  We analyze a one-dimensional spin-string model, in which string oscillators
are linearly coupled to their two nearest neighbors and to Ising spins
representing internal degrees of freedom. String-spin coupling induces a
long-range ferromagnetic interaction among spins that competes with a spin-spin
antiferromagnetic coupling. As a consequence, the complex phase diagram of the
system exhibits different flat rippled and buckled states, with first or second
order transition lines between states. The two-dimensional version of the model
has a similar phase diagram, which has been recently used to explain the
rippled to buckled transition observed in scanning tunnelling microscopy
experiments with suspended graphene sheets. Here we describe in detail the
phase diagram of the simpler one-dimensional model and phase stability using
bifurcation theory. This gives additional insight into the physical mechanisms
underlying the different phases and the behavior observed in experiments.
