• Taiki HagaEmail author
Part of the Springer Theses book series (Springer Theses)


In the first part of this chapter, we provide a brief review for general results concerning phase transitions and critical phenomena in pure systems. We next introduce models for disordered systems and discuss their properties. Especially, we explain the so-called dimensional reduction, which predicts that the critical behavior of disordered systems is identical to that of lower dimensional systems without disorder, as well as its limitation. In the second half of this chapter, we review the dynamics of disordered systems driven out of equilibrium. Finally, we explain the purpose of this study.


Phase transition Disordered systems Dimensional reduction Depinning transition Dynamical reordering transition 


  1. 1.
    Ma S-K (1976) Modern theory of critical phenomena. W. A. Benjamin, IncGoogle Scholar
  2. 2.
    Goldenfeld N (1992) Lectures on phase transitions and the renormalization group. Westview PressGoogle Scholar
  3. 3.
    Wilson KG, Kogut J (1974) The renormalization group and the \(\epsilon \) expansion. Phys Rep 12:75ADSCrossRefGoogle Scholar
  4. 4.
    Zinn-Justin J (1989) Quantum field theory and critical phenomena. Clarendon Press, OxfordzbMATHGoogle Scholar
  5. 5.
    Fishman S, Aharony A (1979) Random field effects in disordered anisotropic antiferromagnets. J Phys C 12:L729ADSCrossRefGoogle Scholar
  6. 6.
    Gennes PG (1984) Liquid-liquid demixing inside a rigid network. Qualitative features. J Phys Chem 88:6469Google Scholar
  7. 7.
    Pitard E, Rosinberg ML, Stell G, Tarjus G (1995) Critical behavior of a fluid in a disordered porous matrix: an ornstein-zernike approach. Phys Rev Lett 74:4361ADSCrossRefGoogle Scholar
  8. 8.
    Villain J (1982) Commensurate-incommensurate transition with frozen impurities. J Physique Lett (France) 43:551CrossRefGoogle Scholar
  9. 9.
    Harris R, Plischke M, Zuckermann MJ (1973) New model for amorphous magnetism. Phys Rev Lett 31:160ADSCrossRefGoogle Scholar
  10. 10.
    Bellini T, Buscaglia M, Chiccoli C, Mantegazza F, Pasini P, Zannoni C (2000) Nematics with quenched disorder: what is left when long range order is disrupted? Phys Rev Lett 85:1008ADSCrossRefGoogle Scholar
  11. 11.
    Bellini T, Buscaglia M, Chiccoli C, Mantegazza F, Pasini P, Zannoni C (2002) Nematics with quenched disorder: how long will it take to heal? Phys Rev Lett 88:245506ADSCrossRefGoogle Scholar
  12. 12.
    Rotunno M, Buscaglia M, Chiccoli C, Mantegazza F, Pasini P, Bellini T, Zannoni C (2005) Nematics with quenched disorder: pinning out the origin of memory. Phys Rev Lett 94:097802ADSCrossRefGoogle Scholar
  13. 13.
    Petridis L, Terentjev EM (2006) Nematic-isotropic transition with quenched disorder. Phys Rev E 74:051707ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Nattermann T (1997) Spin glasses and random fields. In: Young AP (ed). World Scientific, SingaporeGoogle Scholar
  15. 15.
    Ertas D, Kardar M (1994) Critical dynamics of contact line depinning. Phys Rev E 49:R2532ADSCrossRefGoogle Scholar
  16. 16.
    Prevost A, Rolley E, Guthmann C (2002) Dynamics of a helium-4 meniscus on a strongly disordered cesium substrate. Phys Rev B 65:064517ADSCrossRefGoogle Scholar
  17. 17.
    Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, Vinokur VM (1994) Vortices in high-temperature superconductors. Rev Mod Phys 66:1125ADSCrossRefGoogle Scholar
  18. 18.
    Nattermann T, Scheidl S (2000) Vortex-glass phases in type-II superconductors. Adv Phys 49:607ADSCrossRefGoogle Scholar
  19. 19.
    Imry Y, Ma S-K (1975) Random-field instability of the ordered state of continuous symmetry. Phys Rev Lett 35:1399ADSCrossRefGoogle Scholar
  20. 20.
    Aizenman M, Wehr J (1989) Rounding of first-order phase transitions in systems with quenched disorder. Phys Rev Lett 62:2503ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Bricmont J, Kupiainen A (1987) Lower critical dimension for the random-field using model. Phys Rev Lett 59:1829ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Aharony A, Imry Y, Ma S-K (1976) Lowering of dimensionality in phase transitions with random fields. Phys Rev Lett 37:1364ADSCrossRefGoogle Scholar
  23. 23.
