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The spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chainwith ferromagnetic next-nearest-neighbour (NNN) interaction is investigated.The ground state is the Haldane phase for weak NNN interaction, and is theferromagnetic phase for weak antiferromagnetic interaction. We find a series oftopologically distinct spin-gap phases with various magnitudes of edge spinsfor strong NNN interaction. The phase boundaries between these phases aredetermined on the basis of the DMRG calculation with additional spins thatcompensate the edge spins. It is found that each of the exact solutions withshort-range antiferromagnetic correlation on the ferromagnetic-nonmagneticphase boundary is representative of each spin gap phase.

Topological Phases of the Spin-1/2 Ferromagnetic-Antiferromagnetic Alternating Heisenberg Chain with Frustrated Next-Nearest-Neighbour Interaction