DOI: 10.1051/jp1:1996160
J. Phys. I France 6 (1996) 327-355

Complex Exponents and Log-Periodic Corrections in Frustrated Systems

H. Saleur¹ and D. Sornette<sup>2

1</sup>  Departement of Physics and Astronomy, University of Southern California, Los Angeles CA 90089-0484
²  Laboratoire de Physique de la Matière Condensée CNRS URA 190, Université des Sciences, B.P. 70, Parc Valrose, 06108 Nice Cedex 2, France

(Received 16 October 1995, received in final form 14 November 1995, accepted 4 December 1995)

Abstract
Recently, it has been observed that rupture processes in highly disordered media and earthquakes exhibit universal log-periodic corrections to scaling. We argue that such corrections should actually be present in a wide class of disordered systems and provide a theoretical framework to handle them.

At the naivest level, a natural explanation for log-periodic corrections is discrete scale invariance, a notion qualitatively similar to the concept of "lacunarity". However in nature, any such structure would be largely perturbed by disorder. We therefore investigate first the effect of disorder on the log-periodic corrections. Remarkably, we find that they are generally robust. We discuss a variety of disorder associated effects, like renormalization of the log-periodic frequencies.

We then propose a general explanation based on the fact that a discrete fractal is actually a fractal with complex dimension, and then that complex critical exponents should generally be expected in the field theories that describe geometrical systems, because the latter are non unitary. We discuss detailed features of non unitary theorie, and present evidence of complex exponents in lattice animals, a simple geometrical generalization of percolation, which can be argued to be associated with rupture.

Finally, we extend our discussion to more general frustrated systems. We reemphasize that the non-unitarity, generated here by the averaging over disorder, can lead to complex exponents, as were actually found earlier in some expansion approaches. More physically, since replica symmetry breaking is described by an ultrametric tree, it may naturally lead to discrete scale invariance, albeit not in real space but in replica space. We then study a dynamical model describing transitions between states in a hierarchical system of barriers modelling the energy landscape in the phase space of meanfield spinglasses, that leads again to log-periodic corrections.

We conclude by mentioning a few physical cases where we think log-periodic corrections should be observable.

© Les Editions de Physique 1996