One-Dimensional Ising Chain with Competing Interactions: Exact Results and Connection with Other Statistical Models

We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interaction J l , and a kth neighbor antiferromagnetic interaction J~. When Jk /J i = -1 /k , there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N + k l)th term in a generalized Fibonacci sequence defined by, F~ ~)= F ~ l + F } ~ . In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearestneighbor ferroand antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2 • oo strip of the square lattice, and (c) "directed" self-avoiding walks on finite lattice strips.