Pinch points and half moons encode Berry curvature

The realisation that bands of electrons can be classified using topological indices provided both a monumental shock, and an enormous stimulus to research in condensed matter physics. First studied as “Chern Insulators”, in the context of the Integer Quantum Hall effect [1, 2, 3, 4], and later revived in the context of Graphene [5, 6], work on topological bands of electrons has grown to encompass a large and active field of research on topological semi–metals, insulators and superconductors [7, 8]. More recently, magnetic insulators have also entered the stage, through the realization that bands of magnetic excitations can also carry topological indices. A prime example is provided by the “Shastry–Sutherland” magnet SrCu₂(BO₃)₂ [9], where Dzyaloshinski–Moriya interactions both enable triplon excitations to form a dispersing band, and act as a pseudo–magnetic field for these excitations, endowing them with a finite Chern number [10].

Work on topological bands in magnets has evolved into an active subfield in its own right [11, 12, 13, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 21, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], with important themes including the taxonomy of bosonic Chern insulators [21], the analysis of interactions [15, 20, 24], and closing the gap between theory and experiment [42, 43, 44, 45, 46, 47, 18, 48, 49, 50, 51]. This last point, however, presents a serious challenge. In the case of topological insulators, both the quantization of the Hall response, and the character of surface states, provide information about the topology of the underlying electron bands, with surface states easily accessible through tunneling or photoemission experiments [7, 8]. Equivalent surface states exist in topological bands of magnons [13] and triplons [10], but at present there is no established technique for measuring them. Meanwhile, the corresponding thermal Hall and Nernst effects, while sensitive to Berry phases, are not quantized [42, 44, 52], and notoriously difficult to measure. It is therefore of great interest to know whether topological bands of excitations in magnetic insulators have other, more–easily accessible fingerprints in experiment.

In this Letter, we explore signatures of band–topology which are accesible in conventional, bulk, inelastic neutron scattering, in a broad class of frustrated magnets. Building on earlier work [53, 54], we investigate the interplay between the widely observed “pinch–point” and “half–moon” motifs, and the Berry curvature of associated band–excitations. Pinch–points and half–moons arise as a consequence of “fragmentation” — the separation of magnetic degrees of freedom into components with divergence-free (incompressible) and curl-free (irrotational) character [55, 53]. Where bands with divergence–free and curl–free character meet, pinch–point and half–moon features converge on a singular point with incipient Berry curvature of $\pm 2\pi$ [Fig. 1a]. If interactions mix states belonging to these two bands, a gap opens, eliminating the singularity, and endowing each band with a Berry curvature $\pm 2\pi$. This is accompanied by characteristic changes in pinch–point and half–moon features approaching the avoided band–touching [Fig. 1b]. We demonstrate these effects for magnon bands in a model of spin–1/2 moments on the Kagome–lattice, where anisotropic exchange interactions provide the driving force for band topology. None the less, the resulting phenomenology is very general, and we discuss a range of different candidate systems, developing explicit predictions for the Kagome ferromagnet Cu(1,3-bdc) [56].

The Model. In magnets, as in systems with itinerant electrons, topological bands usually originate in spin–orbit coupling. This commonly takes the form of Dzyaloshinski–Moriya (DM) interaction with many of the most widely–studied frustrated lattices having a corner–sharing geometry which permits anisotropic exchange, including DM interactions [12, 13, 10, 57, 14, 15, 58, 16, 17, 18, 19, 21, 22, 23, 24]. As a representative example, we consider a spin–1/2 magnet with anisotropic exchange interactions on the first–neighbor bonds of a kagome lattice [Fig. 2]. The point–group symmetry of this lattice is D_6h [59, 60], and the existence of a mirror plane restricts anisotropic interactions to transverse spin components, so that the most general model is

where $h^{z}$ is an applied magnetic field, $\mathbf{n}_{ij}$ is a unit vector in the direction of the bond $ij$, and

. In what follows we will emphasize DM interactions $D_{z}$, setting $J_{\perp}=J_{z}=J$ and $K_{\perp}=0$. None the less, the conclusions we reach about the relationship between “half moons” and Berry curvature are completely general, and hold regardless of which type of anisotropy is considered. A complete analysis, including $K_{\perp}$, is given in the Supplemental Material.

