Abstract
We use the RG framework set up in [1] to explore the ϕ3 theory with a random field interaction. According to the Parisi-Sourlas conjecture this theory admits a fixed point with emergent supersymmetry which is related to the pure Lee-Yang CFT in two less dimensions. We study the model using replica trick and Cardy variables in d = 8 − ϵ where the RG flow is perturbative. Allowed perturbations are singlets under the Sn symmetry that permutes the n replicas. These are decomposed into operators with different scaling dimensions: the lowest dimensional part, ‘leader’, controls the RG flow in the IR; the other operators, ‘followers’, can be neglected. The leaders are classified into: susy-writable, susy-null and non-susy-writable according to their mixing properties. We construct low lying leaders and compute the anomalous dimensions of a number of them. We argue that there is no operator that can destabilize the SUSY RG flow in d ≤ 8. This agrees with the well known numerical result for critical exponents of Branched Polymers (which are in the same universality class as the random field ϕ3 model) that match the ones of the pure Lee-Yang fixed point according to dimensional reduction in all 2 ≤ d ≤ 8. Hence this is a second strong check of the RG framework that was previously shown to correctly predict loss of dimensional reduction in random field Ising model.
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A. Kaviraj, S. Rychkov and E. Trevisani, Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group, JHEP 03 (2021) 219 [ar**v:2009.10087] [INSPIRE].
A. Kaviraj, S. Rychkov and E. Trevisani, Parisi-Sourlas Supersymmetry in Random Field Models, Phys. Rev. Lett. 129 (2022) 045701 [ar**v:2112.06942] [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].
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 [ar**v: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 [ar**v:1901.08473] [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 [ar**v:1605.05072].
A. Kaviraj, S. Rychkov and E. Trevisani, Random Field Ising Model and Parisi-Sourlas supersymmetry. Part I. Supersymmetric CFT, JHEP 04 (2020) 090 [ar**v:1912.01617] [INSPIRE].
J.L. Cardy, Nonperturbative aspects of supersymmetry in statistical mechanics, Physica D 15 (1985) 123 .
C.-N. Yang and T.D. Lee, Statistical theory of equations of state and phase transitions. I. Theory of condensation, Phys. Rev. 87 (1952) 404 [INSPIRE].
T.D. Lee and C.-N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87 (1952) 410 [INSPIRE].
M.E. Fisher, Yang-Lee Edge Singularity and ϕ3 Field Theory, Phys. Rev. Lett. 40 (1978) 1610 [INSPIRE].
J.L. Cardy, Conformal Invariance and the Yang-lee Edge Singularity in Two-dimensions, Phys. Rev. Lett. 54 (1985) 1354 [INSPIRE].
D.S. Gaunt, The critical dimension for lattice animals, J. Phys. A 13 (1980) L97.
S. Redner, Mean end-to-end distance of branched polymers, J. Phys. A 12 (1979) L239.
T.C. Lubensky and J. Isaacson, Statistics of lattice animals and dilute branched polymers, Phys. Rev. A 20 (1979) 2130 [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.
F. Wegner, Supermathematics and its Applications in Statistical Physics: Grassmann Variables and the Method of Supersymmetry, Springer, Berlin, Germany (2016) [DOI] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP 03 (2019) 163 [ar**v:1612.00809] [INSPIRE].
M. Srednicki, Quantum Field Theory, Cambridge University Press, Cambridge, U.K. (2007) [DOI].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Three loop analysis of the critical O(N) models in 6-ε dimensions, Phys. Rev. D 91 (2015) 045011 [ar**v:1411.1099] [INSPIRE].
J.A. Gracey, Four loop renormalization of ϕ3 theory in six dimensions, Phys. Rev. D 92 (2015) 025012 [ar**v:1506.03357] [INSPIRE].
J.A. Gracey, I. Jack and C. Poole, The a-function in six dimensions, JHEP 01 (2016) 174 [ar**v:1507.02174] [INSPIRE].
F.J. Wegner, The Critical State, General Aspects, in 12th School of Modern Physics on Phase Transitions and Critical Phenomena, Ladek Zdroj Poland, 21–24 June 2001 [INSPIRE].
D.E. Feldman, Critical Exponents of the Random-Field O(N) Model, Phys. Rev. Lett. 88 (2002) 177202 [cond-mat/0010012].
S. Hikami, Conformal Bootstrap Analysis for Single and Branched Polymers, PTEP 2018 (2018) 123I01 [ar**v:1708.03072] [INSPIRE].
J. Cardy, Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications, J. Phys. A 46 (2013) 494001 [ar**v:1302.4279] [INSPIRE].
M. Hogervorst, M. Paulos and A. Vichi, The ABC (in any D) of Logarithmic CFT, JHEP 10 (2017) 201 [ar**v:1605.03959] [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [ar**v:1307.3111] [INSPIRE].
J.D. Miller and K. De’Bell, Randomly branched polymers and conformal invariance, hep-th/9211127 [INSPIRE].
E. Trevisani, The Parisi-Sourlas Uplift and Infinitely Many Solvable Models in 4d, work in progress.
S. Franz, G. Parisi and F. Ricci-Tersenghi, Glassy critical points and the random field ising model, J. Stat. Mech. 2013 (2013) L02001.
J.L. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press, Cambridge, U.K. (1996) [DOI].
H.-P. Hsu, W. Nadler and P. Grassberger, Simulations of lattice animals and trees, J. Phys. A 38 (2005) 775.
T.C. Lubensky and J. Isaacson, Field theory for the statistics of branched polymers, gelation, and vulcanization, Phys. Rev. Lett. 41 (1978) 829.
H. Kleinert and V. Schulte-Frohlinde, Critical properties of ϕ4 theories, World Scientific, Singapore (2001) [DOI].
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Kaviraj, A., Trevisani, E. Random field ϕ3 model and Parisi-Sourlas supersymmetry. J. High Energ. Phys. 2022, 290 (2022). https://doi.org/10.1007/JHEP08(2022)290
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DOI: https://doi.org/10.1007/JHEP08(2022)290