Download Critical Phenomena in Loop Models by Adam Nahum PDF

By Adam Nahum

When just about a continuing section transition, many actual structures can usefully be mapped to ensembles of fluctuating loops, which would symbolize for instance polymer earrings, or line defects in a lattice magnet, or worldlines of quantum particles.
'Loop types' supply a unifying geometric language for difficulties of this kind.
This thesis goals to increase this language in instructions. the 1st a part of the thesis tackles ensembles of loops in 3 dimensions, and relates them to the statistical houses of line defects in disordered media and to severe phenomena in two-dimensional quantum magnets. the second one half issues two-dimensional loop types that lie open air the traditional paradigms: new forms of severe element are chanced on, and new effects given for the common homes of polymer cave in transitions in dimensions.
All of those difficulties are proven to be with regards to sigma types on advanced or genuine projective house, CP^{n−1} or RP^{n−1} -- now and again in a 'replica' restrict -- and this thesis can be an in-depth research of serious behaviour in those box theories.

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D. Coddington, J. Phys. C. D. Mirlin, F. Evers, A. Mildenberger, J. Phys. A: Math. Gen. 36, 3255 (2003) T. A. Fisher, L. Balents, C. Nayak, Phys. Rev. Lett. 81, 4704 (1998) R. Bundschuh, C. Cassanello, D. R. Zirnbauer, Phys. Rev. B 59, 4382 (1999) 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. References 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 23 T. B. A. Fisher, Phys.

15) where the variance of determines λ. 8 Disordered Systems 17 ϕ(x)ϕ(y) = lim ϕ1 (x)ϕ1 (y) , n→0 ϕ(x)ϕ(y) 2 = lim ϕ1 (x)ϕ2 (x)ϕ1 (y)ϕ2 (y) . n→0 The second formula suggests a heuristic interpretation of distinct replicas as separately equilibrated copies of the system, subject to the same disorder realisation. Let us now compare with the ‘replica-like’ field theories for the loop models. To begin with, note that the symmetry group of Eq. 15 is in general the discrete group Sn (permutations) rather than a continuous one.

Q m1 (xm ) .

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