Pinch points and half moons encode Berry curvature

TL;DR

This work establishes a direct link between Berry curvature in topological magnon bands and universal spectral fingerprints—pinch points and half moons—in frustrated magnets. Using a Kagome-lattice spin-1/2 model with anisotropic exchange, Dzyaloshinskii–Moriya interaction, and bond-symmetric exchange, the authors show that a quadratic band touching at the zone center carries Berry curvature $\pm 2\pi$, and that perturbations mixing the intersecting bands open a gap while imprinting finite Berry curvature on the bands. They develop a two-level effective description with a vector $\mathbf{d}(\mathbf{q})$ whose winding yields the curvature, and demonstrate how $S(\mathbf{q},\omega)$ captures these topological features, including the transition from pinch points to gapped, curved bands. The results give concrete, experiment-accessible criteria for inferring topological magnon bands from inelastic neutron scattering in Kagome and related magnets, and they connect to real materials such as Cu(1,3-bdc) and pyrochlores with observed half moons and pinch points. The framework also applies to electron systems on frustrated lattices, suggesting a broad relevance of these spectral fingerprints for topology in quantum matter.

Abstract

"Half moons", distinctive crescent patterns in the dynamical structure factor, have been identified in inelastic neutron scattering experiments for a wide range of frustrated magnets. In an earlier paper [H. Yan et al., Phys. Rev. B 98, 140402(R) (2018)] we have shown how these features are linked to the local constraints realized in classical spin liquids. Here we explore their implication for the topology of magnon bands. The presence of half moons indicates a separation of magnetic degrees of freedom into irrotational and incompressible components. Where bands satisfying these constraints meet, it is at a singular point encoding Berry curvature of $\pm 2π$. Interactions which mix the bands open a gap, resolving the singularity, and leading to bands with finite Berry curvature, accompanied by characteristic changes to half--moon motifs. These results imply that inelastic neutron scattering can, in some cases, be used to make rigorous inference about the topological nature of magnon bands.

Figures (7)

• Figure 1: Systematic modification of pinch--point and half--moon features in a system with topological bands. a) Case with topologically critical bands, considered in Yan2018. Pinch points are inscribed on both flat and dispersing bands, which meet in the zone center. Cuts through the dispersing band at fixed energy reveal crescent--shaped half--moon features. The band--touching point is singular, and has a localized Berry curvature (BC) of $\pm 2\pi$. b) The introduction of interactions which mix states in the two bands opens a gap, eliminating the singular correlations at the band--touching point, and endowing each band with BC of $\pm 2\pi$.
• Figure 2: Kagome lattice, showing geometry of corner--sharing triangles, and convention for labeling sites (A,B,C) within 3--site primitive unit cell. Inset: Black circulating arrows indicate the sense in which bonds are counted for Dzyaloshinskii--Moriya (DM) interactions.
• Figure 3: Relationship between singular features in the dynamical structure factor $S({\bf q}, \omega)$, and Berry curvature in Kagome lattice magnets with and without Dzyaloshinskii--Moriya (DM) interactions. (a) Spin--wave dispersion of Heisenberg antiferromagnet (HAF) on a Kagome lattice in high magnetic field. The colorscale shows how each band contributes to $S({\bf q}, \omega)$. Pinch points, singular for ${\bf q} \to \Gamma$, are inscribed on the flat band at $\omega = 2$, and the dispersing band which touches it. (b) Berry curvature associated with spin--wave bands. Curvature is localised at the topologically--critical band--touching points, shown with red circles, and is zero elsewhere. (c) Dispersion in the presence of finite DM interaction, $D_z =0.1$, showing how the mixing of states between bands opens gaps at band--touching points. (d) Berry curvature generated by DM interaction, through mixing of states. Integrated accross the Brillouin zone (BZ), this leads to bands with the Chern numbers $C = 1,~0,~-1$. (e) Dynamical structure factor $S({\bf q}, \omega)$ at $\omega = 2.0$, for $D_z =0$, showing pinch points inscribed on flat band. Red hexagon denotes the BZ considered in (a)--(d). (f) Equivalent results for $\omega = 2.5$, showing half moons associated with dispersing band. [cf. Fig. \ref{['fig:summary.trivial']}]. (g) Structure factor for $\omega = 1.85$, $D_z =0.1$, showing how the mixing of states between bands eliminates the pinch--point singularity for ${\bf q} \to \Gamma$. (h) Equivalent results for $\omega = 2.0$, showing elimination of singular features on the dispersing band. (i) Equivalent results for $\omega = 2.5$, showing survival of half moons away from zone center. [cf. Fig. \ref{['fig:summary.topological']}]. All results were obtained within linear spin wave (LSW) theory for Eq. \ref{['eq:H']}, for parameters $J = 1$, $g_zh^z = 5$. Results in (e)--(i) have been convoluted with a Gaussian of FHWM = $0.1\ J$ to mimic experimental resolution.
• Figure 4: Half--moon features signalling topological bands in the Kagome lattice ferromagnet Cu[1,3-bdc]. (a) Magnon band structure for parameters taken from fits to experiment Chisnell2015, showing (approximately) flat band at high energies, and gaps to dispersing bands at lower energies. Because of the FM sign of $J$, magnon bands are inverted relative to predictions shown in Fig \ref{['fig:HAF.LSW']}. (b) Corresponding dynamical structure factor $S({\bm q}, \omega)$, plotted on an irreducible wedge of the Brillouin Zone. (c) Half--moon features associated with topological band at intermediate energy, as revealed by $S({\bm q}, \omega = 2.4\ \text{meV})$. (d) Angle--integrated structure factor $S(|\bm{q}|,\omega)$, for comparison with experiment Chisnell2015. (e) $S(|\bm{q}|,\omega = 2.4\ \text{meV})$, showing double peak as a consequence of the half--moon features. Results were calculated within linear spin wave theory for Eq. (\ref{['eq:H']}), with $J = 0.6\ \text{meV}$, $D_z = 0.09\ \text{meV}$, $g_z = 2.2$ and $h = 7\ \text{T}$. Structure factors have been convoluted with a Gaussian of FWHM $0.15\ \text{meV}$ to mimic experimental resolution.
• Figure S1: Relationship between spectral features, Berry curvature and band topology in a model with bond--symmetric exchange anistoropy, $K_\perp$. (a) Spin--wave dispersion, illustrating how the mixing of states between bands opens gaps at band--touching points. The colorscale shows how each band contributes to $S({\bf k}, \omega)$, (b) Berry curvature associated with spin--wave bands. Integrated accross the Brillouin zone (BZ), this leads to bands with the Chern numbers $C = -1,~2,~-1$. (c) Dynamical structure factor $S({\bf k}, \omega)$ at $\omega = 2.02$, showing how the mixing of states between bands eliminates the pinch--point singularity for ${\bf k} \to \Gamma$. Red hexagon denotes the BZ considered in (a)--(b). (d) Equivalent results for $\omega = 2.04$, showing the change of the structure factor pattern. (e) Equivalent results for $\omega = 2.19$, showing survival of half moons away from zone center in the middle band. All results were obtained within linear spin wave (LSW) theory for Eq. \ref{['eq:H_ani']}, for parameters $J_z = J_\perp = 1$, $g_zh^z = 5$, $K_\perp = 0.63,\ D_z =0$. Results in (c)--(e) have been convoluted with a Gaussian of FHWM = $0.03\ J_z$ to mimic experimental resolution.