One-dimensional Potts model, Lee-Yang edges, and chaos.
It is known that the exact renormalization transformations for the one-dimensional Ising model in a field can be cast in the form of the logistic map f(x)=4x(1-x) with x a function of the Ising couplings K and h. The locus of the Lee-Yang zeros for the one-dimensional Ising model in the K,h plane is given by the Julia set of the logistic map. In this paper we show that the one-dimensional q-state Potts model for q> or =1 also displays such behavior. 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 Lee-Yang zeros no longer lie on the unit circle in the complex z=e(h) plane for q not equal 2, but their locus still maps onto the Julia set of the logistic map.
Dolan
Brian P
BP
Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland.
Johnston
D A
DA