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 ($-J_1^F<0$) and antiferromagnetic ($J_1^A>0$) nearest-neighbor (NN) interactions, exhibits the Dimer and spin-$2s$ Haldane phases in the limits $J_1^F\mathrm{/}{J}_1^A\to 0$ and $J_1^F\mathrm{/}{J}_1^A\to \mathrm{\infty }$ respectively. These two phases are understood to be topologically equivalent. Induction of the frustration through the next nearest-neighbor ferromagnetic interaction ($-J_2^F<0$) produces a very rich quantum phase diagram. With frustration, the whole phase diagram is divided into a ferromagnetic (FM) and a nonmagnetic (NM) phase. For $s=1\mathrm{/}2$, the full NM phase is seen to be of Haldane-Dimer type, but for $s>1\mathrm{/}2$, 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 $J_2^F=\frac{1}{2}{J}_1^F$ line in the NM phase, an NN dimer state is the {\it exact} groundstate, provided $J_1^A>J_C=\kappa J_1^F$ where $\kappa \le s+h$ for applied magnetic field $h$. Without magnetic field, the position of $J_C$ is on the FM-NM phase boundary when $s=1\mathrm{/}2$, but for $s>1\mathrm{/}2$, the location of $J_C$ is on the phase separation line between the Haldane-Dimer and spiral phases.