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

Abstract

The recently elucidated structure of the delafossite YCuO${}_{2.5}$ reveals a Cu-O network with nearly independent $\Delta$ chains having different interactions between the $s=1/2$ spins. Motivated by this result, we study the $\Delta$ chain for various ratios $J_{\mathrm{bb}}/J_{\mathrm{bv}}$ 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 $S_{\mathrm{tot}}=0$ and 1, has a gap only in the interval $0.4874(1)\le J_{\mathrm{bb}}/J_{\mathrm{bv}}\le 1.53(1)$. The gap is dispersionless for $J_{\mathrm{bb}}/J_{\mathrm{bv}}=1$, but has increasing $k$-dependence as $J_{\mathrm{bb}}/J_{\mathrm{bv}}$ moves away from unity, related to the instability of dimers in the ground state.

4 pages, 6 figures (revtex twocolumn)

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