Tracking Valley Topology with Synthetic Weyl Paths

Xiying Fan, Tianzhi Xia, Huahui Qiu, Qicheng Zhang, and Chunyin Qiu

Phys. Rev. Lett. 128, 216403 – Published 27 May 2022

Abstract

Inspired by the newly emergent valleytronics, great interest has been attracted to the topological valley transport in classical metacrystals. The presence of nontrivial domain-wall states is interpreted with a concept of valley Chern number, which is well defined only in the limit of small band gap. Here, we propose a new visual angle to track the intricate valley topology in classical systems. Benefiting from the controllability of our acoustic metacrystals, we construct Weyl points in synthetic three-dimensional momentum space through introducing an extra structural parameter (rotation angle here). As such, the two-dimensional valley-projected band topology can be tracked with the strictly quantized topological charge in three-dimensional Weyl crystal, which features open surface arcs connecting the synthetic Weyl points and gapless chiral surface states along specific Weyl paths. All theoretical predictions are conclusively identified by our acoustic experiments. Our findings may promote the development of topological valley physics, which is less well defined yet under hot debate in multiple physical disciplines.

Received 6 February 2022
Accepted 6 May 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.216403

Physics Subject Headings (PhySH)

Acoustic metamaterials

Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiying Fan, Tianzhi Xia, Huahui Qiu, Qicheng Zhang , and Chunyin Qiu ^*

Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China

^*To whom all correspondence should be addressed. cyqiu@whu.edu.cn

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Vol. 128, Iss. 21 — 27 May 2022

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Images

Figure 1
AVMs and synthetic WPs. (a) 2D AVM made of regular triangular scatterers (white) in air background (blue). The rotation angle $θ$ defines the orientation of the anisotropic scatterers. (b) 2D BZ and its 3D extension (with $k_{z}$ mimicked by $θ$ ). The DPs (black spheres) in 2D evolve into WPs (color spheres) in the synthetic 3D BZ, where red and blue label their topological charges of $\pm 1$ . (c) 2D band structures for the AVMs of $θ = 15 °$ and 30°. (d) Top panel: $θ$ evolution of the band-edge frequencies at $K$ , where the sign of band gap ( $2 Δ_{g}$ ) distinguishes topologically distinct AVH phases. Bottom panel: relative deviation of the Berry phase ( $δ$ ) plotted as a function of $θ$ , which becomes notable in the case of big band gap.
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Figure 2
Weyl path interpretation of the valley topology in rigid-boundary systems. (a) Synthetic SBZ projected along the $k_{y}$ direction, plotted with dimensionless momenta ${\tilde{k}}_{x}$ and ${\tilde{k}}_{z}$ . The color spheres highlight the projections of synthetic WPs, and the red circle specifies a Weyl path encircling one projected WP. (b) Surface band (orange) plotted in the one half of SBZ. The blue surfaces sketch the boundaries of projected bulk bands. (c) Projected band structure (red line) along the clockwise Weyl path, which is characterized by the parameter ${\tilde{k}}_{l} = 0 \sim 1$ scaled with circumference. The blue-shaded areas are bulk projections. (d)–(f): Similar to (a)–(c), but for the projection along the $k_{x}$ direction. Note that the projected WPs at $K_{1}$ ( $K_{2}$ ) and ${\bar{K}}_{1}^{'}$ ( ${\bar{K}}_{2}^{'}$ ) are overlapped now. In contrast to (c), the surface states in (f) are no longer gapless since the Weyl path encircles a pair of oppositely charged WPs.
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Figure 3
Observations of the gapless chiral surface states and open surface arcs in synthetic momentum space. (a) Schematic of a rigid-boundary sample used for measuring 1D edge states along the $x$ direction. The green star highlights the sound source near the bottom edge (black line). (b) Measured (color scale) and simulated (yellow line) edge dispersions for the AVM with $θ = 0 °$ . (c) Measured (open circles) and simulated (black line) surface bands along a rectangular loop centered at ${\bar{K}}_{2}$ in the $k_{x} − k_{z}$ SBZ (inset). Each circle corresponds to the peaked frequency for a given surface momentum. (d) Experimental evidence for the topological surface arcs at the synthetic WP frequency $f = 0.507$ . (e) Similar to (c), but for a loop in the $k_{y} − k_{z}$ SBZ (inset).
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Figure 4
Domain-wall systems. (a) Schematics of the 2D domain-wall system (left) and its synthetic extension to 3D (right). (b) Interface dispersion simulated along a circular Weyl path centered at ${\bar{K}}_{2}$ (inset). (c) Measured (color scale) and simulated (yellow line) interface dispersion for a domain-wall system formed by the AVMs with $θ_{1} = 0 °$ and $θ_{2} = 60 °$ . (d) Measured (open circles) and simulated (black line) interface bands along a rectangular Weyl path centered at ${\bar{K}}_{2}$ (inset). (e) Measured and simulated isofrequency contour at the Weyl frequency.
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