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) (-J1F<0) and antiferromagnetic (J1A>0) nearest-neighbor (NN) interactions, exhibits the dimer and spin-2 s Haldane phases in the limitsJ1F/J1A→0andJ1F/J1A→∞respectively. These two phases are understood to be topologically equivalent. Induction of the frustration through the next NN FM interaction (-J2F<0) 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 theJ2F=12J1Fline in the NM phase, an NN dimer state is the exact groundstate, providedJ1A>JC=κJ1Fwhere κ ⩽ s + h for applied magnetic field h . Without magnetic field, the position of J _C is on the FM-NM phase boundary when s = 1/2, but for s > 1/2, the location of J _C is on the phase separation line between the Haldane-dimer and spiral phases.
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