We focus on single–magnon excitations about a state where spins are collinear, either because of FM exchange interactions $J<0$ or because they have been polarised by magnetic field $h^{z}\gg J$. For sufficiently large spin–wave gap (guaranteed by high magnetic field), these will be well–described by a non–interacting theory [62, 15, 20, 24], which we can access through the equation of motion (EoM) $-i\hbar\ \partial_{t}a^{\dagger}_{\lambda,{\bf q}}=[{\mathcal{H}},a^{\dagger}_{\lambda,{\bf q}}]$ for the magnon creation operator

where $\lambda=1,2,3$ is the band index, and all information about band topology is encoded in $\phi_{\lambda,i}$.

Symmetry analysis and Helmholtz decomposition. In order to make connection with singular features in scattering (half moons), we now decompose magnetic excitations $\phi_{\lambda,i}$ into incompressible (divergence–free) and irrotational (curl–free) components. It is convenient to start from irreducible representations (irreps) of spins on the triangular units which form the building–blocks of the Kagome lattice

where the convention for labelling lattice sites is given in Fig. 2. In the long–wavelength limit (${\bm{q}}\to{\bm{0}}\equiv\Gamma$), this allows us to describe magnons, Eq (3), in terms of fields ${\bm{m}}_{\sf E}$ and $\Phi_{{\sf A}_{1}}$. We then employ the Helmholtz–Hodge decomposition [63], writing

where $\nabla\cdot{\bm{m}}_{\sf E}^{\text{\sf curl}}=0$, $\nabla_{\perp}\cdot{\bm{m}}_{\sf E}^{\text{\sf div}}=0$, and $\nabla_{\perp}=(-\partial_{y},\partial_{x})$ is the two-dimensional curl. From the Heisenberg EoM $-i\partial_{t}{\bm{m}}_{\sf E}=[{\mathcal{H}},{\bm{m}}_{\sf E}]$ we find

where $\omega_{0}=h-3J$, $\rho_{S}=\frac{1}{8}J$, and we set $\hbar=1$.

Origin of half moons. In the absence of anisotropic exchange interactions, i.e. for $D_{z}=0$, excitations with incompressible and irrotational character are entirely decoupled [53, 64]. Solving Eq. (6a) by Fourier transformation $-i\partial_{\alpha}\ {\color[rgb]{0.1,0.0499999999999999,0.96}\rightarrow}\ q_{\alpha}$, we find a flat band of incompressible excitations

with corresponding eigenvector

Meanwhile, making use of the identity $\nabla(\nabla\cdot{\bm{m}}_{\sf E}^{\text{\sf div}})=\nabla^{\color[rgb]{0.1,0.0499999999999999,0.96}T}\nabla{\bm{m}}_{\sf E}^{\text{\sf div}}$, we can solve Eq. (6b) to find a quadratically–dispersing band of irrotational excitations

with eigenvector

For ${\bm{q}}\to{\bm{0}}$ the two bands are degenerate, and touch quadratically.

We are now in a position to calculate dynamical structure factors. Writing ${\bm{q}}=q\ (\cos\theta,\sin\theta)$ we find

where $\nu=\pm$ indexes the irrotational and incompressible modes, respectively; $\alpha,\beta=x,y$;

and, for $D_{z}=0$, $f(q)\equiv 1$.

For $D_{z}=0$, the limit $S^{\alpha\beta}_{\nu}({\bm{q}}\to{\bf 0},\omega)$ is a function of $\theta$, and therefore singular. This form of singularity is well–known in the context of spin–liquids, where it is referred to as a “pinch–point” [65, 53, 66]. For the flat band associated with incompressible excitations, the pinch point is directly visible in the dynamical structure factor at $\omega=\omega_{0}$ [53, 64]. Meanwhile, for the dispersing band of irrotational excitations, the pinch–point manifests as crescent–shaped “half–moons” in constant–energy cuts for $\omega>\omega_{0}$ [53]. These effects are illustrated schematically in Fig. 1a. ²²2Here and in Fig. 3, singular behaviour for ${\bf q}\to{\bf 0}$ is obscured by the finite energy resolution used in making plots of $S({\bm{q}},\omega)$..

