The one-dimensional long-range ferromagnetic Ising model with a periodic external field

We consider the one-dimensional ferromagnetic Ising model with very long-range interaction under a periodic, biased and weak external field and prove that at sufficiently low temperatures the model has a unique limiting Gibbs state.

► The ferromagnetic 1D Ising model with a periodic external field is investigated.
► The low temperature Gibbs state is unique under a weak and biased external field.
► Medium-valued external fields may lead to non-uniqueness of Gibbs states.