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Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 476–498 | Cite as

Real Space Migdal–Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment

  • Maria Chiara Angelini
  • Giulio Biroli
Article

Abstract

In this manuscript, in honour of L. Kadanoff, we present recent progress obtained in the description of finite dimensional glassy systems thanks to the Migdal–Kadanoff renormalisation group (MK-RG). We provide a critical assessment of the method, in particular discuss its limitation in describing situations in which an infinite number of pure states might be present, and analyse the MK-RG flow in the limit of infinite dimensions. MK-RG predicts that the spin-glass transition in a field and the glass transition are governed by zero-temperature fixed points of the renormalization group flow. This implies a typical energy scale that grows, approaching the transition, as a power of the correlation length, thus leading to enormously large time-scales as expected from experiments and simulations. These fixed points exist only in dimensions larger than \(d_L>3\) but they nevertheless influence the RG flow below it, in particular in three dimensions. MK-RG thus predicts a similar behavior for spin-glasses in a field and models of glasses and relates it to the presence of avoided critical points.

Keywords

Renormalization group Glassy systems Spin-Glasses 

Mathematics Subject Classification

82Bxx 

Notes

Acknowledgements

We thank C. Cammarota, M. Moore, G. Tarjus, P. Urbani for discussions. We acknowledge support from the ERC Grants NPRGGLASS and from the Simons Foundation (N. 454935, Giulio Biroli).

