  The classical Ising chain is the paradigm for the non-existence of phase
transitions in 1D systems and was solved by Ernst Ising one hundred years ago.
More recently, a decorated two leg Ising ladder has received interest for the
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 to 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.
