Strain-Induced Gauge Field and Landau Levels in Acoustic Structures

The emerging field of topological acoustics that explores novel gauge-field-related phenomena for sound has drawn attention in recent years. However, previous approaches constructing a synthetic gauge field for sound predominantly relied on a periodic system, being unable to form a uniform effective magnetic field, thus lacking access to some typical magnetic-induced quantum phenomena such as Landau energy quantization. Here we introduce strain engineering, previously developed in graphene electronics and later transferred to photonics, into a two-dimensional acoustic structure in order to form a uniform effective magnetic field for airborne acoustic wave propagation. Landau levels in the energy spectrum can be formed near the Dirac cone region. We also propose an experimentally feasible scheme to verify the existence of acoustic Landau levels with an acoustic measurement. As a new freedom of constructing a synthetic gauge field for sound, our study offers a path to previously inaccessible magneticlike effects in traditional periodic acoustic structures.

Figure 1

(a) Top and side views of a unit cell of the acoustic honeycomb lattice. The unit cell consists of two identical resonators at inequivalent sites. Each pair of nearest-neighbor resonators is connected with a thin coupling waveguide. (b) The Dirac cone spectrum in a square area with a side length of $0.4\pi /a$ locates at a corner of the Brillouin zone. (c) Triaxial-strain-induced acoustic Landau levels near the Dirac region. A disk with radius $40a_0$which consists of 11 600 resonators is adopted in the calculation. The strength of the triaxial strain is $q=0.004/a$. The numbers indicate the orders of Landau levels.

Figure 2

(a)–(c) Schematics of the unstrained and strained acoustic lattices with increasing strength of triaxial strain (a) $q=0.000/a$, (b) $q=0.015/a$, and (c) $q=0.030/a$. A disk is adopted in the calculation. It has a radius of $6a_0$ which consists of 262 resonators. (d)–(f) The eigenvalues of the acoustic lattices are plotted in terms of the state number in the ascending order. (d) Without strain, the frequency spectrum shows a continuous behavior. (e),(f) The band gaps emerge and widen with increasing strain.

Figure 3

The logarithmic plots of inverse participation ratio values for the strain strength of (a) $q=0.000/a$, (b) $q=0.015/a$, and (c) $q=0.030/a$, respectively. With increasing IPR values, the eigenstates become more and more localized. (d)–(f) The localization features of the eigenstates with different IPR values for the lattice with a strength of strain (d) $q=0.000/a$, (e) $q=0.015/a$, and (f) $q=0.030/a$. $X$ represents the state number, and $Y$ is ${\log }_{10}(\mathrm{IPR})$. The thermal color indicates the amplitude of the acoustic pressure.

Figure 4

(a)–(c) The excitation of acoustic waves at the edges of the unstrained and strained lattices with the strength of (a) $q=0.000/a$, (b) $q=0.015/a$, and (c) $q=0.030/a$. The operating frequency of the source is $0.281\times 2\pi c/a_0$ (5565 Hz). The black arrow points to the location of the acoustic source. The extended state is excited in panel (a). Highly confined states are excited in panels (b) and (c), because the frequency locates in the band gap between 0th-order and 1st-order Landau levels. (d)–(f) In the aperiodic lattice with a strain strength of $q=0.015/a$, the excitation frequency of the source is (d) $0.237\times (2\pi c/a_0)$, (e) $0.261\times (2\pi c/a_0)$, and (f) $0.293\times (2\pi c/a_0)$, respectively. The intensity profiles show the transition from localization (d) to spreading (e) to localization (b) to spreading (f). The thermal color indicates the amplitude of the acoustic pressure.