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# Pseudomagnetic fields for sound at the nanoscale

Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved March 10, 2017 (received for review September 27, 2016)

## Significance

Unlike electrons, phonons do not feel a magnetic field, because they are not charged. Thus, much of the physics connected with charged particles in magnetic fields is absent for phonons, be it the Lorentz force or the unidirectional transport along the sample edges. Recently, researchers have started to study how one might make phonons akin to electrons in a magnetic field or related topological settings. The first experimental implementations involving coupled pendula or gyroscopes or air currents have already been realized at the macroscopic scale. Here, we describe a design that is well-suited for the nanoscale. It is purely geometric in nature and could be implemented based on an already experimentally demonstrated platform, a simple patterned 2D material.

## Abstract

There is a growing effort in creating chiral transport of sound waves. However, most approaches so far have been confined to the macroscopic scale. Here, we propose an approach suitable to the nanoscale that is based on pseudomagnetic fields. These pseudomagnetic fields for sound waves are the analogue of what electrons experience in strained graphene. In our proposal, they are created by simple geometrical modifications of an existing and experimentally proven phononic crystal design, the snowflake crystal. This platform is robust, scalable, and well-suited for a variety of excitation and readout mechanisms, among them optomechanical approaches.

Unlike electrons, phonons do not feel a magnetic field, because they are not charged. As a consequence, much of the interesting physics connected to the behavior of charged particles in a magnetic field is absent for phonons, be it the Lorentz force or the unidirectional transport along the edges of the sample. In the past two years, researchers have started to study how one might make sound waves behave in ways similar to electrons in a magnetic field or related topological settings. This research promises to pave the way toward transport along edge channels that are either purely unidirectional (1) or helical (2) (i.e., with two “spins” moving in opposite directions), as well as the design of novel zero-frequency boundary modes (3, 4). The first few experimental realizations (1, 2, 4⇓⇓⇓–8) and a number of theoretical proposals (3, 9⇓⇓⇓⇓⇓⇓⇓⇓–18) involve macroscopic setups. These include coupled spring systems (1, 2, 11⇓–13, 19) and circulating fluids (9, 10, 16, 20); for a review see ref. 21. These designs represent important proof-of-principle demonstrations of topological acoustics and could open the door to useful applications at the macroscopic scale. However, they are not easily transferred to the nanoscale, which would be even more important for potential applications. First concepts for Chern insulators or topological insulators at the nanoscale are challenging and have not yet been implemented, because they require either strong laser driving (22) or designs that are hard to fabricate (23).

Fruitful inspiration for an entirely different avenue toward nonreciprocal transport can be found in the physics of electrons propagating on the curved surface of carbon nanotubes (24) and in strained graphene (25⇓⇓–28). It has been discovered that these electrons experience so-called pseudomagnetic fields, whose distribution depends on the strain pattern. Pseudomagnetic fields mimic real magnetic fields but have opposite sign in the two valleys of the graphene band structure and, thus, do not break time-reversal symmetry. This results in helical transport, with two counterpropagating species of excitations. In the past, this concept has already been successfully transferred to engineered carbon monoxide molecular graphene (29) and a photonic waveguide system (30). The idea of pseudomagnetic fields generated by distortions is so powerful because it can be implemented entirely using a purely geometrical approach, without any external driving, which is a crucial advantage at the nanoscale.