    Young AP (1977) On the lowering of dimensionality in phase transitions with random fields. J Phys C 10:L257ADSCrossRefGoogle Scholar
  24. 24.
    Parisi G, Sourlas N (1979) Random magnetic fields, supersymmetry, and negative dimensions. Phys Rev Lett 43:744ADSCrossRefGoogle Scholar
  25. 25.
    Binder K, Young AP (1986) Spin glasses: experimental facts, theoretical concepts, and open questions. Rev Mod Phys 58:801ADSCrossRefGoogle Scholar
  26. 26.
    Mézard M, Parisi G, Virasoro MA (1987) Spin-glasses and beyond. World Scientific, SingaporezbMATHGoogle Scholar
  27. 27.
    Sherrington D, Kirkpatrick S (1975) Solvable model of a spin-glass. Phys Rev Lett 35:1792ADSCrossRefGoogle Scholar
  28. 28.
    De Almeida JRL, Thouless DJ (1978) Stability of the sherrington-kirkpatrick solution of a spin glass model. J Phys A 11:983ADSCrossRefGoogle Scholar
  29. 29.
    Parisi G (1980) The order parameter for spin glasses: a function on the interval 0–1. J Phys A 13:1101ADSCrossRefGoogle Scholar
  30. 30.
    Parisi G (1983) Order parameter for spin-glasses. Phys Rev Lett 50:1946ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    De Dominicis C, Young P (1983) Weighted averages and order parameters for the infinite range Ising spin glass. J Phys A 16:2063ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Mézard M, Parisi G (1991) Replica field theory for random manifolds. J Phys I 1:809Google Scholar
  33. 33.
    Le Doussal P, Wiese KJ (2003) Functional renormalization group at large \(N\) for disordered elastic systems, and relation to replica symmetry breaking. Phys Rev B 68:174202ADSCrossRefGoogle Scholar
  34. 34.
    Grinstein G, Luther A (1976) Application of the renormalization group to phase transitions in disordered systems. Phys Rev B 13:1329ADSCrossRefGoogle Scholar
  35. 35.
    Harris AB (1974) Effect of random defects on the critical behaviour of using models. J Phys C Solid State Phys 7:1671ADSCrossRefGoogle Scholar
  36. 36.
    Ma S-k, Rudnick J (1978) Time-dependent Ginzburg-landau model of the spin-glass phase. Phys Rev Lett 40:589ADSCrossRefGoogle Scholar
  37. 37.
    Dotsenko V, Harris AB, Sherrington D, Stinchcombe RB (1995) Replica symmetry breaking in the critical behavior of the random ferromagnet. J Phys Math Gen 28:3093ADSCrossRefGoogle Scholar
  38. 38.
    Tarjus G, Dotsenko V (2002) Is there a spin-glass phase in the random temperature Ising ferromagnet? J Phys Math Gen 35:1627ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Fisher DS (1985) Sliding charge-density waves as a dynamical critical phenomenon. Phys Rev B 31:1396ADSCrossRefGoogle Scholar
  40. 40.
    Narayan O, Fisher DS (1992) Critical behavior of sliding charge-density waves in 4-\(\epsilon \) dimensions. Phys Rev B 46:11520ADSCrossRefGoogle Scholar
  41. 41.
    Narayan O, Fisher DS (1993) Threshold critical dynamics of driven interfaces in random media. Phys Rev B 48:7030ADSCrossRefGoogle Scholar
  42. 42.
    Chauve P, Giamarchi T, Le Doussal P (2000) Creep and depinning in disordered media. Phys Rev B 62:6241ADSCrossRefGoogle Scholar
  43. 43.
    Le Doussal P, Wiese KJ, Chauve P (2002) Two-loop functional renormalization group theory of the depinning transition. Phys Rev B 66:174201ADSCrossRefGoogle Scholar
  44. 44.
    Tanguy A, Gounelle M, Roux S (1998) From individual to collective pinning: effect of long-range elastic interactions. Phys Rev E 58:1577ADSCrossRefGoogle Scholar
  45. 45.
    Roters L, Hucht A, Lubeck S, Nowak U, Usadel KD (1999) Depinning transition and thermal fluctuations in the random-field Ising model. Phys Rev E 60:5202ADSCrossRefGoogle Scholar
  46. 46.
