Quantum Topological Excitations: from the Sawtooth Lattice to the Heisenberg Chain

The recently elucidated structure of the delafossite YCuO${}_{2.5}$ reveals a Cu-O network with nearly independent $\mathrm{\Delta }$ chains having different interactions between the $s=1\mathrm{/}2$ spins. Motivated by this result, we study the $\mathrm{\Delta }$ chain for various ratios $J_{\mathrm{b}\mathrm{b}}\mathrm{/}{J}_{\mathrm{b}\mathrm{v}}$ of the base-base and base-vertex interactions. By exact diagonalization and extrapolation, we show that the elementary excitation spectrum, which (within numerical error) is the same for total spins $S_{\mathrm{t}\mathrm{o}\mathrm{t}}=0$ and 1, has a gap only in the interval $0.4874(1)\le J_{\mathrm{b}\mathrm{b}}\mathrm{/}{J}_{\mathrm{b}\mathrm{v}}\le 1.53(1)$. The gap is dispersionless for $J_{\mathrm{b}\mathrm{b}}\mathrm{/}{J}_{\mathrm{b}\mathrm{v}}=1$, but has increasing $k$-dependence as $J_{\rm bb}/J_{\rm bv}$ moves away from unity, related to the instability of dimers in the ground state.