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Critical phenomena and spinwaves in fractal dimension

MSC : Pascal Monceau

Laws involving a non integer space dimension appear in a wide range of physical systems (gels, polymers, magnetic domains, percolating structures, electrolytic deposits,…) and biological systems (lungs, neuronal networks,…); geometrical properties of such objects are gathered together under the generic term ‘fractals’. Although the topological properties of fractals are complex, dominated by structural discontinuities and irregular shapes, a hierarchical organization can be brought out from their observation at different scales, the self similarity. The stimulating questions I am interested in involve the fundamental physics field of phase transitions occurring in these objects whose exhibit a multi-scale heterogeneity. These questions are studied by means of Monte Carlo simulations in the framework of the ferromagnetic Ising or Potts models on generic fractal networks called Sierpinski fractals.

If the self similarity is deterministic, it turns out that the second order para-ferromagnetic transitions occurring in such a case are described in the framework of weak universality: a universality class does not only depend on the order parameter dimension, the space dimension, and the interaction range, but also on topological details of the fractal structure.

If the self similarity is random, the critical behavior is described by a new universality class and, although the space dimensionality is not integer, lack of self-averaging properties exhibit some features very close to the ones of a random fixed point associated with a relevant disorder.

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Configurations of spin state computed by Monte-Carlo simulation of the 14-states Potts model (each color is associated to a spin state) on a fractal with a dimension close to 1,89.
Left : High temperature para-magnetic phase.
Right : Critical temperature. Note the occurrence of clusters at different scales constrained by the underlying network, himself discrete scale invariant.


[1] P. Monceau, Phys. Rev. B 74, 094416, (Sept. 2006). Critical behavior of the ferromagnetic q-state Potts model in fractal dimensions : Monte Carlo simulations on Sierpinski and Menger fractal structures.

[2] P. Monceau, Physica A 379, 559 (Jun. 2007). First order phase transitions of the Potts model in fractal dimension.

[3] P. Monceau and J. C. S. Lévy, Phys. Lett. A 374, 1872 (Apr. 2010). Spin waves in deterministic fractals.

[4] P. Monceau, Phys. Rev. E. 84, 051132 (Nov. 2011). Critical behaviour of the Ising model on random fractals.

[5] P. Monceau and J. C. S. Lévy, Physica E. 44, 1697, (Apr 2012). Effects of deterministic and random discrete scale invariance on spin wave spectra.

[6] P. Monceau, Phys. Rev. E. 86, 061123 (Dec. 2012). Effects of random and deterministic discrete scale invariance on the critical behavior of the Potts model.

[7] P. Monceau, in Complexité et désordre ; Eléments de réflexion, Edited by J. C. S. Lévy, Edp sciences, collection Grenoble Sciences (Nov. 2015).

[8] P. Monceau, in Magnetic Structures of 2D and 3D Nanoparticles. Properties and applications, Pan Stanford Publishing (Dec. 2015) . Magnetic properties of nanostructures in non-integer dimensions.

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