Topological phases in acoustic and mechanical systems

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The study of classical wave physics has been reinvigorated by incorporating the concept of the geometric phase, which has its roots in optics, and topological notions that were previously explored in condensed matter physics. Recently, sound waves and a variety of mechanical systems have emerged as excellent platforms that exemplify the universality and diversity of topological phases. In this Review, we introduce the essential physical concepts that underpin various classes of topological phenomena realized in acoustic and mechanical systems: Dirac points, the quantum Hall, quantum spin Hall and valley Hall effects, Floquet topological phases, 3D gapless states and Weyl crystals.

Key points

  • Acoustic and mechanical systems are versatile platforms to study a wide range of topological phases that were first investigated in condensed matter physics.

  • Topological phenomena that can be observed include Dirac points and analogues of the quantum Hall effect, quantum spin Hall effect, valley Hall effect, Floquet topological phases, gapless states and Weyl systems.

  • Because classical acoustic systems are different from condensed matter systems (for example, they lack Kramers degeneracy), new approaches are needed to realize topological phases.

  • Schemes of symmetry breaking in phononic crystals play a key role in the realization of these topological phases, and their consequences and limitations are discussed.

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Fig. 1: Geometric phase in a 1D phononic crystal.
Fig. 2: Dirac cone dispersion.
Fig. 3: Breaking time-reversal symmetry.
Fig. 4: Acoustic and mechanical analogues of the quantum spin Hall effect and valley Hall effect.
Fig. 5: Acoustic Weyl semimetal.

Change history

  • 15 April 2019

    This article has been corrected to add a missing image credit to the caption of Fig. 4. The credit line of Fig. 4 now reads “Panel d is adapted from ref.78, CC-BY-4.0 and an image courtesy of Dr Miniaci, Swiss Federal Laboratories for Materials Science and Technology.”


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The authors are grateful to R.-Y. Zhang for proofreading the manuscript. G.M. thanks W.-J. Chen, Shubo Wang and Changqing Xu for helpful discussions. G.M. is supported by the Hong Kong Research Grants Council (grant no. RGC-ECS 22302718, ANR-RGC A-HKUST601/18 and CRF C6013-18GF), the National Natural Science Foundation of China (grant no. 11802256) and the Hong Kong Baptist University through FRG2/17-18/056. M.X. is supported by the startup funding of Wuhan University and the US National Science Foundation (grant no. CBET-1641069). C.T.C. is supported by the Hong Kong Research Grants Council (grant no. AoE/P-02/12).

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Weyl points

Point degeneracies on the band structure, with local properties describable by the Weyl Hamiltonian. They can be regarded mathematically as a magnetic monopole in momentum space.

Fermi arcs

In classical wave crystals, a type of isofrequency cut present as arcs in the momentum space. In condensed matter physics, the cut of eigen-spectra is at the Fermi energy.

Adiabatic cyclic evolution

A cyclic change in a parameter that is slow enough that the physical system remains in its instantaneous eigenstate, the eigenvalue of which is not degenerate with any other eigenvalue of the Hamiltonian.

Berry phase

A phase difference acquired over an adiabatic cyclic evolution, resulting solely from the geometric properties of the parameter space of the system Hamiltonian.

Poincaré sphere

A graphical representation of all the polarization states of a pure state of light on the surface of a 3D sphere.

Berry connection

The local gauge potential for Berry curvature \({\boldsymbol{A}}=i\left\langle {u}_{{\boldsymbol{R}}}\left|{\nabla }_{{\boldsymbol{R}}}\right|{u}_{{\boldsymbol{R}}}\right\rangle \); \(\left|{u}_{{\boldsymbol{R}}}\right\rangle \) is the instantaneous eigenstate of the system with parameter R. Viewable as a momentum-space analogy to the vector potential for a magnetic field.

Surface impedance

The surface property of a truncated bulk for waves. For acoustic waves, surface impedance is defined as the ratio of the sound pressure variation to the local fluid velocity.

Good quantum number

The eigenvalue of an operator that remains unchanged as the system evolves. In the example in the main text, the system changes from a boundary-less periodic system to a truncated bulk.

Berry curvature

A local gauge field defined as \({\boldsymbol{\Omega }}=\nabla \times {\boldsymbol{A}}\), where A is the Berry connection.

Kramers pairs

In time-reversal-symmetric systems with half-integer total spins, every energy eigenstate is at least two-fold degenerate. This degeneracy is protected by time-reversal symmetry, and the two eigenstates form a Kramers pair.

Hofstadter model

A mathematical model describing the behaviour of electrons in a magnetic field in a 2D lattice. The energy levels form a fractal set.

Duality symmetry

Symmetry such that Maxwell equations of a source-free system are invariant under εμ, με, EB and B→−E, where ε is the dielectric constant, μ is the permeability, E is the electric field and B is the magnetic field.

Lamb modes

Two types of mode of waves in a solid plate, one symmetric about the plate mid-plane (characterized by breathing vibrations) and one antisymmetric (characterized by bending vibrations).

Poynting vectors

Vector fields representing the direction and amplitude of local energy flow. In acoustics, the vector field is defined as S = Pv, where P is the pressure variation and v is the local fluid velocity.

Time-ordering operator

A mathematical representation of a procedure that orders the product of a series of operators according to the time sequence of these operations.

Screw symmetry

A combination of rotation about an axis and a translation parallel to that axis that leaves a crystal unchanged.

Nodal surface

Two or more states form a nodal surface when they are degenerate over a continuous range of momenta that form a surface in the momentum space.

Isostatic lattices

Lattices consisting of point masses connected by rigid bonds or central-force springs, in which the number of bonds equals the total number of degrees of freedom of the masses.

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Ma, G., Xiao, M. & Chan, C.T. Topological phases in acoustic and mechanical systems. Nat Rev Phys 1, 281–294 (2019) doi:10.1038/s42254-019-0030-x

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