Observation of a chiral wave function in the twofold-degenerate quadruple Weyl system BaPtGe

Haoxiang Li, Tiantian Zhang, A. Said, Y. Fu, G. Fabbris, D. G. Mazzone, J. Zhang, J. Lapano, H. N. Lee, H. C. Lei, M. P. M. Dean, S. Murakami, and H. Miao

Phys. Rev. B 103, 184301 – Published 4 May 2021

Abstract

Topological states in quantum materials are defined by bulk wave functions that possess nontrivial topological invariants. While edge modes are widely presented as signatures of nontrivial topology, how bulk wave functions can manifest explicitly topological properties remains unresolved. Here, using high-resolution inelastic x-ray spectroscopy (IXS) combined with first principles calculations, we report experimental signatures of chiral wave functions in the bulk phonon spectrum of BaPtGe, which we show to host a previously undiscovered twofold-degenerate quadruple Weyl node. The chirality of the degenerate phononic wave function yields a nontrivial phonon dynamical structure factor, $S (Q, ω)$ , along high-symmetry directions, that is in excellent agreement with numerical and model calculations. Our results establish IXS as a powerful tool to uncover topological wave functions, providing a key missing ingredient in the study of topological quantum matter.

Received 29 October 2020
Accepted 16 April 2021

DOI:https://doi.org/10.1103/PhysRevB.103.184301

Physics Subject Headings (PhySH)

Phonons Symmetry protected topological states Topological phases of matter

Topological materials

Inelastic light scattering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Haoxiang Li^1,*, Tiantian Zhang^2,3,*,†, A. Said^4,*, Y. Fu⁵, G. Fabbris ⁴, D. G. Mazzone ⁶, J. Zhang¹, J. Lapano¹, H. N. Lee ¹, H. C. Lei^5,‡, M. P. M. Dean ⁷, S. Murakami ^2,3, and H. Miao ^1,§

¹Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
²Department of Physics, Tokyo Institute of Technology, Okayama, Meguro-ku, Tokyo 152-8551, Japan
³Tokodai Institute for Element Strategy, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama, Kanagawa 226-8503, Japan
⁴Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA
⁵Department of Physics and Beijing Key Laboratory of Opto-Electronic Functional Materials and Micro-devices, Renmin University of China, Beijing, China
⁶Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
⁷Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA

^*These authors contributed equally to this work.
^†ttzhang@stat.phys.titech.ac.jp
^‡hlei@ruc.edu.cn
^§miaoh@ornl.gov

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Vol. 103, Iss. 18 — 1 May 2021

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Images

Figure 1
Twofold-degenerate quadruple Weyl node and chiral wave functions. (a), (b) Crystal structure and Brillouin zone (BZ) of BaPtGe, respectively. The high-symmetry points in the three-dimensional BZ and the projected two-dimensional BZ are shown in (b). (c) TQW node with $C = + 4, - 4$ for each band. (d) Pseudospin derived from Eq. (1) at the eight symmetry-related $R$ points. Colored arrows represent pseudospin directions. Zoomed pseudospin textures near time-reveal-related momenta $(0.5, - 0.5, - 0.5)$ and $(- 0.5, 0.5, 0.5)$ in reciprocal lattice units (r.l.u.) are showing opposite chirality. (e) DFT-calculated phonon dispersion of BaPtGe. Solid circles at the $Γ$ points correspond to the theoretically predicted TQW. Dashed rectangles at the $Γ$ and $R$ points are corresponding to threefold and fourfold double Weyl nodes, which are zoomed in (f)–(g).
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Figure 2
DFT-calculated dynamical structural factor. (a)–(c) DFT-calculated $S^{DFT} (Q, ω), R (Q, ω),$ and $I (Q, ω)$ along the high-symmetry $R_{1} (- 0.5, 2.5, 2.5) - Γ (0, 3, 3) - R_{2} (0.5, 3.5, 3.5)$ direction, respectively. $R (Q, ω)$ and $I (Q, ω)$ considers only the real and imaginary parts of $e_{d}^{j} (q)$ . (a)–(c) are shown in the same colorscale.
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Figure 3
Directional chiral wave function of the TQW. (a)–(c) Experimental $S^{exp} (Q, ω)$ results along the three high-symmetry directions [001], [011], and [111], where $X_{1} = (0, 3, 2.5), X_{2} = (0, 3, 3.5), M_{1} = (0, 2.5, 2.5), M_{2} = (0, 3.5, 3.5), R_{1} = (- 0.5, 2.5, 2.5), R_{2} = (0.5, 3.5, 3.5)$ . Experimental resolution convoluted $S_{r}^{DFT} (Q, ω), R_{r} (Q, ω)$ are shown in (d)–(f) and (g)–(i), respectively. The convolution function is determined by measuring the elastic scatting from plexiglass, which possesses a pseudo-voigt line shape with $Δ E \sim 1.4 meV$ . Panels (d)–(i) are shown in the same colorscale.
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Figure 4
$Q$ -dependent phonon intensity. (a)–(c) The extracted peak intensities from the experimentally determined $S^{exp} (Q, ω)$ (orange dots) and calculated $S_{r}^{DFT} (Q, ω)$ (blue dashed lines) along [001], [011], and [111] directions, respectively. The curve near 6 meV corresponds to the TQW. (d)–(f) $Q$ -dependent peak intensities of the TQW that are extracted from $S^{exp} (Q, ω), S_{r}^{DFT} (Q, ω), R_{r}^{DFT} (Q, ω)$ , respectively. Vertical bars in panels (a)–(c) denote errors of the peak positions estimated based on the energy resolution and counting statistics. Error bars in panels (d)–(f) denote one standard deviation assuming Poissonian counting statistics in the measured $S^{exp} (Q, ω)$ intensity. Panels (g) and (h) depict the chiral atomic motion of the TQW with $C = + 4$ and $- 4$ , respectively. The results are calculated from the simplified model along the [111] direction. Colored arrows are pointing to the directions of Pt motions at a given time. Chirality is determined by the (g) clockwise and (h) anticlockwise motions.
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