Anomalous Topological Edge States in Non-Hermitian Piezophononic Media

Penglin Gao, Morten Willatzen, and Johan Christensen

Phys. Rev. Lett. 125, 206402 – Published 13 November 2020

Abstract

The bulk-boundary or bulk-edge correspondence is a principle relating surface confined states to the topological classification of the bulk. By marrying non-Hermitian ingredients in terms of gain or loss with media that violate reciprocity, an unconventional non-Bloch bulk-boundary correspondence leads to unusual localization of bulk states at boundaries—a phenomenon coined non-Hermitian skin effect. Here, we numerically employ the acoustoelectric effect in electrically biased and layered piezophononic media as a solid framework for non-Hermitian and nonreciprocal topological mechanics in the MHz regime. Thanks to a non-Hermitian skin effect for mechanical vibrations, we find that the bulk bands of finite systems are highly sensitive to the type of crystal termination, which indicates a failure of using traditional Bloch bands to predict the wave characteristics. More surprisingly, when reversing the electrical bias, we unveil how topological edge and bulk vibrations can be harnessed either at the same or opposite interfaces. Yet, while bulk states are found to display this unconventional skin effect, we further discuss how in-gap edge states in the same instant, counterintuitively are able to delocalize along the entire layered medium. We foresee that our predictions will stimulate new avenues in echo-less ultrasonics based on exotic wave physics.

Received 18 July 2020
Accepted 21 October 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.206402

Physics Subject Headings (PhySH)

Topological phase transition

Topological insulators

Condensed Matter, Materials & Applied PhysicsParticles & Fields

Authors & Affiliations

Penglin Gao ^1,*, Morten Willatzen^2,3, and Johan Christensen ^1,†

¹Department of Physics, Universidad Carlos III de Madrid, ES-28916 Leganès, Madrid, Spain
²Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
³Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, 30 Xueyuan Road, Haidian District, Beijing 100083, China

^*Corresponding author. pgao@pa.uc3m.es
^†Corresponding author. johan.christensen@uc3m.es

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Vol. 125, Iss. 20 — 13 November 2020

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Images

Figure 1
Mechanical SSH model with nonreciprocal coupling. (a) A piezophononic non-Hermitian system, electrically biased with electric field $E_{0}$ . The atomic chain illustrates how the continuum system maps to a non-Hermitian SSH model. (b) Modal space, showing the physical meaning of the complex modal solutions $k = k^{'} + i k^{''}$ . (c) The imaginary parts of the two modal solutions, $k_{1}^{''}$ and $k_{2}^{''}$ , are plotted as a function of $E_{0}$ for frequencies of interest. The dashed lines indicate the Cherenkov threshold where either $k_{1}^{''} = 0$ or $k_{2}^{''} = 0$ , indicating perfect one-way transmission.
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Figure 2
Breakdown of the conventional bulk-edge correspondence. (a) A group of non-Hermitian band diagrams are plotted in the complex frequency plane as a function of the Bloch wave number $k_{B}$ . As indicated in panel (b), the selected phases I, II, III, and IV are marked at specific attenuation levels $k_{1}^{''} = 0$ , 200, 266, and $300 m^{- 1}$ , respectively. Computed eigenfrequency spectra for a finite array (containing $N = 50$ unit cells) when subject to periodic (c) and free (d) boundary conditions. For clarity, the in-gap edge states in (d) are highlighted by red dots.
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Figure 3
(a) Phase diagram comprising two variables, the impedance ratio $Z_{A} / Z_{B}$ and attenuation $k_{1}^{''}$ (in units $m^{- 1}$ ) in a forward-biased crystal. The dashed and solid lines mark the phase transition points calculated under periodic and free boundary conditions, respectively. Background colors are used to differentiate the gapped (pink) from the gapless (blue) phase when $N = 50$ . The purple vertical dashed line corresponds to the phases highlighted in Fig. 2 for the piezophonic medium made of InSb and GaAs (constant impedance). (b) Numerically calculated eigenfields for the selected phases as indicated in panel (a). (c) A surface plot mapping the magnitude of the transfer matrix determinant $| \det (T) |$ .
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Figure 4
Non-Hermitian topology and anomalous edge states when the electric field is reversed. (a) A complete phase diagram plotted against the impedance ratio $Z_{A} / Z_{B}$ and attenuation $k_{2}^{''}$ (in units $m^{- 1}$ ). (b) Numerically calculated eigenfields for the selected phases as indicated in panel (a). (c) Calculated eigenfields of the in-gap edge states vs $k_{2}^{''}$ [dashed line in panel (a)] illustrating unusual transitions of the field confinement.
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Figure 5
Numerically calculated eigenfrequency spectra subject to a variety of different boundary conditions: free-free (left panels), fixed-free (middle panels), and free-fixed (right panels). Top and bottom panels correspond to finite arrays of lengths $N = 50$ and $N = 100$ , respectively. The in-gap edge states are highlighted by red dots.
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