Colloquium: Topologically protected transport in engineered mechanical systems

Mechanical vibrations are being harnessed for a variety of purposes and at many length scales, from the macroscopic world down to the nanoscale. The considerable design freedom in mechanical structures allows one to engineer new functionalities. In recent years, this has been exploited to generate setups that offer topologically protected transport of vibrational waves (topological phonon transport), both in the solid state and in fluids. Borrowing concepts from electronic physics and being cross fertilized by concurrent studies for cold atoms and electromagnetic waves, this field of topological transport in engineered mechanical systems offers a rich variety of phenomena and platforms. In this Colloquium, a unifying overview of the various ideas employed in this area is provided, different approaches and experimental implementations are summarized, and the challenges as well as the prospects are commented upon.

Figure 1

Topological transport. (a) Schematic of a periodic crystal with two different sublattices. (b) Fictitious band structure of (a) featuring two bands with a band gap in the middle. (c) Topological features can be inspected by investigating, for each band, the map from the Brillouin zone (torus) to the Hilbert space. Here it is visualized as a Bloch sphere. Each pole represents a Bloch wave localized on a sublattice. (d) Topologically protected edge state featuring transport in only one direction (for a Chern insulator). The transport is immune to backscattering even at irregularities.

Figure 2

Topological transport in a Chern insulator. (a) Domain-wall geometry. Also indicated are the Chern number of the lower band for the two domains and the edge-state propagation direction (arrow). (b) Topological band structure for the domain-wall geometry shown in (a). (c) Band structure of a smoothly deformed system, with the same resulting net number of right movers at any frequency. (d) Robustness against deformations of the domain wall. (e) Global properties of the Berry connection. The shade of green (gray) identifies a BZ region over which ${\mathcal{A}}_n(\mathbf{k})$ varies smoothly. Each region is defined by fixing the phase of the Bloch waves on a particular sublattice. For the topological case, the crosses indicate quasimomenta whose Bloch waves have zeros at the sublattice used to define the gauge in the light-shaded region.

Figure 3

(a) Top sketch: spectrum of a finite-size time-symmetric topological insulator. Each pair of degenerate levels corresponds to a pair of counterpropagating edge waves (bottom sketch). Any coupling is forbidden because it would violate Kramers degeneracy. (b) Trivial and (c) topological band structures in a semi-infinite plane geometry. The degeneracies at the time-symmetric high-symmetry points $\mathrm{\Gamma }$ and $X$ correspond to Kramers doublets. The trivial and topological band structures, respectively, can be smoothly modified into a gapped band structure and a band structure with a single edge state connecting the two bulk bands without lifting any Kramer doublets.

Figure 4

Edge states generated via an engineered Dirac Hamiltonian. (a) Edge excitations can travel around arbitrarily shaped corners. The change of direction involves only a small momentum transfer $\mathrm{\Delta }p$. Backscattering requires large $\mathrm{\Delta }p$ and thus is suppressed. (b) Edge channel localized along a domain wall separating two domains governed by the Dirac Hamiltonian Eq. (3) with opposite values $\pm m_{\mathrm{bk}}$ of the mass parameter $m$. (c) The two domains share the same gapped Dirac-cone band structure. (d) Straight domain-wall geometry. The valley Chern numbers $C_V$ are also indicated. (e) Top sketch: flux $C_W$ of the Berry curvature through a closed surface surrounding the Weyl (Dirac) point (red). The flux lines are shown in black. $C_W$ is equal to the difference of the fluxes through two semi-infinite planes in different domains (bottom sketch) or, equivalently, the difference of the local valley Chern numbers. (f) Band structure for the geometry and chiral charge displayed in (d) and (e) (left sketch) and for the time-reversed Dirac Hamiltonian (right sketch). The edge states obey the bulk-boundary correspondence.

Figure 5

Graphical timeline of works on topological transport in vibrational systems. We depict both theoretical proposals and experimental works. Each panel is labeled with the name of the first author. The icons indicate whether time-reversal symmetry is broken and whether the model employed in the work is based mainly on the coupling of discrete localized modes (pendulum symbol), the acoustics in gases or liquids (acoustic wave), or the elastic vibrations in solids (vibrating plate). The length scale is shown for experimental implementations. Whenever theoretical works propose concrete designs that can work at the nanoscale, this is indicated; likewise, we indicate when concepts by their nature are geared toward macroscopic or mesoscopic scales. From top left to bottom right (ordered according to first appearance in e-print form or journal submission date): [117], [111], [144], [157], [103], [132], [143], [101], [64], ([65], [99], [114], [122], [109], [37], [23], [51], [80], [92], [16], [91], [169], [1], [130], [134], [108], [97], [15], [33], [138], [161], [160], [20], [167], and [96].

