Manipulation of Weyl Points in Reciprocal and Nonreciprocal Mechanical Lattices

Mingsheng Tian, Ivan Velkovsky, Tao Chen, Fengxiao Sun, Qiongyi He, and Bryce Gadway

Phys. Rev. Lett. 132, 126602 – Published 21 March 2024

Abstract

We introduce feedback-measurement technologies to achieve flexible control of Weyl points and conduct the first experimental demonstration of Weyl type I-II transition in mechanical systems. We demonstrate that non-Hermiticity can expand the Fermi arc surface states from connecting Weyl points to Weyl rings, and lead to a localization transition of edge states influenced by the interplay between band topology and the non-Hermitian skin effect. Our findings offer valuable insights into the design and manipulation of Weyl points in mechanical systems, providing a promising avenue for manipulating topological modes in non-Hermitian systems.

Received 22 August 2023
Accepted 22 February 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.126602

Physics Subject Headings (PhySH)

Acoustic metamaterials

Non-Hermitian systems Weyl semimetal

Condensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

Authors & Affiliations

Mingsheng Tian ^1,2, Ivan Velkovsky ², Tao Chen ², Fengxiao Sun ¹, Qiongyi He ^1,3,4,*, and Bryce Gadway ^2,†

¹State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
²Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA
³Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
⁴Hefei National Laboratory, Hefei, 230088, China

^*qiongyihe@pku.edu.cn
^†bgadway@illinois.edu

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Vol. 132, Iss. 12 — 22 March 2024

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Images

Figure 1
Schematic diagram of experimental mechanical lattices and Weyl band structure. (a) Illustration of mechanical setup based on measurements and feedback forces. The mechanical arrays consist of $N$ unit cells, each containing sublattice $a$ and sublattice $b$ , marked in blue and orange, respectively. (b) Phase diagram for WP1 and WP2 as a function of modulation phase difference $Δ ϕ$ and detuning frequency ratio $ω_{b} / ω_{a}$ . (c) Band structure of Weyl type-I for periodic mechanical arrays, corresponding to the orange triangle marked in (b) with $Δ ϕ = π$ and $ω_{b} / ω_{a} = 1 / 2$ . The isoenergy contours of the lower band are shown at the bottom of each figure, where we marked the position and chirality of four WPs with signs $+$ and $- .$ (d) The corresponding band structure for truncated mechanical arrays, showing the bulk states (transparent gray) and Fermi arc surface states (blue and orange colors). (e),(f) Band structure at type I-II transition point at the red-cross point of (b). (g),(h) Type II Weyl band structure at the blue star of (b). In all plots, we take $ω_{a} = 1.5 j$ , $k = 0$ , and $λ_{1} = λ_{2} = 2 j$ .
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Figure 2
Observation of Weyl type I-II transition and Fermi arcs. (a)–(c) Experimentally measured projected band structure in truncated mechanical arrays, showing the case of WP1 (a), the transition point (b), and WP2 (c) at nontrivial cuts of $θ = 0.45 π$ . (d) The projected WP1 ( $Δ ϕ = π$ ) band structure dependence on $θ$ at a special cut of $ϕ = π / 2$ passing through WPs, where the two edge states appear as arcs connecting a pair of WPs with different chiralities. Here the orange and blue solid curves represent the two numerical calculated edge states. In experiments, we take $λ_{1} / 2 π = λ_{2} / 2 π = 60 mHz$ , $j / 2 π = 30 mHz$ , $ω_{a} / 2 π = 45 mHz$ , $ω_{b} / 2 π = 22.5 mHz$ , and $2 N = 18$ . The experimental error bars for the edge mode frequencies (typically smaller than the data points) are the standard error based on five measurements.
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Figure 3
Direct observation of the Fermi arc surface states. (a) Numerically calculated mode profiles in the trivial region ( $ϕ = π / 2$ and $θ = 0.3 π$ ). The inset shows the band structure as a function of $ϕ$ when $θ = 0.3 π$ . (b)–(c) Experimentally observed energy distribution of all oscillators evolving with time, where we initially shake either the first (b) or last (d) oscillator. The brightness in these plots reflects the amount of normalized energy at each oscillator. (d)–(f) Numerically calculated (d) and experimentally observed (e),(f) results in the nontrivial region with $θ = 0.45 π$ , showing the presence of edge states. The orange and blue curves represent the two edge states. Here, all figures are plotted in the WP1 region ( $Δ ϕ = π$ ), and the parameters used in experiments remain consistent with those presented in Fig. 2.
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Figure 4
The impacts of non-Hermiticity. (a) The Weyl ring of non-Hermitian band structure at $Δ ϕ = π$ . (b) The periodic- and open boundary spectra at $ϕ = θ = 0.4 π$ . (c) The wave functions of the edge modes for different $θ$ at $ϕ = 0.4 π$ , showing an edge state localization transition from one side to the other. (d) TDR-NHDR diagram of edge state as a function of $θ$ and $ϕ$ for WP1 ( $Δ ϕ = π$ ), WP1-WP2 transition point ( $Δ ϕ = π / 3$ ), and WP2 ( $Δ ϕ = 0$ ). In all plots, we take $ω_{a} = 1.5 j$ , $k = 0$ , $ω_{b} / ω_{a} = 0.5$ , and $λ_{2} = 2.5 λ_{1} = 2 j$ .
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Figure 5
Edge localization transition on Fermi arc from the competition of band topology and non-Hermiticity. We exhibit the numerical mode profiles, band structure, and the experimental time evolution of normalized oscillator energies with $θ = 0.3 π$ (a)–(c) and $θ = 0.45 π$ (d)–(f), respectively. The surface states exhibit a localization transition, moving from one edge to the opposite edge. Here, we take $λ_{1} / 2 π = 24$ , $λ_{2} / 2 π = 60 mHz$ , and other parameters are same as those depicted in Fig. 3.
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