Emergence of Long Range Order in the XY Model on Diluted Small World Networks
Part of the
Springer Proceedings in Complexity
book series (SPCOM)
We study the XY model on diluted Small World networks, i.e Small World networks whose number of links scales with the system size N links ∼N γ , 1<γ<2. Starting from the regular lattice topology, we first concentrate on the behaviour varying the dilution parameter γ: for low values, the system does not exhibit a phase transition; while for γ approaching 2 a second order transition of the magnetisation arises since the system is near the HMF regime. Hence γ c =1.5 appears to be a critical value: an energy range is observed in which the magnetisation shows important fluctuations and does not reach the equilibrium state. We then take in account the model on a Small World network: for the latter, we have chosen the Watts-Strogatz model, whose topology is parametrized by the rewiring probability p, 0<p<1. We performed microcanonical simulations of the dynamics and we highlight the presence of a second order phase transition appearing even for very low p and γ, when the topology is still near the regular lattice one. Moreover we observe a dependence of the critical energy ϵ c on the rewiring probability p.
KeywordsSystem Size Small World Small World Network Average Path Length Regular Network
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The authors would like to thank A. Barrat for fruitful discussions and S. De Nigris is grateful to A. Machens for informing her of the ECCS 2012 conference. S. De Nigris is financially supported by DGA/DS/MRIS.
Albert R, Jeong H, Barabasi AL (1999) The diameter of world wide web. Nature 401:130–131
Lago-Fernandez LF, Huerta R, Corbacho F, Siguenza JA (2000) Fast response and temporal coherent oscillations in small-world networks. Phys Rev Lett 84:2758–2761
Dorogovtsev AV, Goltsev SN, Mendes JFF (2008) Critical phenomena in complex networks. Rev Mod Phys 80. doi: 10.1103/RevModPhys.80.1275
Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442
Halsey TC (1985) Topological defects in the fully frustrated xy model and in 3he-a films. J Phys C, Solid State Phys 18:2437
Toner J, Tu Y (1995) Long-range in a two-dimensional dynamical XY model: how birds fly together. Phys Rev Lett 75:4326–4329
McLachlan RI, Atela P (1992) The accuracy of symplectic integrators. Nonlinearity 5:541–562
Ciani A, Ruffo S, Fanelli D (2011) Long-range interactions and diluted networks. Springer, Berlin
Kosterlitz JM, Thouless DJ (1973) Ordering, metastability and phase transitions in two-dimensional systems. J Phys C, Solid State Phys 6:1181–1203
Kim BJ, Hong H, Holme P, Jeon GS, Minnhagen P, Choi MY (2001) XY model in small-world networks. Phys Rev E 64:056135
Medvedyeva K, Holme P, Minnhagen P, Kim BJ (2003) Dynamical critical behaviour of the XY model in small-world networks. Phys Rev E 67:036118
Campa A, Dauxois T, Ruffo S (2009) Statistical mechanics and dynamics of solvable models with long-range interactions. Phys Rep 480:57–159
Leyvraz F, Ruffo S (2002) Ensemble inequivalence in systems with long-range interactions. J Phys A, Math Gen 35:285–294
© Springer International Publishing Switzerland 2013