Pseudospins and Topological Effects of Phonons in a Kekul\'e Lattice

Pseudospins and Topological Effects of Phonons in a Kekulé Lattice

Yizhou Liu, Chao-Sheng Lian, Yang Li, Yong Xu, and Wenhui Duan

Phys. Rev. Lett. 119, 255901 – Published 22 December 2017

Abstract

The search for exotic topological effects of phonons has attracted enormous interest for both fundamental science and practical applications. By studying phonons in a Kekulé lattice, we find a new type of pseudospin characterized by quantized Berry phases and pseudoangular momenta, which introduces various novel topological effects, including topologically protected pseudospin-polarized interface states and a phonon pseudospin Hall effect. We further demonstrate a pseudospin-contrasting optical selection rule and a pseudospin Zeeman effect, giving a complete generation-manipulation-detection paradigm of the phonon pseudospin. The pseudospin and topology-related physics revealed for phonons is general and applicable for electrons, photons, and other particles.

Received 21 August 2017

DOI:https://doi.org/10.1103/PhysRevLett.119.255901

Physics Subject Headings (PhySH)

Phonons Thermal conductivity Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yizhou Liu^1,2,3, Chao-Sheng Lian^1,2, Yang Li^1,2, Yong Xu^1,2,3,*, and Wenhui Duan^1,2,4,†

¹State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China
²Collaborative Innovation Center of Quantum Matter, Beijing 100084, People’s Republic of China
³RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
⁴Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China

^*yongxu@mail.tsinghua.edu.cn
^†dwh@phys.tsinghua.edu.cn

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Vol. 119, Iss. 25 — 22 December 2017

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Figure 1
(a) A honeycomb lattice with Kekulé distortion having force constants $f_{1, 2, 3}$ (colored red, gray, and green) with nearest neighbors and (b) the Brillouin zone. The black and red dashed lines correspond to a unit cell and $\sqrt{3} \times \sqrt{3}$ supercell, respectively. (c) Schematic of Dirac phonons formed by linear band crossings between the longitudinal acoustic (“LA”) and longitudinal optical (“LO”) branches at the $K$ and $K^{'}$ points. The four eigenmodes ( $| A, K ⟩$ , $| B, K ⟩$ , $| A, K^{'} ⟩$ , and $| B, K^{'} ⟩$ ) are displayed with information about circular polarizations and phases. (d)–(f) Phonon dispersions ( $θ = π / 4$ ) for $m_{K} = 0$ and $m_{K} = \pm 0.05$ , respectively. A band gap opens at the $Γ$ when $m_{K} \neq 0$ , showing two doubly degenerate bands $E_{1}$ and $E_{2}$ whose band orders are opposite between $m_{K} > 0$ and $m_{K} < 0$ . (g)–(k) Berry curvature distribution of $E_{1}$ and $E_{2}$ bands for pseudospin up and pseudospin down. The phonon pseudoangular momenta $j_{ph}$ at the $Γ$ are labeled.
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Figure 2
(a) Schematic of a nanoribbon structure including an interface with opposite $m_{K}$ on the two sides, which has topologically protected pseudospin-polarized interface states. An armchair nanoribbon including 63 atom dimer lines with $| m_{K} | = 0.05$ was considered, whose edge atoms in the uppermost and lowermost three lines were fixed with infinite atomic masses for eliminating trivial edge states. In the transport calculations, the transport system included a center part together with two semi-infinite contacts that were all composed of the armchair nanoribbon. Disorders were introduced over the whole center part that contains two zigzag chains (or one unit cell of the nanoribbon). (b) Phonon dispersion of the nanoribbon, showing in-gap interface states that are pseudospin-momentum locked. (c) Phonon transmission as a function of frequency $ω$ for $Δ = 0$ , 0.1, 0.2.
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Figure 3
(a) Schematic of a Kekulé lattice under a TRS-breaking field [e.g., Coriolis ( $Ω$ ) or magnetic ( $B$ ) field]. (b) Phonon dispersion ( $m_{K} = 0.1$ ) of $E_{1}$ and $E_{2}$ bands with a TRS-breaking mass of $m_{T} = 0.05$ , showing the phonon pseudospin Zeeman splitting. The pseudospin polarization was calculated by projecting wave functions onto pseudospin eigenstates. (c) A topological phase diagram for varying $m_{K}$ and $m_{T}$ , which is divided into four parts characterized by different pseudospin Chern numbers ( $C_{↑}$ , $C_{↓}$ ). (d) Topological interface states between two distinct phases depicted in (c), calculated for interfaces between two semi-infinite regions ( $| m_{K} | = | m_{T} | = 0.05$ ) by the NEGF method.
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Figure 4
(a) Schematic of a phonon pseudospin Hall effect under a band-edge gradient $E_{eff}$ (black arrow). The Hall currents are opposite for pseudospin up (red curve) and pseudospin down (blue curve). (b) Schematic of pseudospin-selective phonon generation by infrared absorption and Stokes Raman scattering. Circularly polarized photons (left-handed $σ_{-}$ or right-handed $σ_{+}$ ) mainly excite phonons of one pseudospin that conserves the (pseudo)angular momentum. (c) By selectively generating phonons of one pseudospin (e.g., using Raman spectroscopy), a net phonon Hall current (blue curve) could be observed.
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Physical Review Letters

Pseudospins and Topological Effects of Phonons in a Kekulé Lattice

Yizhou Liu, Chao-Sheng Lian, Yang Li, Yong Xu, and Wenhui Duan

Phys. Rev. Lett. 119, 255901 – Published 22 December 2017

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