Journal
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL
Volume 37, Issue 25, Pages 6455-6475Publisher
IOP PUBLISHING LTD
DOI: 10.1088/0305-4470/37/25/003
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We calculate equilibrium solutions for Ising spin models on 'small world' lattices, which are constructed by superimposing random and sparse Poissonian graphs with finite average connectivity c onto a one-dimensional ring. The nearest neighbour bonds along the ring are ferromagnetic, whereas those corresponding to the Poissonian graph are allowed to be random. Our models thus generally contain quenched connectivity and bond disorder. Within the replica formalism, calculating the disorder-averaged free energy requires the diagonalization of replicated transfer matrices. In addition to developing the general replica symmetric theory, we derive phase diagrams and calculate effective field distributions for two specific cases: that of uniform sparse long-range bonds (i.e. 'small world' magnets), and that of +/-J random sparse long-range bonds (i.e. 'small world' spin glasses).
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