References

  1. 1.
    Mezard, M., Parisi, G., Virasoro, M.A. (eds.): Spin Glass Theory and Beyond. World Scientific Press, Singapore (1987)Google Scholar
  2. 2.
    Fisher, D.S., Huse, D.A.: Absence of many states in realistic spin glasses. J. Phys. A 20, L1005 (1987)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Fisher, D.S., Huse, D.A.: Equilibrium behavior of the spin-glass ordered phase. Phys. Rev. B 38, 386 (1988)ADSCrossRefGoogle Scholar
  4. 4.
    Bray, A.J., Moore, M.A.: Nonanalytic magnetic field dependence of the magnetisation in spin glasses. J. Phys. C 17, L463 (1984)ADSCrossRefGoogle Scholar
  5. 5.
    Bray, A.J., Moore, M.A.: Scaling theory of the ordered phase of spin glasses. In: van Hemmen, J.L., Morgenstern, I. (eds.) Heidelberg Colloquium on Glassy Dynamics. Lecture Notes in Physics, vol. 275. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  6. 6.
    Kirkpatrick, T.R., Thirumalai, D., Wolynes, P.G.: Scaling concepts for the dynamics of viscous liquids near an ideal glassy state. Phys. Rev. A 40, 1045 (1989)ADSCrossRefGoogle Scholar
  7. 7.
    Wolynes, P.G., Lubchenko, V. (eds.): Structural Glasses and Super-Cooled Liquids. Wiley, Hoboken (2012)Google Scholar
  8. 8.
    Chandler, D., Garrahan, J.P.: Dynamics on the way to forming glass: bubbles in space-time. Annu. Rev. Phys. Chem. 61, 191–217 (2010)CrossRefGoogle Scholar
  9. 9.
    Parisi, G.: A sequence of approximated solutions to the SK model for spin glasses. J. Phys. A 13, 115–121 (1980)ADSCrossRefGoogle Scholar
  10. 10.
    Parisi, G.: A function on the interval 0–1. J. Phys. A 13, 1101–1112 (1980)ADSCrossRefGoogle Scholar
  11. 11.
    Kadanoff, L.P.: Variational principles and approximate renormalization group calculations. Phys. Rev. Lett. 34, 1005 (1975)ADSCrossRefGoogle Scholar
  12. 12.
    Tarjus, G., Tissier, M.: Nonperturbative functional renormalization group for random-field models: the way out of dimensional reduction. Phys. Rev. Lett. 93, 267008 (2004)ADSCrossRefGoogle Scholar
  13. 13.
    Tarjus, G., Tissier, M.: Quasi-long-range order, and criticality in random-field models. Phys. Rev. Lett. 96, 087202 (2006)ADSCrossRefGoogle Scholar
  14. 14.
    Tissier, G., Tissier, M.: Field Ising model. Phys. Rev. Lett. 107, 041601 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    Migdal, A.A.: Phase transitions in gauge and spin-lattice systems. Sov. Phys. JETP 42, 743 (1976)ADSGoogle Scholar
  16. 16.
    Berker, A.N., Ostlund, S.: Renormalisation-group calculations of finite systems: order parameter and specific heat for epitaxial ordering. J. Phys. C 12, 4961 (1979)ADSCrossRefGoogle Scholar
  17. 17.
    Cao, M.S., Matcha, J.: Migdal–Kadanoff study of the random-field Ising model. Phys. Rev. B 48, 3177 (1993)ADSCrossRefGoogle Scholar
  18. 18.
    Boettcher, S.: Low-temperature excitations of dilute lattice spin glasses. Europhys. Lett. 67, 453 (2004)ADSCrossRefGoogle Scholar
  19. 19.
    Boettcher, S.: Stiffness exponents for lattice spin glasses in dimensions \(d= 3,\ldots,6\). Eur. Phys. J. B 38, 83 (2004)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Boettcher, S.: Phys. Rev. Lett. 95, 197205 (2005)ADSCrossRefGoogle Scholar
  21. 21.
    Gardner, E.: A spin glass model on a hierarchical lattice. J. Physique 45, 1755 (1984)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Antenucci, F., Crisanti, A., Leuzzi, L.: Renormalization group on hierarchical lattices in finite dimensional disordered Ising and Blume–Emery–Griffiths models. J. Stat. Phys. 155, 909 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45, 79 (1980)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Gross, D.J., Mezard, M.: The simplest spin glass. Nucl. Phys. B 240, 431 (1984)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Franz, S., Parisi, G., Ricci-Tersenghi, F.: Mosaic length and finite interaction-range effects in a one-dimensional random energy model. J. Phys. A 41, 324011 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Caltagirone, F., Ferrari, U., Leuzzi, L., Parisi, G., Rizzo, T.: Ising M–p-spin mean-field model for the structural glass: continuous versus discontinuous transition. Phys. Rev. B 83, 104202 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    Gross, D.J., Kanter, O., Sompolinsky, H.: Mean-field theory of the Potts glass. Phys. Rev. Lett. 55, 304 (1985)ADSCrossRefGoogle Scholar
