Abstract
Quenched disorder is very important but notoriously hard. In 1979, Parisi and Sourlas proposed an interesting and powerful conjecture about the infrared fixed points with random field type of disorder: such fixed points should possess an unusual supersymmetry, by which they reduce in two less spatial dimensions to usual non-supersymmetric non- disordered fixed points. This conjecture however is known to fail in some simple cases, but there is no consensus on why this happens. In this paper we give new non-perturbative arguments for dimensional reduction. We recast the problem in the language of Conformal Field Theory (CFT). We then exhibit a map of operators and correlation functions from Parisi-Sourlas supersymmetric CFT in d dimensions to a (d − 2)-dimensional ordinary CFT. The reduced theory is local, i.e. it has a local conserved stress tensor operator. As required by reduction, we show a perfect match between superconformal blocks and the usual conformal blocks in two dimensions lower. This also leads to a new relation between conformal blocks across dimensions. This paper concerns the second half of the Parisi-Sourlas conjecture, while the first half (existence of a supersymmetric fixed point) will be examined in a companion work.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics, Cambridge University Press (1996).
Y. Imry and S.-k. Ma, Random-Field Instability of the Ordered State of Continuous Symmetry, Phys. Rev. Lett.35 (1975) 1399 [INSPIRE].
A. Aharony, Y. Imry and S.K. Ma, Lowering of Dimensionality in Phase Transitions with Random Fields, Phys. Rev. Lett.37 (1976) 1364 [INSPIRE].
G. Parisi and N. Sourlas, Random Magnetic Fields, Supersymmetry and Negative Dimensions, Phys. Rev. Lett.43 (1979) 744 [INSPIRE].
G. Parisi and N. Sourlas, Critical Behavior of Branched Polymers and the Lee-Yang Edge Singularity, Phys. Rev. Lett.46 (1981) 871 [INSPIRE].
D.C. Brydges and J.Z. Imbrie, Branched polymers and dimensional reduction, Annals Math.158 (2003) 1019 [math-ph/0107005].
J. Cardy, Lecture on Branched Polymers and Dimensional Reduction, cond-mat/0302495.
J.Z. Imbrie, Lower Critical Dimension of the Random Field Ising Model, Phys. Rev. Lett.53 (1984) 1747 [INSPIRE].
J.Z. Imbrie, The Ground State of the Three-dimensional Random Field Ising Model, Commun. Math. Phys.98 (1985) 145 [INSPIRE].
N.G. Fytas, V. Martin-Mayor, M. Picco and N. Sourlas, Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions, Phys. Rev. Lett.116 (2016) 227201 [arXiv:1605.05072].
N.G. Fytas, V. Martin-Mayor, M. Picco and N. Sourlas, Restoration of Dimensional Reduction in the Random-Field Ising Model at Five Dimensions, Phys. Rev.E 95 (2017) 042117 [arXiv:1612.06156] [INSPIRE].
N.G. Fytas, V. Martín-Mayor, G. Parisi, M. Picco and N. Sourlas, Evidence for Supersymmetry in the Random-Field Ising Model at D = 5, Phys. Rev. Lett.122 (2019) 240603 [arXiv:1901.08473] [INSPIRE].
A. Kaviraj, S. Rychkov and E. Trevisani, Random field Ising model and Parisi-Sourlas supersymmetry II. Renormalization Group, work in progress.
G. Parisi and N. Sourlas, Supersymmetric Field Theories and Stochastic Differential Equations, Nucl. Phys.B 206 (1982) 321 [INSPIRE].
B. Mcclain, A. Niemi, C. Taylor and L.C.R. Wijewardhana, Superspace, dimensional reduction, and stochastic quantization, Nucl. Phys.B 217 (1983) 430 [INSPIRE].
T. Castellani and A. Cavagna, Spin-glass theory for pedestrians, J. Stat. Mech.2005 (2005) P05012 [cond-mat/0505032].
J.L. Cardy, Nonperturbative aspects of supersymmetry in statistical mechanics, PhysicaD 15 (1985) 123.
J.L. Cardy and A.J. Mckane, Field theoretical approach to the study of Yang-Lee and Griffiths singularities in the randomly diluted Ising model, Nucl. Phys.B 257 (1985) 383 [INSPIRE].
