Haldane and dimer phases in a frustrated spin chain: an exact groundstate and associated topological phase transition.

A Heisenberg spin-s chain with alternating ferromagnetic (FM) ([Formula: see text]) and antiferromagnetic ([Formula: see text]) nearest-neighbor (NN) interactions, exhibits the dimer and spin-2s Haldane phases in the limits [Formula: see text] and [Formula: see text] respectively. These two phases are understood to be topologically equivalent. Induction of the frustration through the next NN FM interaction ([Formula: see text]) produces a very rich quantum phase diagram. With frustration, the whole phase diagram is divided into a FM and a nonmagnetic (NM) phase. For s = 1/2, the full NM phase is seen to be of Haldane-dimer type, but for s > 1/2, a spiral phase comes between the FM and the Haldane-dimer phases. The study of a suitably defined string-order parameter and spin-gap at the phase boundary indicates that the Haldane-dimer and spiral phases have different topological characters. We also find that, along the [Formula: see text] line in the NM phase, an NN dimer state is the exact groundstate, provided [Formula: see text] where κ ⩽ s + h for applied magnetic field h. Without magnetic field, the position of J