Chaotic Behaviour of Renormalisation Flow in a Complex Magnetic Field

Abstract

It is demonstrated that decimation of the one dimensional Ising model, with periodic boundary conditions, results in a non-linear renormalisation transformation for the couplings which can lead to chaotic behaviour when the couplings are complex. The recursion relation for the couplings under decimation is equivalent to the logistic map, or more generally the Mandelbrot map. In particular an imaginary external magnetic field can give chaotic trajectories in the space of couplings. The magnitude of the field must be greater than a minimum value which tends to zero as the critical point T=0 is approached, leading to a gap equation and associated critical exponent which are identical to those of the Lee-Yang edge singularity in one dimension.