In this paper, we show how to engineer pseudomagnetic fields for sound waves at the nanoscale. In addition, it turns out that our design will be realizable in a platform that has already been fabricated and reliably operated in experiments, the snowflake phononic crystal (31). That platform has the added benefit of being a well-studied optomechanical system, which, as we will show, can also provide powerful means of excitation and readout. Besides presenting our nanoscale design we also put forward a general approach to pseudomagnetic fields for Dirac quasiparticles based on the smooth breaking of the appropriate point group (the

## Results

### Dirac Equation and Gauge Fields.

The 2D Dirac Hamiltonian in the presence of a gauge field

In a condensed matter setting, the Dirac Hamiltonian describes the dynamics of a particle in a honeycomb lattice, or certain other periodic potentials, within a quasimomentum valley (i.e., within the vicinity of a lattice high-symmetry point in the Brillouin zone). In this context, **1** is defined in two different valleys mapped into each other via the time-reversal symmetry operator

### Dirac Phonons in the Snowflake Phononic Crystal.

Finite element method (FEM) mechanical simulations of a silicon thin-film snowflake crystal are presented in Fig. 1. Throughout this paper, we restrict our attention to the modes that are even under the mirror symmetry *SI Appendix*). The mechanical band structure is shown in Fig. 1*C*. It features a large number of Dirac cones at the high-symmetry point *C*). In other words, the mass

In preparation of this, we use the snowflake radius as a knob to engineer a pair of Dirac cones that are spectrally well-isolated from other bands and have a large velocity. The snowflake crystal can be viewed as being formed by an array of triangular membranes arranged on a honeycomb lattice and connected through links (Fig. 1*B*). In principle, we could choose a situation where the links are narrow (large snowflake radius *A*). This observation explains the emergence (see *r* = 180 nm plot of Fig. 1*E*) of a group of three bands, separated from the remaining bands by complete band gaps, and supporting large-velocity Dirac cones. The triplet of isolated bands can be fitted well by a Kagome lattice tight-binding model with nearest-neighbor and next-nearest-neighbor hopping. The Kagome lattice model would be entirely sufficient to guide us in the engineering of the desired gauge fields. However, we prefer to pursue a more fundamental and general approach based on the symmetries of the underlying snowflake crystal.

### Identifying the Dirac Pseudospin by the Symmetries.

The snowflake thin-film slab crystal has *B* are usually referred to as essential degeneracies. They are preserved if the point group includes at least the

The Dirac Hamiltonian [**1**] for a given valley *SI Appendix*). In other words, the quasi-angular momentum

In our phononic Dirac system, the eigenstates *B*). Three snapshots of the instantaneous displacement field for the state with *D*–*F*. By definition of a quasi-angular momentum eigenstate with *B* and *D*–*F*). (For a detailed explanation see *SI Appendix*.) Below, we will take advantage of our insight into the symmetries of the pseudospin eigenstates to engineer a local force field which selectively excites unidirectional waves.

### Pseudomagnetic Fields and Symmetry Breaking.

A crucial step toward the engineering of a pseudomagnetic field is the implementation of a spatially constant vector potential

A perturbation that breaks the *C*). This can be identified with the appearance of a constant gauge field **1**]. As such, the connection between changes in the microscopic structure of the phononic metamaterial and the resulting gauge field can be obtained from FEM simulations by extracting the quasi-momentum shift of the Dirac cones. We emphasize that, in this context,

In the snowflake phononic crystal, we can achieve the desired type of symmetry breaking (breaking *A*). If only one of the arms is changed, symmetry requires that the vector potential *D*. For the Dirac cones associated with the Kagome lattice, our FEM simulations show that the cone displacement grows linearly with the length changes, as long as these remain much smaller than the average arm length

### Phononic Landau Levels and Edge States in a Strip.

We can test these concepts by implementing a constant phononic pseudomagnetic field in an infinite snowflake crystal strip, where we can directly test our simplified description against full microscopic simulations. The strip is of finite width

