Chaotic Behaviour of Renormalisation Flow in a Complex Magnetic Field

TL;DR

This paper shows that in a one-dimensional Ising model with a complex magnetic field, the renormalisation flow can become chaotic, revealing complex dynamics similar to the logistic map and linking to Lee-Yang singularities.

Contribution

It demonstrates that the renormalisation flow of the 1D Ising model with complex couplings can exhibit chaos, connecting magnetic field effects to known singularities.

Findings

Chaotic trajectories occur with sufficiently large imaginary magnetic field.

The chaos is described by the logistic and Mandelbrot maps.

Critical behavior matches Lee-Yang edge singularity.

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.