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

A Heisenberg spin-schain with alternating ferromagnetic (FM) (-J1F<0) and antiferromagnetic (J1A>0) nearest-neighbor (NN) interactions, exhibits the dimer and spin-2sHaldane 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. Fors= 1/2, the full NM phase is seen to be of Haldane-dimer type, but fors> 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 theexactgroundstate, providedJ1A>JC=κJ1Fwhereκ⩽s+hfor applied magnetic fieldh. Without magnetic field, the position ofJ_Cis on the FM-NM phase boundary whens= 1/2, but fors> 1/2, the location ofJ_Cis on the phase separation line between the Haldane-dimer and spiral phases.