  28. 28.
    Angelini, M.C., Biroli, G.: Super-Potts glass: a disordered model for glass-forming liquids. Phys. Rev. B 90, 220201(R) (2014)ADSCrossRefGoogle Scholar
  29. 29.
    Angelini, M.C., Biroli, G.: Real space renormalization group theory of disordered models of glasses. arXiv preprint. arXiv:1604.03717 (2016)
  30. 30.
    Marinari, E., Mossa, S., Parisi, G.: Glassy Potts model: a disordered potts model without a ferromagnetic phase. Phys. Rev. B 59, 8401 (1999)ADSCrossRefGoogle Scholar
  31. 31.
    Fernandez, L.A., et al.: Critical properties of the four-state commutative random permutation glassy Potts model in three and four dimensions. Phys. Rev. B 77, 104432 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    Takahashi, T., Hukushima, K.: Evidence of a one-step replica symmetry breaking in a three-dimensional Potts glass model. Phys. Rev. E 91, 020102 (2015)ADSCrossRefGoogle Scholar
  33. 33.
    Drossel, B., Bokil, H., Moore, M.A.: Spin glasses without time-reversal symmetry and the absence of a genuine structural glass transition. Phys. Rev. E 62, 7690 (2000)ADSCrossRefGoogle Scholar
  34. 34.
    Marinari, E., Parisi, G., Ricci-Tersenghi, F., Ruiz-Lorenzo, J.J.: Numerical simulations of spin glass systems. In: Young, A.P. (ed.) Spin Glasses and Random Fields. World Scientific, Singapore (1997)Google Scholar
  35. 35.
    Marinari, E., Parisi, G., Ricci-Tersenghi, F., Ruiz-Lorenzo, J., Zuliani, F.: Replica symmetry breaking in short-range spin glasses: theoretical foundations and numerical evidences. J. Stat. Phys. 98, 973 (2000)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Berthier, L., Young, A.P.: Time and length scales in spin glasses. J. Phys. Condens. Matter 16, S729 (2004)ADSCrossRefGoogle Scholar
  37. 37.
    Janus Collaboration: Nature of the spin-glass phase at experimental length scales. J. Stat. Mech. P06026 (2010)Google Scholar
  38. 38.
    Moore, M.A., Bokil, H., Drossel, B.: Evidence for the droplet picture of spin glasses. Phys. Rev. Lett. 81, 4252 (1998)ADSCrossRefGoogle Scholar
  39. 39.
    Bokil, H., Bray, A.J., Drossel, B., Moore, M.A.: Comment on general method to determine replica symmetry breaking transitions. Phys. Rev. Lett. 82, 5174 (1999)ADSCrossRefGoogle Scholar
  40. 40.
    Bokil, H., Bray, A.J., Drossel, B., Moore, M.A., Bokil, H., et al.: Reply. Phys. Rev. Lett. 82, 5177 (1999)ADSCrossRefGoogle Scholar
  41. 41.
    Drossel, B., Bokil, H., Moore, M.A., Bray, A.J.: The link overlap and finite size effects for the 3D Ising spin glass. Eur. Phys. J. B 13, 369 (2000)ADSzbMATHCrossRefGoogle Scholar
  42. 42.
    de Almeida, J.R.L., Thouless, D.J.: Stability of the Sherrington–Kirkpatrick solution of a spin glass model. J. Phys. A 11, 983 (1978)ADSCrossRefGoogle Scholar
  43. 43.
    Janus Collaboration: Dynamical transition in the \(D=3\) Edwards–Anderson spin glass in an external magnetic field. Phys. Rev. E 89, 032140 (2014)Google Scholar
  44. 44.
    Janus Collaboration: The three-dimensional Ising spin glass in an external magnetic field: the role of the silent majority. J. Stat. Mech. P05014 (2014)Google Scholar
  45. 45.
    Larson, D., Katzgraber, H.G., Moore, M.A., Young, A.P.: Spin glasses in a field: three and four dimensions as seen from one space dimension. Phys. Rev. B 87, 024414 (2013)ADSCrossRefGoogle Scholar
  46. 46.
    Bray, A.J., Roberts, S.A.: Renormalisation-group approach to the spin glass transition in finite magnetic fields. J. Phys. C 13, 5405 (1980)ADSCrossRefGoogle Scholar
  47. 47.
    Moore, A., Bray, A.J.: Disappearance of the de Almeida–Thouless line in six dimensions. Phys. Rev. B 83, 224408 (2011)ADSCrossRefGoogle Scholar
  48. 48.
