Anti-Parity-Time Symmetry in a Su-Schrieffer-Heeger Sonic Lattice

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Anti-Parity-Time Symmetry in a Su-Schrieffer-Heeger Sonic Lattice

Bolun Hu, Zhiwang Zhang, Zichong Yue, Danwei Liao, Yimin Liu, Haixiao Zhang, Ying Cheng, Xiaojun Liu, and Johan Christensen

Phys. Rev. Lett. 131, 066601 – Published 7 August 2023

Abstract

The Su-Schrieffer-Heeger (SSH) model is an important cornerstone in modern condensed-matter topology, yet it is the simplest one-dimensional (1D) tight binding approach to dwell into the characteristics of spinless electrons in chains of staggered bonds. Moreover, the chiral symmetry assures that its surface-confining states pin to zero energy, i.e., they reside midgap in the energy dispersion. Symmetry is also an attribute related to artificial media that are subject to parity $P$ and time-reversal $T$ operations. This non-Hermitian family has been thoroughly nourished in a wave-based context, where anti- $P T$ ( $A P T$ ) symmetric systems are the youngest belonging members, permitting refractionless optics, inverse $P T$ -symmetry breaking transition, and asymmetric mode switching. Here, we report the first extension of $A P T$ symmetry in an acoustic setting by endowing a SSH lattice with gain and loss components. We show that the in-gap topological defect state hinges on the non-Hermitian phase, in that the broken symmetry suppresses it, yet when $P T$ or $A P T$ symmetry is intact, it is observed with either damped or evanescent decay, respectively. Our experiments showcase how the non-Hermitian SSH lattice serves as a utile platform to investigate topological properties across various $P T$ symmetric phases using sound.

Received 14 January 2023
Accepted 6 July 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.066601

Physics Subject Headings (PhySH)

Acoustic metamaterials Topological phase transition

Metamaterials Topological materials

Condensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

Authors & Affiliations

Bolun Hu^1,2, Zhiwang Zhang^1,*, Zichong Yue¹, Danwei Liao ¹, Yimin Liu¹, Haixiao Zhang ¹, Ying Cheng^1,†, Xiaojun Liu ^1,3,‡, and Johan Christensen ^4,§

¹Department of Physics, MOE Key Laboratory of Modern Acoustics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
²School of Science, Jiangsu Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, Jiangnan University, Wuxi 214122, China
³College of Aerospace Engineering, Chongqing University, Chongqing 400044, China
⁴IMDEA Materials Institute, Calle Eric Kandel, 2, 28906 Getafe, Madrid, Spain

^*zhangzhiwang@nju.edu.cn
^†chengying@nju.edu.cn
^‡liuxiaojun@nju.edu.cn
^§johan.christensen@imdea.org

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Issue

Vol. 131, Iss. 6 — 11 August 2023

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Images

Figure 1
Non-Hermitian phase diagram. (a) Schematics of the non-Hermitian SSH chain and corresponding SC composed of the coupled sublattices $A$ and $B$ . (b) Non-Hermitian phase diagram of the SSH chain while modulating the non-Hermiticity and coupling strengths. (c),(d) Real (c) and imaginary (d) band diagrams calculated by varying the non-Hermiticity factor $β$ with $| v - w | / | v + w | = 0.5287$ as highlighted in (b). (e)–(g) Theoretical and simulated band diagrams with $β = 0.02$ (e), 0.06 (f), and 0.10 (g) in the $P T$ symmetric, $P T$ symmetry broken and $A P T$ symmetric phases, respectively. In (c)–(g), the ratio of coupling strength is fixed at $| v - w | / | v + w | = 0.5287$ as highlighted by the gray dashed line in (b).
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Figure 2
Non-Hermiticity modulation in an acoustic cavity. (a) Structural schematic and the effective lumped-circuit model of a Hermitian acoustic cavity. (b) Effective lumped-circuit models where the non-Hermiticity is introduced through a metafluid, which is experimentally realized (c),(d) thanks to an active CNT lid coating. (e),(f) Simulated spectral amplitude modulation $Δ p$ vs the non-Hermiticity factor $β$ of the metafluid (e), and vs the electrical signal modulation $S$ applied to the CNT film (f). (g) Corresponding detected pressure modulation spectrum with the electrical signal strength from (f).
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Figure 3
Measured dispersion across various $P T$ symmetric phases. (a) Photograph of the experimental setup. The inset shows the arrangements of non-Hermitian sites in the finite NHSC. (b) CNT film mounted of the top lid of the cavity and its corresponding thermogram. (c),(d) Simulated (c) and experimentally measured (d) band diagrams with the non-Hermiticity factor $β$ ranging from 0.02 to 0.10 as indicated. Yellow solid lines represent the theoretical results.
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Figure 4
Non-Hermitian topological in-gap defect state. (a) Photograph of the experimental setup with the introduced defect site, which is composed of a passive cavity as highlighted by the pink box. The inset shows the position of the introduced defect site in the finite NHSC. (b)–(d) Calculated eigenfrequencies of the NHSC in the $P T$ symmetric [ $β = 0$ , (b)], $P T$ symmetry broken [ $β = 0.06$ , (c)], and $A P T$ symmetric phases [ $β = 0.12$ , (d)]. The insets show the pressure field distributions of the eigenstates marked in subfigures. (e)–(g) Frequency-dependent spatial profiles of the pressure amplitude fields measured in each cavity. Insets: magnifications of the defect and bulk states at the frequencies marked by the dashed lines.
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