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

TL;DR

This study investigates the excitation spectrum of the Δ chain model with varying interaction ratios, revealing a gapped spectrum within a specific parameter range and dispersion characteristics depending on the ratio.

Contribution

It provides a detailed analysis of the Δ chain's excitation spectrum for different interaction ratios using exact diagonalization, highlighting the conditions for a gap and dispersion behavior.

Findings

The excitation spectrum is gapped only within a specific ratio range.

The spectrum is dispersionless at equal interactions (ratio=1).

The spectrum develops k-dependence as the ratio deviates from unity.

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 $J_{\rm bb}/J_{\rm bv}$ 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 $S_{\rm tot}=0$ and 1, has a gap only in the interval $0.4874(1) \leq J_{\rm bb}/J_{\rm bv} \leq 1.53(1)$. The gap is dispersionless for $J_{\rm bb}/J_{\rm bv}=1$, 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.