Band topology has been a celebrated line of research in condensed matter physics since the discovery of topological insulators over a decade ago. It has now been developed to a stage that covers a variety of electronic as well as non-electronic systems, the latter including electromagnetic and mechanical waves in artificial structures. While the topological classical waves has enriched our fundamental understanding of band topology and led to far-reaching technical applications, their natural quantum counterparts, namely topological bosonic excitations in real crystals, have so far not been investigated beyond a handful of examples. This is partly because solid-state scattering spectroscopies, which are the most powerful methods for detecting bosonic quasiparticles in crystals, are very demanding measurements to do.

With the hope of using inelastic neutron scattering spectroscopy to promote this research field, ICQM member Prof. Yuan Li and his group, together with collaborators led by Prof. Chen Fang at the Institute of Physics, Chinese Academy of Sciences, undertook a series of efforts in developing theoretical motives, experimental methods, and analytical expertise to cope with the problem. They had a specific goal in mind: to search for bosonic analogues of “topological semimetals”. Compared to bosonic analogues of topological insulators, the wanted “semimetal” band structures have topological observables already in the bulk, namely band-crossing points or lines with linear dispersions in all directions, and are therefore much more suitable for neutron scattering which is intrinsically a bulk probe.

The material under spotlight was Cu3TeO6. Below 61 K, this material becomes an antiferromagnet, where the spins on the Cu2+ ions form a (nearly) bipartite structure with half of them pointing in the same direction and opposite to the other half. By using linear spin-wave theory to calculate magnon excitations from the antiferromagnetic ground state, Prof. Li and coworkers discovered that the magnons would always have linear band crossings, and further analysis showed that the crossings are topological – they carry topological charges which are whole numbers that do not depend on model details, but only on the symmetries. The required symmetry is PT, time reversal followed by space inversion, which is arguably the simplest symmetry element in a magnetic group. If a global spin-rotation U(1) symmetry is also present, the topological band crossings take on the form of Dirac points (Fig. 1a), whereas upon the removal of the U(1) symmetry, the Dirac points expand into nodal lines (Fig. 1b). Both the Dirac points and the nodal lines are new to the “zoo” of topological band crossings predicted for specific materials. Moreover, the high crystal symmetry of Cu3TeO6 (space group #206, Fig. 1c) ensures that the P-point of the Brillouin zone will always host Dirac points (in the presence of U(1) symmetry), which is convenient for experimental detection. These theoretical results were published in December 2017 in Physical Review Letters, where Yuan Li and his student Chenyuan Li were respectively a corresponding author and a co-first author.

Figure 1. (a) Magnon band structure of Cu3TeO6, calculated with a J1-J2 model (J2 = 0.134 J1 > 0), with several Dirac points (D1-D3) in the Brillouin zone. (b) The Brillouin zone, and schematics showing that as the U(1) symmetry is removed, the Dirac points will generally expand into nodal lines or nodal rings. (c) The J1-J2 magnetic interaction network displayed in a body-centered cubic unit cell. Adapted from [1]. |

Subsequently, inelastic neutron scattering experiments were performed on a large array of co-aligned single crystals to determine the magnon band structure. The experiments yielded very clear-cut magnon signals in the four-dimensional momentum-energy space. In order to enable a close-knit comparison between the experimental spectra and the theoretical calculations, the team spent much effort to model the magnetic interactions – they concluded on the existence of leading magnetic interactions both between the nearest-neighbor Cu2+, due to their close distance, and between the ninth-nearest-neighbor Cu2+, due to a relatively straight exchange pathway between them. The success of the model in describing the experimental data can be seen from Fig. 2a-b, where even the finest details in the experiment can be reproduced by the calculations. This had in turn facilitated the experimental verification of the band topology. Indeed, not only did the experimental data show a Dirac-cone-like dispersion at the P-point (Fig. 2c), as expected according to the general theory, the intensity patterns near the Dirac point were also nearly identical to those in the calculation (Fig. 2d-e). Such intensity patterns, called the dynamical structure factor, or S(Q,w), contain key information about the magnons’ wave functions, so the similarity to the theoretical prediction can be considered a direct verification of the topological structures of the wave functions.

Figure 2. (a) and (b) Experimental and calculated intensity patterns (color scale: blue to red, from low to high) along high-symmetry lines in the Brillouin zone centered at (1, 1, 2). Black lines indicate the theoretical dispersions. Orange and magenta dashed-line boxes in (a) indicate the regions where the high-resolution data in (c) and in (d-e) are collected, respectively. (c) Sub-resolution visualization of the linear dispersions near the highest-energy Dirac point at the P-point. The data have been symmetrized around P. (d) and (e) Intensity patterns along an H-P-H momentum trajectory, and the corresponding theoretical calculation. Black lines in (d) indicates the theoretical dispersions, and magenta dashed lines illustrate a characteristic intensity envelope, asymmetric with respect to P, which is very similar between the experiment and the calculation. Adapted from [2]. |

The discovery of topological magnon excitations in a real crystal enriches our understanding of band topology for bosons and sheds new light on a large class of antiferromagnetic materials. Moreover, it demonstrates the strength of inelastic neutron scattering for the determination of band topology – one can visualize truly four-dimensional dispersions and the underlying quasiparticle wave functions. Finally, the band topology is expected to give rise to other novel experimental consequences that are yet to be uncovered in future work.

The above experimental results were published online recently in Nature Physics. The project was financially supported by the National Natural Science Foundation of China, the Ministry of Science and Technology of China, and the Strategic Priority Research Program of CAS under several different grants. The neutron scattering experiments were performed at the MLF, J-PARC, Japan, under a user program. The PKU students involved in this project are: Weiliang Yao and Lichen Wang (PhD students; contribution: experiments), Chenyuan Li (undergraduate student who will join MIT in Sept. 2018; contribution: theory and analysis), Shangjie Xue (undergraduate student who will join MIT in Sept. 2018; contribution: neutron experiment), and Yang Dan (undergraduate student now in PhD program at UIUC; contribution: sample growth).

- K. Li et al., Phys. Rev. Lett. 119, 247202 (2017).
- W. Yao et al., Nat. Phys., DOI:10.1038/s41567-018-0213-x (2018).