Bifurcation in correlation length of the Ising model on a ‘Toblerone’ lattice

The classical Ising chain is a paradigm for the non-existence of phase transitions in one-dimensional systems and was solved by Ernst Ising 100 years ago. More recently, a decorated two-leg Ising ladder has received interest due to its curious thermodynamics that resemble a phase transition; a sharp peak in the specific heat at low, but finite temperature. We use this model to reveal a bifurcation in the correlation lengths due to a crossing of the sub-leading eigenvalues of the transfer matrix, which results in two distinct length scales necessary to describe the decay of correlations. We discuss this phenomenon in the context of the geometric frustration in the model. We also provide additional results to aid in the understanding of the curious thermodynamics of the model through a study of the magnetic susceptibilities.