Longitudinal magnons in large-$S$ easy-axis magnets

TL;DR

The paper develops and compares two analytic frameworks to understand longitudinal magnons with $S^z=\pm 2S$ in large-$S$ easy-axis magnets: (i) a strong-coupling mapping to an effective spin-$1/2$ XXZ model and (ii) a multiboson spin-wave theory that treats multipolar and dipolar excitations on equal footing. For $S=1$ antiferromagnets and ferromagnets, it provides quantitative predictions for the $L$-magnon dispersion, its gap, and its stability, including lifetimes when the mode enters the two-magnon continuum, and it derives exact two-particle results for the FM case to benchmark the approaches. The work demonstrates that longitudinal magnons can be coherent and long-lived in a broad $J/D$ range, elucidating the interplay between anisotropy, bound-state formation, and decay channels, with implications for experimental observations in FeI$_2$, FePS$_3$, FePSe$_3$, and related materials. By detailing how cubic and quartic interactions shape the spectra, lifetimes, and bound-state structure, the study provides a systematic framework to explore high-rank spin-tensor order phenomena and multipolar excitations in 2D quantum magnets.

Abstract

Longitudinal magnons are a distinct type of multipolar excitations in magnetic materials with large spins $S\ge 1$ and strong easy-axis anisotropy. These excitations have angular momentum $S^z = \pm 2S$ and can be viewed as a propagating full spin reversal. We study longitudinal magnons for the nearest-neighbor Heisenberg ferromagnet and antiferromagnet on a square lattice with large single-ion anisotropy. In the strong-coupling limit, we derive an effective spin-1/2 model including two leading contributions in $J/D$. The effective model provides a simple description of the longitudinal magnon dynamics. For $S=1$, we compare results from several theoretical approaches that include the effective spin-1/2 model, the linked-cluster expansion, the multiboson spin-wave theory, and, for a ferromagnet, an exact two-particle solution. Among these approaches, the multiboson spin-wave theory provides the decay rate of longitudinal magnons and describes evolution of the excitation spectra from strong to weak anisotropy.

Figures (5)

• Figure 1: Schematic representation of the magnetic structure (a) and low-lying excitations (b-f) in the spin-$S$ easy-axis ferromagnet. The top row shows a chart for different quantum states of a single spin.
• Figure 2: Energy gaps for $T$-magnons (a) and $L$-magnons (b) for the $S=1$ square-lattice antiferromagnet as a function of $J/D$. Legend abbreviations are as follows: HA denotes the harmonic approximation, EH is for the effective-Hamiltonian approach, LC-4 is for the fourth-order linked-cluster data, LC-12 is for the twelfth-order linked-cluster results of Ref. Oitmaa08, and MB is for the multiboson theory including nonlinear quantum corrections. The lightly-shaded region in (b) indicates the two-magnon continuum.
• Figure 3: The longitudinal magnon spectra computed by different methods for the antiferromagnet with $J/D=1/5$ (a), 1/2 (b), and 1 (c). Legend abbreviations are the same as in Fig. \ref{['Gaps']}. Lightly shadowed regions denote the two-magnon continuum. Panel (c) includes the $L$-magnon decay rate $\Gamma_\mathbf{k}$.
• Figure 4: Number of bound states for different total momenta $\bf k$ for the easy-axis ferromagnet on a square lattice. The horizontal line represents a path in the Brillouin zone. The vertical line spans a range of $|J|/D$ values. The number of bound states varies for each of the shaded regions and is indicated by $N_B$. The inset zooms in on the region close to $\mathbf{k}=(\pi,0)$ with $\alpha = 0.986$
• Figure 5: The two-magnon spectrum for the $S=1$ square-lattice ferromagnet. Legend abbreviations are SIBS for the single-ion bound state ($L$-magnon), EBS for the exchange bound state, both from the exact solution, EH for the effective Hamiltonian approach and MB for the multiboson spin-wave theory. Panels (b) and (c) include the $L$-magnon decay rate $\Gamma_\mathbf{k}$.