Real-projective-plane hybrid-order topological insulator realized in phononic crystals

Pengtao Lai, Jien Wu, Zhenhang Pu, Qiuyan Zhou, Jiuyang Lu, Hui Liu, Weiyin Deng, Hua Cheng, Shuqi Chen, and Zhengyou Liu

Phys. Rev. Applied 21, 044002 – Published 1 April 2024

Abstract

The manifold of the fundamental domain of the Brillouin zone is always considered to be a torus. However, under the synthetic gauge field, the Brillouin manifold can be modified by the projective symmetries, resulting in unprecedented topological properties. Here, we realize a real-projective-plane hybrid-order topological insulator in a phononic crystal by introducing the $Z_{2}$ gauge field. Such an insulator hosts two momentum-space nonsymmorphic reflection symmetries, which change the Brillouin manifold from a torus to a real projective plane. These symmetries can simultaneously lead to a Klein-bottle insulator and quadrupole insulator phases in different bulk gaps. The nonsymmorphic reflection symmetries on Brillouin real-projective-plane, edge states of the Klein-bottle insulator, and corner states of the quadrupole insulator are observed. These results evidence the hybrid-order topology on the Brillouin manifold beyond the torus, and enrich the topological physics.

Received 29 November 2023
Revised 7 March 2024
Accepted 15 March 2024

DOI:https://doi.org/10.1103/PhysRevApplied.21.044002

Physics Subject Headings (PhySH)

Synthetic gauge fields Topological phases of matter

Acoustic metamaterials Phononic crystals Topological insulators

Z_2 symmetry

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Pengtao Lai^1,∥, Jien Wu^2,∥, Zhenhang Pu³, Qiuyan Zhou³, Jiuyang Lu³, Hui Liu¹, Weiyin Deng ^3,*, Hua Cheng^1,†, Shuqi Chen ^1,4,‡, and Zhengyou Liu^3,5,§

¹The Key Laboratory of Weak Light Nonlinear Photonics, Ministry of Education, Smart Sensing Interdisciplinary Science Center, School of Physics and TEDA Institute of Applied Physics, Nankai University, Tianjin 300071, China
²School of Physics and Optoelectronics, South China University of Technology, Guangzhou, Guangdong 510640, China
³Key Laboratory of Artificial Micro- and Nanostructures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
⁴The Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
⁵Institute for Advanced Studies, Wuhan University, Wuhan 430072, China

^*dengwy@whu.edu.cn
^†hcheng@nankai.edu.cn
^‡schen@nankai.edu.cn
^§zyliu@whu.edu.cn
^∥These authors contributed equally to this work.

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Vol. 21, Iss. 4 — April 2024

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Images

Figure 1
Acoustic real-projective-plane HOTI protected by the MSNS reflection symmetries $M_{x}^{g}$ and $M_{y}^{g}$ . (a),(b) Unit cells of tight-binding model and PC. Red (blue) tubes represent negative (positive) couplings. (c) Schematics of MSNS reflection symmetries. (d) Bulk dispersion of PC with KBI and QI gaps. (e) Isofrequency contours (black curves) of PC at $5.5 kHz$ in the first BZ. The reduced BZ (enclosed by the red and blue arrows) can be regarded as a real projective plane. (f) Phase diagram determined by $Z_{2}$ invariants $(w_{x}, w_{y})$ of KBI gap and corner charge $Q_{c}$ of QI gap in the $l_{x} − l_{y}$ plane. The white (gray) line denotes the phase boundary of KBI (QI). The red star represents the hybrid-order topological phase with specific parameters used in (b),(d).
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Figure 2
Observation of bulk properties of acoustic real-projective-plane HOTI. (a) Photograph of PC sample. The enlarged sample shows the detailed configuration of unit cell enclosed by green lines. (b) Calculated Wannier bands $v_{y}$ and $v_{x}$ of the first band of PC, which indicates the $Z_{2}$ invariants $(w_{x}, w_{y}) = (1, 0)$ . (c) Calculated Wannier sector polarizations $P_{y}^{v_{x}}$ and $P_{x}^{v_{y}}$ of the lowest two bands of PC, revealing the quantized quadrupole moment. (d) Measured (color map) and calculated (white lines) dispersions along the Γ–X–M–Γ lines. (e) Measured and calculated isofrequency contours at $5.5 kHz$ .
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Figure 3
Observation of edge states in the KBI gap. (a) Calculated projected dispersions along the x direction. The colored lines represent a pair of edge states with a nonlocal twist in the KBI gap, and the color denotes the degree of localization at the top (blue) and bottom (yellow) boundaries. (b) Calculated projected dispersion along the y direction. There is no edge state in the KBI gap. (c) Eigenfield distributions of edge modes at $S_{1} (k_{x} = - π / a)$ and $S_{2} (k_{x} = 0)$ points marked in (a), which are connected by $M_{y}^{g}$ symmetry. (d) Measured edge dispersions (color maps) excited at the centers of the top and bottom boundaries, respectively. The black lines are the calculated edge dispersions in (a).
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Figure 4
Observation of corner states in the QI gap. (a) Schematic of square-shaped PC sample. (b) Eigenfrequency spectrum. The blue and red dots denote the edge and corner modes in the QI gap, respectively. Inset: enlarged region for corner states. (c) Measured pressure field distribution at $5.65 kHz$ , demonstrating the existence of corner states. (d) Measured response spectra at the positions $C_{1}$ , $C_{2}$ , and $C_{3}$ , which are normalized by the maximum value.
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