Figure 6

Graphical timeline of works (continued). From top left to bottom right: [173], [34], [79], ([168], [148], [18], [90], [136], [142], [93], [29], [170], [123], [84], [150], ([159], [120], [85], [140], [165], and [104].

Figure 7

Classification of different experimental works based on the frequency of the phononic edge channel and the size of the unit cell. The different categories of mechanical systems are discrete macroscopic systems, acoustic systems, and elastic systems. The numerical labels refer to the indices of the respective works in the timeline of Figs. 5 and 6 and show a trend toward on-chip small-fingerprint high-frequency devices.

Figure 8

Main differences between (a) electrons and (b) phonons from the perspective of topological transport. Left column: electronic transport can be affected by magnetic fields or spin-orbit coupling, while the transport of neutral phonons requires different approaches. Middle column: while electronic transport typically involves only states at the Fermi level, vibrations can be excited at any frequency in the band structure. Right column: electronic topological transport may exploit the microscopic crystal structure, while phononic transport is engineered on larger length scales (nanoscopic up to macroscopic).

Figure 9

Chern insulators. (a) Haldane model. Excitations hop on a honeycomb lattice between nearest-neighbor and next-nearest-neighbor sites. Regions pierced by the same magnetic flux ${\mathrm{\Phi }}_1$, ${\mathrm{\Phi }}_2$, or ${\mathrm{\Phi }}_3=-6({\mathrm{\Phi }}_1+{\mathrm{\Phi }}_2)$ are highlighted in the same color. All next-nearest-neighbor transitions in the direction indicated by the arrow have the probability amplitude $J_2\exp (i\varphi )$, with $\varphi =2\pi ({\mathrm{\Phi }}_1+2{\mathrm{\Phi }}_2)/{\mathrm{\Phi }}_0$ and ${\mathrm{\Phi }}_0$ the magnetic-flux quantum. (b) Band structure (right sketch) of the semi-infinite translationally invariant strip configuration (left sketch; infinitely extended in the horizontal direction). The topological edge mode (red curve) connects subsequent bulk bands. (c)–(f) Different mechanisms to break the time-reversal symmetry in topologically protected phononic systems: (c) The rotating blue cylinder generates a fluid flow that results in a different sound speed along the two circulation directions of the acoustic waves, thus breaking time-reversal symmetry ([65]; [103]; [157]; [23]). (d) Precession of a spinning gyroscope, part of a lattice connected with springs ([101]; [143]; [97]). (e) Rotation of a lattice to create a Coriolis force acting on spring-coupled masses ([64]; [144]). (f) Time-dependent modulation of lattice sites in a chiral fashion, such as using optomechanical interactions and suitable illumination or other mechanisms ([111]; [37]; [29]; [90]).

Figure 10

Phononic spin-Hall Hamiltonians. (a) Two-copy version of the Hofstadter model. The phases $\pm \varphi$ acquired by a pseudospin up and down hopping in the direction of the arrows for the three types of links (with varying shading) are indicated. Overall a pseudospin up (down) going around any plaquette acquires the Berry phase $2\pi /3$ ($-2\pi /3$). (b) The two modes $x$, $y$ were realized by [132] as two one-dimensional pendula. (c) Definition of the matrix $K$. The unit cells are highlighted in gray. (d) Band structure (right sketch) of the semi-infinite strip configuration (left sketch) featuring topological edge states (red and orange arrows).

Figure 11

Valley Hall effect. (a) Effect of different symmetries on the Dirac cones. The degeneracy splits on breaking the ${\mathcal{C}}_2$ or ${\mathcal{C}}_v$ symmetry of the original ${\mathcal{C}}_6$ or ${\mathcal{C}}_{3v}$ crystal. The shading of the resulting hyperbolic bands encodes the valley Chern number. (b) Left sketch: translationally invariant strip including two domains of opposite mass, with a helical edge channel arising at the domain interface (with the propagation direction depending on the valley index). Right sketch: the distinct valley Chern number $C_{\mathrm{K}}$ of the band for each domain. (c) Array of triangular scatterers for the acoustic waves discussed by [80]. The scatterers are rotated to break the mirror symmetry $C_v$, which opens a gap in the Dirac cones. (d) Implementation of the valley Hall effect in an optomechanical crystal ([120]). The silicon slab consists of patterned holes of two different sizes, which are used to manipulate the Dirac cones. The vibrations are measured optically via the photonic-crystal cavity mode.