Exactly the same phenomenology is found in direct calculations from the microscopic model Eq. (1). In Fig. 3 we present results of linear spin wave (LSW) theory, showing the flat band and dispersing bands found for $J=1$, $h=5$, $D_{z}=0$ [Fig. 3a]. The pinch point on the flat band is exhibited in Fig. 3e, while half moons associated with the dispersing band are shown in Fig. 3f. Details of these calculations are given in the Supplemental Materials.

Opening of gap and elimination of singularity. The new feature which arises in the presence of anisotropic exchange interactions is the mixing of incompressible and irrotational excitations. Solving Eq. (6a), Eq. (6b) for $D_{z}\neq 0$, we find that the quadratic band–touching at ${\bf q}={\bf 0}$ is replaced by a gap

between bands with dispersion

Hybridization also eliminates singular correlations for ${\bf q}\to{\bf 0}$; the dynamical structure factor is given by Eq. (LABEL:eq:S.q.omega), but with

where $q_{0}=\sqrt{|\Delta/\rho_{S}|}\sim\sqrt{|D_{z}/J|}$ determines the scale at which correlations cross over from a pinch–point structure at large $q$, to a non–singular behaviour for $q\to 0$.

As a consequence the pinch–points (half–moons) observed for $D_{z}=0$, vanish for $q\ll q_{0}$. None the less, half–moon features remain visible for $q\gg q_{0}$, and can also be resolved on the formerly–flat band associated with incompressible excitations, as a consequence of its finite dispersion. This phenomenology is illustrated schematically in Fig. 1b, and for LSW calculations in Fig. 3c and Fig. 3g–3i

Berry phase effects and band–topology. The Helmholtz decomposition, Eq. (5) also has important implications for band topology. In the absence anisotropic exchange interactions, each of the eigenvectors Eq. (8) and Eq. (10), winds exactly once about the singular point, ${\bm{q}}={\bm{0}}$. As a consequence, their quadratic band–touching is topologically critical, encoding Berry curvature of $\pm 2\pi$ at the singularity. Introducing an interaction which mixes excitations with incompressible and irrotational character, such as $D_{z}$, resolves the singularity at $\Gamma$, opening a gap, and injecting Berry curvature of $\pm 2\pi$ into each band.

We can explore these effects in more detail by noting that the EoM, Eq. (6), describe an effective two–level system

where $\mathbb{I}_{2}$ is a $2\times 2$ identity matrix, $\bm{\sigma}$ is a vector of Pauli matrices, $\epsilon_{0}=\omega_{0}+\frac{\rho_{S}}{2}q^{2}$, and

The Berry curvature associated with $\bm{d}({\bm{q}})$ is

This curvature integrates to $\pm 2\pi$, and is concentrated in the vicinity of ${\bm{q}}={\bm{0}}$. In the absence of other sources of Berry curvature, this will endow magnon bands with integer Chern number $C=\pm 1$. This phenomenology, linked to a quadratic band–touching, should be contrasted with the widely–studied case of linear band–crossings (Dirac cones), which encode Berry curvature $\pm\pi$ [68].

It is also possible to calculate both Berry curvature and Chern numbers within LSW theory. Results for $D_{z}=0.1$ are shown in Fig. 3d. Once the third band is taken into account, magnon bands also receive a net contribution of Berry phase $2\times\pm\pi$ from the linear band–crossings at $K$ and $K^{\prime}$ [Fig. 3b]. For $D_{z}>0$, $K_{\|}=0$, this leads to bands with Chern numbers $C=-1,0,1$.

Inference of Berry Curvature from spectral features. We now address the question of what we could have inferred about the topology of magnon bands from universal features in $S({\bm{q}},\omega)$ alone. In a model with only two Magnon bands, the observation of half–moons approaching an avoided band–touching would be enough to completely characterise band topology, up to the sign of Chern numbers $C=\pm 1$. However the Kagome–lattice model we consider supports three band of magnons, and we must also take into account contributions to Berry phase coming from the third band; in particular the linear band–crossings at $K$ and $K^{\prime}$.

