Critical Phenomena in Loop Models by Adam Nahum

By Adam Nahum

When as regards to a continual section transition, many actual structures can usefully be mapped to ensembles of fluctuating loops, which would signify for instance polymer jewelry, or line defects in a lattice magnet, or worldlines of quantum particles.
'Loop versions' 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 homes of line defects in disordered media and to severe phenomena in two-dimensional quantum magnets. the second one half issues two-dimensional loop versions that lie outdoors the normal paradigms: new forms of serious 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 relating to sigma versions on advanced or actual projective area, CP^{n−1} or RP^{n−1} -- from time to time in a 'replica' restrict -- and this thesis is usually an in-depth research of severe behaviour in those box theories.

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97(2); see Ref. [18] for further details. Recent results for a (2+1)-dimensional bilayer SU (3) magnet, which should be in the same universality class, are also consistent with a continuous transition [20]. 534(9)) as was first demonstrated for the loop model on the diamond lattice [3]. A continuous transition at n = 1 is in agreement with naive expectations: a first-order transition would require that the extended and short-loop phases coexisted at pc , and that interfaces between them cost a nonzero surface tension; this seems to be ruled out by the independence of the nodes.

In three dimensions there is a richer variety of possibilities, associated with percolation-like transitions for line defects [28]. The second class contains idealised problems for which interactions may be neglected. Perhaps the most important are Anderson transitions for free fermions in disordered environments. These systems may be modelled by free field theories with random mass terms, gauge fields etc. Finally, there are problems in which both interactions and disorder must be taken into account, but where the role of interactions is more complex than in the dilute magnet mentioned above.

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