Crossover from an incommensurate singlet spiral state with a vanishingly small spin gap to a valence-bond solid state in dimerized frustrated ferromagnetic spin chains

Motivated by the magnetic properties of the spin-chain compounds ${\mathrm{LiCuSbO}}₄\ensuremath{\equiv}{\mathrm{LiSbCuO}}₄$ and ${\mathrm{Rb}}₂{\mathrm{Cu}}₂{\mathrm{Mo}}₃{\mathrm{O}}₁₂$, we study the ground state of the Heisenberg chain with dimerized nearest-neighbor ferromagnetic (FM) (${J}₁,{J}₁^{\ensuremath{'}}<0$) and next-nearest-neighbor antiferromagnetic (${J}₂>0$) couplings. Using the density-matrix renormalization group technique and spin-wave theory, we find a first-order transition between a fully polarized FM and an incommensurate spiral state at $2\ensuremath{\alpha}=\ensuremath{\beta}/(1+\ensuremath{\beta})$, where $\ensuremath{\alpha}$ is the frustration ratio ${J}₂/|{J}₁|$ and $\ensuremath{\beta}$ the degree of dimerization ${J}₁^{\ensuremath{'}}/{J}₁$. In the singlet spiral state the spin-gap is vanishingly small in the vicinity of the FM transition, corresponding to a situation of ${\mathrm{LiCuSbO}}₄$. For larger $\ensuremath{\alpha}$, corresponding to ${\mathrm{Rb}}₂{\mathrm{Cu}}₂{\mathrm{Mo}}₃{\mathrm{O}}₁₂$, and smaller $\ensuremath{\beta}$ there is a crossover from this frustration induced incommensurate state to an Affleck-Lieb-Kennedy-Tasaki-type valence-bond solid state with substantial spin gaps.