Experimental characterization of three-band braid relations in non-Hermitian acoustic lattices

Letter
Open Access

Experimental characterization of three-band braid relations in non-Hermitian acoustic lattices

Qicheng Zhang, Luekai Zhao, Xun Liu, Xiling Feng, Liwei Xiong, Wenquan Wu, and Chunyin Qiu

Phys. Rev. Research 5, L022050 – Published 5 June 2023

Abstract

The nature of complex eigenenergy enables unique band topology in non-Hermitian (NH) lattices. Recently, there has been fast growing interest in the elusive winding and braiding topologies of the NH single and double bands, respectively. Here, we explore the even more intricate NH multiband topology and present an experimental characterization of the three-band braid relations by acoustic systems. Based on a concise tight-binding lattice model, we design a ternary cavity-tube structure equipped with a highly controllable unidirectional coupler, through which acoustic NH Bloch bands are experimentally reproduced in a synthetic space. We identify the NH braid relations from the global evolution of the eigenvalues and acoustic states, including a noncommutative braid relation $σ_{1} σ_{2} \neq σ_{2} σ_{1}$ and a swappable braid relation $σ_{1} σ_{2} σ_{1} = σ_{2} σ_{1} σ_{2}$ . Our results could promote the understanding of NH Bloch band topology and pave the way toward designing practical devices for manipulating acoustic states.

Received 19 December 2022
Accepted 19 April 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.L022050

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Acoustic metamaterials Topological phases of matter

Non-Hermitian systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Qicheng Zhang , Luekai Zhao, Xun Liu , Xiling Feng, Liwei Xiong, Wenquan Wu , 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

^*Corresponding author: cyqiu@whu.edu.cn

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Issue

Vol. 5, Iss. 2 — June - August 2023

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Images

Figure 1
Fundamental braid relations of NH three-band systems. The red, blue, and purple lines represent three separable complex energy bands braided in the ( $E_{R}, E_{I}, k$ ) space. (a) Braid elements $σ_{1}$ and $σ_{2}$ . (b) Noncommutative braid relation (NBR) $σ_{1} σ_{2} \neq σ_{2} σ_{1}$ . (c) Swappable braid relation (SBR) $σ_{1} σ_{2} σ_{1} = σ_{2} σ_{1} σ_{2}$ .
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Figure 2
One-dimensional NH lattice model and three-band braids. (a) A sketch of the three-band lattice model, where $t_{0} = 1, t_{m}, t_{n} \in R$ , and $m, n > 0$ are assumed for simplicity. (b) Phase diagram exemplified for $m = 1$ and $n = 2$ . The dots mark the band braids to be realized in our experiments. (c),(d) A trivial and a nontrivial three-band braids, respectively. (e),(f) The associated manifestations on Riemann surfaces (real parts), where $E P_{1}$ ( $E P_{2}$ ) represents the EP formed by the lower (higher) two Riemann sheets. The eigenvalues, the EPs, and their branch cuts are projected to the $α - β$ plane, on which the black dot is the starting point and the arrow indicates the evolution of the momentum.
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Figure 3
Experimental setup and acoustic realization of the elementary braids $σ_{1}$ and $σ_{2}$ . (a) Experimental setup. The acoustic cavities 1, 2, and 3 emulate three orbitals and the narrow tubes in between mimic the reciprocal coupling $t_{0}$ . A controllable unidirectional coupler (UC), consisting of a microphone $D_{UC}$ , an amplifier, a phase shifter, and a loudspeaker $S_{UC}$ , is introduced between the cavities 1 and 3 to generate the unidirectional complex coupling $κ = ρ e^{i θ}$ . A source $S$ and a detector $D$ are used to excite and detect acoustic transmission responses, respectively, together with a detector $R$ for phase reference. The inset exemplifies how the complex coupling $κ$ is retrieved by fitting the transmission response $| S_{13} (ω) |$ . (b),(c) Experimental results for the elementary braids $σ_{1}$ and $σ_{2}$ , including the designed $κ$ (left), the band braids (middle), and their projections on $Ω_{R} - k$ plane (insets), and associated acoustic states (right). To facilitate observation, the initial and final eigenfrequencies (states) are linked into triangles. All experimental data (dots, circles) match well the theoretical predictions (lines).
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Figure 4
Experimental characterization of the NBR $σ_{1} σ_{2} \neq σ_{2} σ_{1}$ . The results of the band braids $σ_{1} σ_{2}$ (a) and $σ_{2} σ_{1}$ (b) indicate inequivalent braiding consequences in both the eigenvalues (left) and acoustic states (right).
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Figure 5
Experimental characterization of the SBR $σ_{1} σ_{2} σ_{1} = σ_{2} σ_{1} σ_{2}$ . The results of the band braids $σ_{1} σ_{2} σ_{1}$ (a) and $σ_{2} σ_{1} σ_{2}$ (b) identify equivalent braiding consequences in both the eigenvalues (left) and acoustic states (right).
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