Existence of two distinct valence bond solid states in the dimerized frustrated ferromagnetic J1−J1′−J2 chain

We study the frustrated and dimerized ferromagnetic-antiferromagnetic ${J}₁\text{\ensuremath{-}}{J}₁^{\ensuremath{'}}\text{\ensuremath{-}}{J}₂$ chain using the density-matrix renormalization group method. Based on numerical calculations of the second derivative of energy, spin gap, spin-spin correlations, string order parameter (SOP), and entanglement spectrum (ES), we obtain the ground-state phase diagram for a wide range of ${J}₁^{\ensuremath{'}}/{J}₁$ and ${J}₂/|{J}₁|$ values. This phase diagram reveals a ferromagnetic phase and two distinct valence bond solid (VBS) phases. The first VBS phase, referred to as ${\mathcal{D}}₃$-VBS, is typified by the formation of valence bonds between third-neighbor spin-1/2's, persisting as a continuation from the ${J}₁^{\ensuremath{'}}/{J}₁=1$ limit. Alternatively, the second VBS phase, referred to as mixed-VBS, exhibits a coexistence of both second- and third-neighbor valence bonds, interpreted as a continuation from the ${J}₁^{\ensuremath{'}}/{J}₁=0$ case. Remarkably, both VBS states are identified as being of Haldane type, marked by a finite SOP and twofold ES degeneracy. Unexpectedly, our analysis uncovers a significant enhancement of the valence bond stability at the boundary of the two VBS phases. This study provides an empirical demonstration of a nontrivial quantum phase transition between different topological VBS states in spin-1/2 chains. Moreover, we find that the ground state of the relevant quasi-one-dimensional material ${\mathrm{LiCuSbO}}₄$ is classified as the ${\mathcal{D}}₃$-VBS state. Collectively, these results mark a substantial stride forward in our comprehension of quantum phase transitions and topological states.