinite on intervals, to appear.Google Scholar
  52. [52]
    M.-D. Choi, T.-Y. Lam and B. Reznick, Real zeros of positive semidefinite forms. I, Math. Z. 171 (1980) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    E.M. Aylward, S.M. Itani and P.A. Parrilo, Explicit SOS decompositions of univariate polynomial matrices and the Kalman-Yakubovich-Popov lemma, IEEE Conf. Decis. Contr. (2007) 5660.Google Scholar
  54. [54]
    M.J. Todd, A study of search directions in primal-dual interior-point methods for semidefinite programming, Optim. Method. Softw. 11 (1999) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    M. Kojima, S. Shindoh and S. Hara, Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices, SIAM J. Optim. 7 (1997) 86.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    R.D. Monteiro, Primal-dual path-following algorithms for semidefinite programming, SIAM J. Optim. 7 (1997) 663.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM J. Optim. 2 (1992) 575.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    Z. Liu and L. Vandenberghe, Low-rank structure in semidefinite programs derived from the KYP lemma, IEEE Decis. Contr. (2007) 5652.Google Scholar
  59. [59]
    M.F. Anjos and S. Burer, On handling free variables in interior-point methods for conic linear optimization, SIAM J. Optim. 18 (2007) 1310.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    K. Kobayashi, K. Nakata and M. Kojima, A conversion of an SDP having free variables into the standard form SDP, Comput. Optim. Appl. 36 (2007) 289.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Std. 45 (1950) 225.MathSciNetCrossRefGoogle Scholar
  62. [62]
    T. Granlund and the GMP Development Team, GNU MP: the GNU Multiple Precision Arithmetic Library, ed. 5.0.5 (2012),
  63. [63]
    M. Nakata, The MPACK (MBLAS/MLAPACK); a multiple precision arithmetic version of BLAS and LAPACK, ed. 0.6.7 (2010),
  64. [64]
    C++ Standards Committee Library Working Group and other contributors, BOOST C++ libraries,
  65. [65]
  66. [66]
    OpenMP Architecture Review Board, OpenMP application program interface, version 3.0 (2008),
  67. [67]
    M. Hasenbusch, Finite size scaling study of lattice models in the three-dimensional Ising universality class, Phys. Rev. B 82 (2010) 174433 [arXiv:1004.4486].ADSCrossRefGoogle Scholar
  68. [68]
    P. Borwein et al., The inverse symbolic calculator (1995),
  69. [69]
    F. Lenz, The ratio of proton and electron masses, Phys. Rev. 82 (1951) 554.ADSCrossRefGoogle Scholar
  70. [70]
    M.F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE].
  71. [71]
    F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) archipelago, arXiv:1504.07997 [INSPIRE].
  72. [72]
    M. Caselle, G. Costagliola and N. Magnoli, Numerical determination of the operator-product-expansion coefficients in the 3D Ising model from off-critical correlators, Phys. Rev. D 91 (2015) 061901 [arXiv:1501.04065] [INSPIRE].ADSGoogle Scholar
  73. [73]
    L. Iliesiu et al., Bootstrapping 3D fermions, to appear.Google Scholar
  74. [74]
    S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].ADSGoogle Scholar
  77. [77]
    M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonUSA

Personalised recommendations