1D Potts, Yang-Lee Edges and Chaos

Dolan, Brian P. and Johnston, D. A. (2000) 1D Potts, Yang-Lee Edges and Chaos. (Preprint)

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Abstract

It is known that the (exact) renormalization transformations for the one-dimensional Ising model in field can be cast in the form of a logistic map f(x) = λx(1 − x) with λ = 4 and x a function of the Ising couplings K and h. Remarkably, the line bounding the region of chaotic behaviour in x is precisely that defining the Yang-Lee edge singularity in the Ising model. The generalisation of this relation between the edge singularity and chaotic behaviour to other models is an open question. In this paper we show that the one dimensional q-state Potts model for q ≥ 1 also displays such behaviour. A suitable combination of couplings (which reduces to the Ising case for q = 2) can again be used to define an x satisfying f(x) = 4x(1 − x). The Yang-Lee zeroes no longer lie on the unit circle in the complex z = exp(h) plane for q ≠ 2, but their locus is still reproduced by the boundary of the chaotic region in the logistic map.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 13 Jun 2018 13:05
Last Modified: 17 Jul 2018 12:53
URI: http://dair.dias.ie/id/eprint/435

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