On flat bands in the $J_1$-$J_2$-$J_3$ XXZ sawtooth chain

TL;DR

This work addresses flat-band magnons in a generalized J1-J2-J3 XXZ sawtooth chain with Dzyaloshinskii-Moriya interactions and axial anisotropy. By deriving the exact one-magnon spectrum, the authors obtain algebraic flat-band constraints on seven parameters (J1,J2,J3,D1,D2,D3,Δ1), construct the corresponding localized magnon states, and map the DM parameters to Katsura-Nagaosa-Balatsky magnetoelectric variables to explore electric-field driven flat bands. They classify multiple solution families, show that Δ2 and Δ3 do not enter the flat-band constraints, and demonstrate that KNB-type flat bands can arise beyond the previously studied φ=0 or φ=θ cases, including in uniform-coupling regimes via DM tuning; the linear spin-wave theory confirms the equivalence with a tight-binding picture. These results provide a versatile framework for realizing flat-band magnon physics in frustrated spin chains, with implications for magnetization plateaus, localized-magnon states, and magnetoelectric control in both materials and quantum simulators.

Abstract

We consider a generalization of the XXZ model on the sawtooth spin chain with Dzyaloshinskii-Moriya interactions in which all exchange constants (symmetric, antisymmetric and axial anisotropy) are different for the three different bonds of each triangle. We derive and resolve algebraic constraints on the exchange constants ensuring the appearance of a flat band in the one-magnon spectrum. The properties of the corresponding flat magnon bands and localized magnon states are analyzed. We further construct the mapping of the flat-band conditions for the Dzyaloshinskii-Moriya constants onto the Katsura-Nagaosa-Balatsky parameters. Based on the mapping the possibility of the electric field driven flat bands with aid of the magnetoelectric coupling is examined.

Figures (7)

• Figure 1: (Color online) Sawtooth chain with different couplings on each bond of the triangle. The filled circles show the lattice sites occupied by spins. The exchange coupling, the XXZ anisotropy and DM interaction along the basal line (black) are $J_1$, $\Delta_1$ and $D_1$, respectively. Left (red) and right (blue) bonds along the zigzag line feature the parameters $J_2$, $\Delta_2$ and $D_2$ (red) and $J_3$, $\Delta_3$ and $D_3$ (blue), respectively.
• Figure 2: (Color online) Zero-temperature exact diagonalization magnetization plots for $J_1=1, J_2=1, J_3=3, D_1=1, D_2=2, D_3=-1$ and for $\Delta_1=\Delta_2=\Delta_3=1$.
• Figure 3: (Color online) The one-magnon spectrum for $J_1=1, J_2=1, J_3=3, D_1=1, D_2=2, D_3=-1$ and for $\Delta_1=\Delta_2=\Delta_3=1$ with the lower flat band.
• Figure 4: (Color online) Zero-temperature exact diagonalization magnetization plots for $J_1=1, J_2=2, J_3=3, \Delta_1=\Delta_2=\Delta_3=1$ and for the values of DM interactions obtained according to Eq. (\ref{['DDD_1']}), $D_1= D_2=D_3=1$.
• Figure 5: (Color online) Zero-temperature magnetization curves for finite system with uniform exchange couplings, $J_1=J_2=J_3=1$ and $\Delta_1=1$ with the values of DM interaction corresponding to the first line of the Table \ref{['Tab:1']}, $D_1=0, D_2=\sqrt 3, D_3=-\sqrt 3$. DM interactions can be implemented via electric field driven flat band sc_KNB. Note the absence of the plateau at $M=0$.