Discrete Scale Invariance and Its Logarithmic Extension
Nuclear Physics B 655 [FS] (2003) 342–352 www.elsevier.com/locate/npe Discrete scale invariance and its logarithmic extension N. Abed-Pour a, A. Aghamohammadi b,d,M.Khorramic,d, M. Reza Rahimi Tabar d,e,f,1 a Department of Physics, IUST, P.O. Box 16844, Tehran, Iran b Department of Physics, Alzahra University, Tehran 19834, Iran c Institute for Advanced Studies in Basic Sciences, P.O. Box 159, Zanjan 45195, Iran d Institute for Applied Physics, P.O. Box 5878, Tehran 15875, Iran e CNRS UMR 6529, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice cedex 4, France f Department of Physics, Sharif University of Technology, P.O. Box 9161, Tehran 11365, Iran Received 28 August 2002; received in revised form 27 January 2003; accepted 4 February 2003 Abstract It is known that discrete scale invariance leads to log-periodic corrections to scaling. We investigate the correlations of a system with discrete scale symmetry, discuss in detail possible extension of this symmetry such as translation and inversion, and find general forms for correlation functions. 2003 Published by Elsevier Science B.V. PACS: 11.10.-z; 11.25.Hf 1. Introduction Log-periodicity is a signature of discrete scale invariance (DSI) [1]. DSI is a symmetry weaker than (continuous) scale invariance. This latter symmetry manifests itself as the invariance of a correlator O(x) as a function of the control parameter x, under the scaling x → eµx for arbitrary µ. This means, there exists a number f(µ)such that O(x) = f(µ)O eµx .
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