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The spin-$1/2$ Heisenberg chain with a ferromagnetic first-neighbor exchangecoupling $J_1$ and an antiferromagnetic second-neighbor $J_2$ has a Haldanedimer ground state with an extremely small spin gap. Thus, the ground state isreadily altered by perturbations. Here, we investigate the effects of XXZexchange magnetic anisotropy of both the easy-axis and easy-plane types and analternation in $J_1$ on the ground state, the spin gap, and magnetic propertiesof the frustrated ferromagnetic spin-$1/2$ chain. It is found that there aretwo distinct dimerized spin-gap phases, in one of which the spin gap and themagnetic susceptibility are extremely small around the SU(2) symmetric case andin the other they are moderately large far away from the SU(2) symmetric case.A small alternation in the amplitude of $J_1$ rapidly shortens the pitch ofspin correlations towards the four-spin periodicity, as in the limit of$J_1/J_2\to0$. These effects are not sufficient to quantitatively explainoverall experimentally observed magnetic properties in thequasi-one-dimensional spin-gapped magnetoelectric cuprateRb$_2$Cu$_2$Mo$_3$O$_{12}$ that exhibits ferroelectricity stabilized by amagnetic field. Our results are also relevant to Cs$_2$Cu$_2$Mo$_3$O$_{12}$,where the ferromagnetic intrachain and antiferromagnetic interchain order hasrecently been found, in a single chain level. We also reveal the nature ofsymmetry-protected topological phase transitions in the model by mapping ontoeffective spin-1 chain models.

Roles of easy-plane and easy-axis XXZ anisotropy and bond alternation on a frustrated ferromagnetic spin-$1/2$ chain