    Charbonneau, P., Yaida, S.: A nontrivial critical fixed point for replica-symmetry-breaking transitions. Preprint. arXiv:1607.04217 (2016)
  49. 49.
    Angelini, M.C., Biroli, G.: Spin glass in a field: a new zero-temperature fixed point in finite dimensions. Phys. Rev. Lett. 114, 095701 (2015)ADSCrossRefGoogle Scholar
  50. 50.
    Mézard, M., Parisi, G.: The Bethe lattice spin glass revisited. Eur. Phys. J. B 20, 217–233 (2001)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    Bray, A.J., Moore, M.A.: Nonanalytic magnetic field dependence of the magnetisation in spin glasses. J. Phys. C 17, L613 (1984)ADSCrossRefGoogle Scholar
  52. 52.
    Southern, B.W., Young, A.P.: Real space rescaling study of spin glass behaviour in three dimensions. J. Phys. C 10, 2179 (1977)ADSCrossRefGoogle Scholar
  53. 53.
    Bray, A.J., Moore, M.A.: Scaling theory of the random-field Ising model. J. Phys. C 18, L927 (1985)ADSCrossRefGoogle Scholar
  54. 54.
    De Dominicis, C., Giardina, I.: Random Fields and Spin Glasses. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  55. 55.
    Tarjus, G., Kivelson, S.A., Nussinov, Z., Viot, P.: The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessment. J. Phys. Condens. Matter 17, R1143 (2005)ADSCrossRefGoogle Scholar
  56. 56.
    Dyre, J.: Colloquium: the glass transition and elastic models of glass-forming liquids. Rev. Mod. Phys. 78, 953 (2006)ADSCrossRefGoogle Scholar
  57. 57.
    Berthier, L., Biroli, G.: Theoretical perspective on the glass transition and amorphous materials. Rev. Mod. Phys. 83, 587 (2011)ADSCrossRefGoogle Scholar
  58. 58.
    Singh, Y., Stoessel, J.P., Wolynes, P.G.: Hard-sphere glass and the density-functional theory of aperiodic crystals. Phys. Rev. Lett. 54, 1059 (1985)ADSCrossRefGoogle Scholar
  59. 59.
    Kurchan, J., Parisi, G., Zamponi, F.: Exact theory of dense amorphous hard spheres in high dimension I. The free energy. J. Stat. Mech. P10012 (2012)Google Scholar
  60. 60.
    Kurchan, J., Parisi, G., Urbani, P., Zamponi, F.: Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition. J. Phys. Chem. B 117, 12979 (2013)CrossRefGoogle Scholar
  61. 61.
    Charbonneau, P., Kurchan, J., Parisi, G., Urbani, P., Zamponi, F.: Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution. J. Stat. Mech. P10009 (2014)Google Scholar
  62. 62.
    Morone, F., Parisi, G., Ricci-Tersenghi, F.: Large deviations of correlation functions in random magnets. Phys. Rev. B 89, 214202 (2014)ADSCrossRefGoogle Scholar
  63. 63.
    Stevenson, J.D., Walczak, A.M., Hall, R.W., Wolynes, P.G.: Microscopic calculation of the free energy cost for activated transport in glass-forming liquids. J. Chem. Phys. 129, 194505 (2008)ADSCrossRefGoogle Scholar
  64. 64.
    Balog, I., Tarjus, G.: Activated dynamic scaling in the random-field Ising model: a nonperturbative functional renormalization group approach. Phys. Rev. B 91, 214201 (2015)ADSCrossRefGoogle Scholar
  65. 65.
    Moore, M.A., Drossel, B.: p-Spin model in finite dimensions and its relation to structural glasses. Phys. Rev. Lett. 89, 217202 (2002)ADSCrossRefGoogle Scholar
  66. 66.
    Angelini, M.C., Parisi, G., Ricci-Tersenghi, F.: Ensemble renormalization group for disordered systems. Phys. Rev. B 87, 134201 (2013)ADSCrossRefGoogle Scholar
  67. 67.
    Berges, J., Tetradis, N., Wetterich, C.: Non-perturbative renormalization flow in quantum field theory and statistical physics. Phys. Rep. 363, 223 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Kloss, T., Canet, L., Wschebor, N.: Nonperturbative renormalization group for the stationary Kardar–Parisi–Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1, and 3+1 dimensions. Phys. Rev. E 86, 051124 (2012)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Dipartimento di Fisica, Ed. Marconi“Sapienza” Università di RomaRomeItaly
  2. 2.Institut Physique Théorique (IPhT) CEA Saclay, and CNRS URA 2306Gif-Sur-YvetteFrance
  3. 3.Laboratoire de Physique StatistiqueEcole Normale SupérieureParisFrance

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