Figure 12

Zone folding. (a) Left sketch: unit cell of an array of rod-shaped scatterers for acoustic waves used by [33]. Right sketch: description of the Dirac cone in $k$ space for both the primitive (blue inner) and a larger (red outer) real-space unit cell, indicating the folding back to the $\mathrm{\Gamma }$ point. (b) Breaking the original discrete translational symmetry ${\mathcal{T}}_a$ by making the central rod smaller induces a gap in the Dirac cone. (c) Domain-wall configuration consisting of two domains with an inverted band arrangement. The radius of the central rod relative to that of its surroundings is varied in the two domains. The helical edge states appear in the bulk band gap of the strip band structure. (d) Schematic implementation of the zone-folding scheme for elastic waves on a patterned plate, as used in the experiment of [18]. The distance $w$ of the etched holes (white dots) from the blue center unetched region is varied to break the translational symmetry.

Figure 13

Accidental degeneracy of Dirac cones. (a) Unit cell of an array of rods with ${\mathcal{C}}_{6v}$ symmetry acting as scatterers for acoustic waves ([51]). (b) Left sketch: the ratio of the diameter to the periodicity is varied to observe the band inversion via a double Dirac cone at the $\mathrm{\Gamma }$ point. As in the zone-folding scheme, helical edge states appear in the bulk band gap. (c),(d) Metal plates patterned with holes of two sizes: large holes are used to obtain the Dirac cones at the $\mathbf{K}$ point for modes that are symmetric and antisymmetric about the $x$-$y$ plane, as suggested by [99]. The smaller holes [the inset in (c)] are used to engineer the dispersion for making the two cones identical. One can pattern partial holes to break the mirror symmetry $M_z$, and thereby gap the double Dirac cones ([96]).

Figure 14

Pseudomagnetic fields. (a) Upon breaking the ${\mathcal{C}}_3$ point-group symmetry, the Dirac cones shift in the Brillouin zone. The displacement from the valleys is given by $\mathbf{A}=(A_x,A_y)$ (here $A_y=0$). (b) Scheme proposed by [16] to displace the Dirac cones in a phononic crystal. The horizontal arm length $L(y)$ of the snowflake-shaped hole is modified in a spatially dependent way, thereby inducing a spatial dependence $A_x(y)$. (c) Left sketch: translationally invariant strip with smooth boundaries. The gray scale encodes the vector potential $A_x$ (for an out-of-plane constant magnetic field $\mathbf{B}=B_z{\mathbf{e}}_z$). Middle sketch: the position dependence of the mass parameter $m$ defines the smooth boundaries. Right sketch: band structure in the vicinity of the $\mathbf{K}$ point. The bulk band gap between the $n=0$ and $n=-1$ Landau levels is highlighted by shading. Away from the $\mathbf{K}$ point the Landau levels turn into gapless edge states. The semiclassical orbits for three modes are displayed in (c).

Figure 15

Challenges for topological phonon transport. (a) Effects of geometrical disorder in time-reversal symmetric systems. Disorder coupling opposite pseudospin directions also breaks Kramers degeneracy (top sketch), and thus could induce backscattering (bottom bottom sketch). (b) Spectral isolation. Phonon band structure of an actual crystal featuring multiple Dirac cones at the $\mathbf{K}$ point in the Brillouin zone. The spectrally isolated Dirac cone is marked by a circle. (c),(d) Backscattering for edge states based on Dirac engineering. (c) Transport with tight vs weak transverse confinement. The reflection is guaranteed to be small only if the transverse localization length $\xi$ exceeds the lattice spacing $a$. The reflection also depends on the domain-wall orientation (the thickness of the arrows represents the wave’s intensity). (d) Backscattering for a smooth domain wall and two values of the carrier frequency $\omega$. The reflection is enhanced at certain locations (the short red segments) that depend on both the frequency and the domain-wall orientation $\phi$ (relative to the microscopic lattice).

Figure 16

Applications of topological waveguides. (a) The advantage of topological waveguides is that the transport is robust against sharp turns, permitting compact devices and some insensitivity to fabrication imperfections. (b) Topological delay lines constructed by elongating the path length with multiple loops between source and detector. (c) Topological directional antenna in a time-reversal preserved topological waveguide. The path between the source and the detector through the waveguide is indicated by the arrow. (d) Nonreciprocal amplifier. The signal is amplified when it travels from the source to the detector, but any noise going the other way will be deamplified, thus protecting the source.