The universal features associated with linear band–crossings are already well–characterised, comprising a single arc in $S({\bm{q}},\omega)$, which swaps orientation above and below the (avoided) band–touching point [70]. This phenomenology can be found in LSW calculations for Eq. (1), described in the [Supplemental Materials], and it is possible to infer Berry curvature $\pm\pi$ at $K$ and $K^{\prime}$ from spectral features near these points.

Combining these results, we can safely infer the existence of topological magnon bands in the Kagome lattice model, Eq. (1), from spectral features alone ³³3An alternative line of reasoning, leading to the same conclusion, is given in Ref. 60.. None the less, the sign of the contribution to Berry phase coming from each band–touching/bond–crossing point depends on details of interactions, and cannot be inferred solely from the existence of half–moons/arcs in $S({\bm{q}},\omega)$. For this reason, additional information is required to assign a unique set of Chern numbers to these bands. This could sought from thermal Hall measurements, which are sensitive to the sign of Berry curvature [42, 44] or, where models are sufficiently well–constrained, from fits to non–universal features in scattering.

Application to experiment. A wide range of frustrated magnets are candidates for topological band structure, and we now consider what might be learned from half moon features observed in inelastic neutron scattering. One example is Cu(1,3-bdc), a layered metal–organic framework (MOF) which realises a structurally–perfect Kagome lattice [72]. Neutron scattering experiments on Cu(1,3-bdc) [69] reveal a ferromagnetic (FM) ground state, and dispersing magnon bands which are well–described by model with first–neighbour FM exchange and DM interactions. In Fig. 4c we present LSW predictions for Cu(1,3-bdc), for parameters taken from experiment. Half–moon features are resolved near to an avoided band–touching at $\omega\sim 2.5\ \text{meV}$. These lead to a characteristic double peak in predictions for the angle–integrated structure factor $S(|{\bm{q}}|,\omega)$, providing a sharp experimental signature of Berry phase $\pm 2\pi$ within the magnon bands.

The bilayer breathing Kagome magnet Ca₁₀Cr₇O₂₈ [73] is also of note. This system has a mixture of FM and AF interactions; saturates at a magnetic field of approximately $1\ \text{T}$; and is well–described by a model which supports pinch–point and half moon features [74]. The pyrochlore magnet Lu₂V₂O₇ is another strong candidate, and features redolant of half–moons, approaching a flat band, have already been observed in experiment [75].

Another interesting avenue to explore would be correlations within topological bands of electrons. The phenomenology of half moons and Chern numbers developed in this Letter applies equally to electrons, and “Kagome metals” such as Fe₃Sn₂ [76] and CoSn [77] are already under study; similar topological bands are argued to be possible in conducting MOF’s [78]. In this context, it would be interesting to revisit, for example, ARPES experiments on CoSn [77], where a flat band is observed a little below the Fermi energy, and reinterpret data in the terms of [Eq. (LABEL:eq:S.q.omega)].

Conclusions. Topological bands of excitations are a ubiquitous feature of systems with both itinerant and localised electrons, originating in the Berry curvature of the underlying band states. In this Letter, we have shown that the existence of pinch–points and half–moons in dynamical structure factors approaching an (avoided) quadratic band–touching point guarantees a net source of Berry curvature $\pm 2\pi$. The link between Berry curvature and spectral features is provided by the effective two–level model, Eq. (16), which describes the coupling between excitations with incompressible and irrorational character, and is characterised by a vector ${\bm{d}}$ [Eq. (18)], that winds twice around the band–touching point. These results imply a robust connection between singular features in scattering and Berry curvature [Fig. 1] which can, in some cases, be used to unambiguously determine the topological nature of bands from spectral features alone. This picture is applicable to magnon bands in a wide range of frustrated magnets, and is also relevant to topological bands of electrons on frustrated lattices [79].