### Background

Entanglement is a phenomenon where the measurement of one observable affects the outcome of another observable, even if the two observables are separated by great distances. A notable entangled system is the exotic spin liquid — a modern state of matter where quantum spins interact — which possesses non-trivial fractionalized excitations called spinons, quasiparticles that carry spin but not charge. However, surprisingly, such excitations are not only restricted to highly entangled systems but can even exist in more-common magnetically ordered insulators. The surprise is due to the conventional wisdom in the study of magnetic insulators where if one was able to identify the spin arrangement, that would have been sufficient to specify the state of a system. But as history teaches us, there are always examples that defy conventional wisdom. These include the electronic models of the graphene-family, photons in a periodic arrangement of dielectric materials that are governed by classical physics, and (as seen in this study) in a ‘common’ magnetically ordered antiferromagnetic system.

The existence of so-called Weyl magnons is of particular interest, as these ‘Weyl’ types of excitations have previously only been observed in electronic and in photonic systems. A magnon is the quantized collective excitation of magnetic/spin structures (analogous to phonons in crystal vibrations). A ‘Weyl’ excitation emerges from a point about which two unique energy bands cross, and about that point the energy dispersion scales linearly with momentum. However, unlike their fermionic and photonic siblings, Weyl magnons are highly tunable and can be manipulated by laboratory magnetic fields. This opens up an amazing potential for Weyl magnons in electronics and thermoelectrical applications, as will be discussed. Natural questions one may ask are: where can Weyl magnons be realized? Are they present in any particular material? The answers to these questions are precisely what are answered in this study where Weyl magnons are predicted to exist in a real and physical antiferromagnetic breathing pyrochlore lattice.

### Study Design and Methodology

The researchers studied the existence of Weyl magnons by focusing on a minimal spin model (i.e. a model which encapsulates the essential features of the system) on a breathing pyrochlore lattice. The breathing pyrochlore lattice is a pyrochlore structure (composed of corner sharing tetrahedra), but lacks inversion symmetry. This lattice structure is present in numerous compounds such as LiGaCr4O8 and LiInCr4O8. The authors employed linear spin-wave theory to study the low-lying excitations/disturbances in the magnetic structure (‘spin waves’), and explored the vast parameter space of the model by also incorporating the effects of quantum fluctuations (fluctuations associated with the quantum nature of the magnet), in addition to studying their robustness to magnetic fields.

### Key Findings

The spin wave analysis demonstrated that the phase diagram (which depicts the possible phases/states that exist) could be split into three regions (Figure 1). Regions I and II possess the same magnetic ordering (non-collinear antiferromagnet), while region III has a different magnetic ordering. However, although regions I and II have the same ground state, their magnetic excitations are distinct as only region I has the type of magnon energy dispersion that allows Weyl points to be realized. Importantly, the Weyl points are topologically robust and exist over a finite region in the phase diagram. One may wonder why it has taken so long for such excitations to be realized. The reason is that Weyl points are shown to exist only when a subtle symmetry of the system is broken: combination of charge conjugation and inversion symmetry. In the common collinear antiferromagnet (where the magnetic moments align such that neighbouring spins point in opposite directions, along a given direction/axis), this symmetry is preserved and so Weyl magnons are not observed. But, the realistic model here does indeed break the required symmetry and thus gives rise to these unusual excitations.

**Figure 1: Region I’s phase diagram and dispersion indicating the presence of Weyl magnons.**

**(a) Phase diagram of minimal spin model. Region I is where the Weyl magnons exist.**

**(b) Spin wave energy dispersion with the location of the Weyl point (circled in red).**

Secondly, the authors demonstrated that not only are the Weyl points well defined in the presence of magnetic fields (as unlike their electrical counterparts, Weyl magnons are neutral) but they can also be manipulated in momentum space by the application of magnetic fields along different directions (Figure 2):

**Figure 2: Manipulation of Weyl points under the application of a magnetic field. The red and blue indicate opposite chirality (“handedness”) Weyl points. The magnetic field is increasing progressively from figures (a) ͢ (f).**

The above figure indicates the ease of manipulation of the Weyl magnons, where not only can the magnons be moved around in momentum space (a) – (e), but they can also be made to disappear (f).

Finally, there are numerous experimental signatures of these excitations. In addition to neutron scattering and optical spectroscopic measurements, Weyl magnons can also give rise to the thermal Hall effect, where the application of a magnetic field perpendicular to the direction of a thermal gradient gives rise to a thermal gradient in the orthogonal direction. This is not only an interesting signature of these excitations, but is also potentially important from an applications point of view.

### Conclusions and Implications

The study predicts the presence of topologically non-trivial excitations known as Weyl magnons in a realistic system, which is not necessarily highly entangled. The study also provides numerous experimental signatures and effects that can be measured to guide the experimental search for these excitations. Due to their ease of manipulability under the influence of readily available magnetic fields, and the demonstration that a thermal gradient (across a solid) can be generated via these magnons, Weyl magnon materials have great potential in electronics and thermoelectrical applications. Materials that in response to magnetic fields create electrical voltage gradients across the solid are already prevalent in today’s society in their use as probes of magnetic fields (magnetometers) and electromagnetic sensors. Moreover, they also serve as effective electronic alternatives to mechanical switches in modern day electronics and computing. Could these Weyl magnon materials be used in such applications (perhaps thermoelectrical devices)? It is certainly possible, and more importantly they could be better than what we have today. This positivity stems from the fact that Weyl magnons, discussed in this study, are housed in antiferromagnetic materials. This is beneficial from an applications point of view as antiferromagnets are invisible to stray magnetic fields. Thus, antiferromagnets possess a distinct robustness and stability, which allows them to eclipse the abilities of today’s ferromagnetic counterparts. Stability of the system, as well as ease of manipulability of their Weyl excitations, makes the materials discussed in this study to be a bright and exciting part of our future. Although it has taken a while for such exotic excitations to be realized, due to their remarkable properties they will soon be part of our everyday lives.

*Li, F., Li, Y., Kim, Y.B., Balents, L., Yu, Y. and Chen, G. (2016). https://www.nature.com/articles/ncomms12691. Nature Communications, 7, p.12691.*

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