J.L. Cardy, Nonperturbative effects in a scalar supersymmetric THEORY, Phys. Lett.125B (1983) 470 [INSPIRE].
A. Klein and J.F. Perez, Supersymmetry and dimensional reduction: a nonperturbative proof, Phys. Lett.125B (1983) 473 [INSPIRE].
A. Klein, L.J. Landau and J.F. Perez, Supersymmetry and the Parisi-Sourlas dimensional reduction: a rigorous proof, Commun. Math. Phys.94 (1984) 459 [INSPIRE].
O.V. Zaboronsky, Dimensional reduction in supersymmetric field theories, hep-th/9611157 [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].
S. Hikami, Conformal Bootstrap Analysis for Single and Branched Polymers, PTEP2018 (2018) 123I01 [arXiv:1708.03072] [INSPIRE].
S. Hikami, Dimensional Reduction by Conformal Bootstrap, PTEP2019 (2019) 083A03 [arXiv:1801.09052] [INSPIRE].
V. Hussin and L.M. Nieto, From Super Lie Algebras to Supergroups: Matrix Realizations and the Factorisation Problem, in Noncompact Lie Groups and Some of Their Applications, E.A. Tanner and R. Wilson, eds., pp. 367–372, Springer Netherlands, Dordrecht (1994) [DOI].
V.G. Kac, Lie Superalgebras, Adv. Math.26 (1977) 8 [INSPIRE].
K. Coulembier, The orthosymplectic supergroup in harmonic analysis, J. Lie Theory23 (2013) 55 [arXiv:1202.0668].
P.H. Dondi and P.D. Jarvis, Diagram and Superfield Techniques in the Classical Superalgebras, J. Phys.A 14 (1981) 547 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP07 (2016) 018 [arXiv:1603.05551] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal field theory, JHEP08 (2019) 066 [arXiv:1807.02522] [INSPIRE].
R.C. King, Branching Rules for Classical Lie Groups Using Tensor and Spinor Methods, J. Phys.A 8 (1975) 429 [INSPIRE].
R.J. Farmer and P.D. Jarvis, Representations of Orthosymplectic Superalgebras. 2. Young Diagrams and Weight Space Techniques, J. Phys.A 17 (1984) 2365 [INSPIRE].
A. Kupiainen and A. Niemi, Dimensional Reduction of Symmetries, Phys. Lett.130B (1983) 380 [INSPIRE].
E. Witten, Notes On Supermanifolds and Integration, arXiv:1209.2199 [INSPIRE].
J. Polchinski, Scale and Conformal Invariance in Quantum Field Theory, Nucl. Phys.B 303 (1988) 226 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
M. Hogervorst, Dimensional Reduction for Conformal Blocks, JHEP09 (2016) 017 [arXiv:1604.08913] [INSPIRE].
M. Billò, V. Gon¸calves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
J.L. Cardy, Anisotropic Corrections to Correlation Functions in Finite Size Systems, Nucl. Phys.B 290 (1987) 355 [INSPIRE].
D. Simmons-Duffin, The Conformal Bootstrap, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., 1–26 June 2015, pp. 1–74 (2017) [DOI] [arXiv:1602.07982] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks, JHEP09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N ) vector models, JHEP06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev.D 87 (2013) 106004 [arXiv:1303.1111] [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, Phys. Rev.D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal Limit for Conformal Blocks in d Dimensions, JHEP08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
H. Osborn. personal communication.
Z. Komargodski and D. Simmons-Duffin, The Random-Bond Ising Model in 2.01 and 3 Dimensions, J. Phys.A 50 (2017) 154001 [arXiv:1603.04444] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
P. Arvidsson, Manifest superconformal covariance in six-dimensional (2, 0) theory, JHEP03 (2006) 076 [hep-th/0602193] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys.336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \)symmetry in six dimensions, JHEP05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th edition, Oxford University Press (2002).
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1912.01617
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Kaviraj, A., Rychkov, S. & Trevisani, E. Random Field Ising Model and Parisi-Sourlas supersymmetry. Part I. Supersymmetric CFT. J. High Energ. Phys. 2020, 90 (2020). https://doi.org/10.1007/JHEP04(2020)090
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2020)090