    Rosso A, Krauth W (2001) Origin of the roughness exponent in elastic strings at the depinning threshold. Phys Rev Lett 87:187002ADSCrossRefGoogle Scholar
  47. 47.
    Rosso A, Krauth W (2002) Roughness at the depinning threshold for a long-range elastic string. Phys Rev E 65:025101(R)ADSCrossRefGoogle Scholar
  48. 48.
    Grüner G (1988) The dynamics of charge-density waves. Rev Mod Phys 60:1129ADSCrossRefGoogle Scholar
  49. 49.
    Fisher DS (1998) Collective transport in random media: from, superconductor to earthquakes. Phys Rep 301:113ADSCrossRefGoogle Scholar
  50. 50.
    Urbach JS, Madison RC, Markert JT (1995) Interface depinning, self-organized criticality, and the barkhausen effect. Phys Rev Lett 75:276ADSCrossRefGoogle Scholar
  51. 51.
    Zapperi S, Cizeau P, Durin G, Stanley HE (1998) Dynamics of a ferromagnetic domain wall: avalanches, depinning transition, and the barkhausen effect. Phys Rev B 58:6353ADSCrossRefGoogle Scholar
  52. 52.
    Kagan YY (2002) Seismic moment distribution revisited: I. Statistical results. Geophys J Int 148:520ADSCrossRefGoogle Scholar
  53. 53.
    Reichhardt C, Olson Reichhardt CJ (2017) Depinning and nonequilibrium dynamic phases of particle assemblies driven over random and ordered substrates: a review. Rep Prog Phys 80:026501ADSCrossRefGoogle Scholar
  54. 54.
    Koshelev AE, Vinokur VM (1994) Dynamic melting of the vortex lattice. Phys Rev Lett 73:3580ADSCrossRefGoogle Scholar
  55. 55.
    Moon K, Scalettar RT, Zimanyi GT (1996) Dynamical phases of driven vortex systems. Phys Rev Lett 77:2778ADSCrossRefGoogle Scholar
  56. 56.
    Ryu S, Hellerqvist M, Doniach S, Kapitulnik A, Stroud D (1996) Dynamical phase transition in a driven disordered vortex lattice. Phys Rev Lett 77:5114ADSCrossRefGoogle Scholar
  57. 57.
    Dominguez D, Gronbech-Jensen N, Bishop AR (1997) First-order melting of a moving vortex lattice: effects of disorder. Phys Rev Lett 78:2644ADSCrossRefGoogle Scholar
  58. 58.
    Giamarchi T, Le Doussal P (1996) Moving glass phase of driven lattices. Phys Rev Lett 76:3408ADSCrossRefGoogle Scholar
  59. 59.
    Le Doussal P, Giamarchi T (1998) Moving glass theory of driven lattices with disorder. Phys Rev B 57:11356ADSCrossRefGoogle Scholar
  60. 60.
    Balents L, Marchetti MC, Radzihovsky L (1998) Nonequilibrium steady states of driven periodic media. Phys Rev B 57:7705ADSCrossRefGoogle Scholar
  61. 61.
    Yaron U, Gammel PL, Huse DA, Kleiman RN, Oglesby CS, Bucher E, Batlogg B, Bishop DJ, Mortensen K, Clausen K, Bolle CA, De La Cruz F (1994) Neutron diffraction studies of flowing and pinned magnetic flux lattices in \(2H{-}\rm {NbSe}_2\). Phys Rev Lett 73:2748ADSCrossRefGoogle Scholar
  62. 62.
    Yaron U, Gammel PL, Huse DA, Kleiman RN, Oglesby CS, Bucher E, Batlogg B, Bishop DJ, Mortensen K, Clausen KN (1995) Structural evidence for a two-step process in the depinning of the superconducting flux-line lattice. Nature 376:753ADSCrossRefGoogle Scholar
  63. 63.
    Pardo F, De La Cruz F, Gammel PL, Bucher E, Bishop DJ (1998) Observation of smetic and moving-bragg-glass phases in flowing vortex lattices. Nature 396:348ADSCrossRefGoogle Scholar
  64. 64.
    Araki T (2012) Dynamic coupling between a multistable defect pattern and flow in nematic liquid crystals confined in a porous medium. Phys Rev Lett 109:257801ADSCrossRefGoogle Scholar
  65. 65.
    Sengupta A, Tkalec U, Ravnik M, Yeomans JM, Bahr C, Herminghaus S (2013) Liquid crystal microfluidics for tunable flow shaping. Phys Rev Lett 110:048303ADSCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of PhysicsKyoto UniversityKyotoJapan

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