The treatment of the boundaries merits special consideration. We recall that particles in pseudomagnetic fields are guaranteed to emulate their topologically protected counterparts (i.e., charged particles in real magnetic fields) only when the system parameters are varied smoothly such that no intervalley coupling is introduced. Clearly, sharp boundaries can lead to such undesired coupling. Indeed, it has been shown for graphene, in the case of real magnetic fields, that the edge states localized close to sharp armchair boundaries are not valley-polarized (37). Such valley mixing would arise also in the present system if sharp boundaries are chosen. Here, we address this challenge by engineering smooth boundaries. A notable advantage of our platform is that this task is completely straightforward. Any smooth gradient of snowflake parameters near the boundaries of the sample will lead to well-defined edge states that are spatially separated from the physical system boundary. In general, this could involve both a potential gradient as well as a gradient in the effective mass (gap). In our simulations, we will display results obtained for a smooth mass gradient, whose details do not matter for the qualitative behavior. This gives rise to magnetic edge states that can be easily addressed, because they have only negligible spatial overlap with their nonmagnetic counterparts. A Dirac mass term appears upon breaking the *B*, with the displacement varying smoothly in the interval

By changing the snowflake arm lengths we can displace the Dirac cones only over a finite range of quasi-momenta. In our simulations *D*. Using Eq. **2** and the definition of the magnetic length

In Fig. 3 we display the phonon band structure and the phonon wave functions (mechanical displacement fields) extracted from finite-element numerical simulations as a function of the quasi-momentum **2**; see Fig. 3 *A* and *B*). The Landau plateaus extend over a quasi-momentum interval of width *C*, where we show the displacement field of the central Landau level. A zoom-in of this field (Fig. 3*D*) reveals that, at the lattice scale, it displays the same intensity pattern as the bulk pseudospin eigenstate *F*. This behavior is also predicted by the effective Dirac description where the central Landau level is indeed a pseudospin eigenstate with

Having demonstrated that we can implement a constant phononic pseudomagnetic field in the bulk, we now discuss the resulting physics at the boundary. Each Landau level gives rise to an edge state in the region of the smooth boundaries. The typical behavior of the wavefunction is shown (for *C*. For increasing quasi-momenta *A* and *B*. Although this may seem trivial, it is by no means what happens if we were to consider sharp boundaries. In fact, both for our system as well as for graphene in a pseudomagnetic field (28), sharp boundaries lead to a peculiar feature. In these cases, for zig-zag edges, there is a different number of edge states on the opposite sides of a strip. The two edges of the strip are not on equal footing because a pseudomagnetic field changes sign under a *SI Appendix*). This localization is induced by the intervalley coupling at the sharp edges and can be prevented by using smooth boundaries, leading to the robust behavior displayed in Fig. 3.

A characteristic feature of the band structure of a finite system with smooth boundaries in a pseudomagnetic field is that there is a band gap without edge states immediately below or above the *A* and *B*). In our approach, the qualitative behavior of the edge states originating from the **1**).

### Transport in a Finite Geometry and Disorder.

Any pseudomagnetic field that is realized without explicit time-reversal symmetry breaking necessarily gives rise to helical transport, where the chirality depends on some artificial spin degree of freedom (i.e., the valley). A central question in this regard is the robustness against short-range disorder. To assess this, we have studied numerically transport in a finite geometry. As presented in Fig. 4, we consider a sample of hexagonal shape with smooth boundaries in the presence of a constant pseudomagnetic field (we choose the symmetric gauge for the vector potential *A*. Its frequency is chosen to lie inside the bulk band gap separating the *B*–*D*). There is a phase delay of

It turns out to be most efficient (and entirely sufficient) to implement the numerical simulations for these rather large finite-size geometries with the help of a tight-binding model on a Kagome lattice (*SI Appendix*). The parameters of that model can be matched to full FEM simulations that have been performed for the translationally invariant case. This allows us to systematically study the effects of disorder. In the presence of moderate levels of smooth disorder, which does not couple the two valleys, the nature of the underlying magnetic field (pseudo vs. real) will not manifest itself and the transport will largely be immune to backscattering. Here, we focus instead on lattice-scale disorder that can lead to scattering with large momentum transfer that couples the two valleys and thereby leads to backscattering. We emphasize that short-range disorder will, in practice, introduce backscattering in any purely geometric approach to acoustic helical transport. In particular, this also includes acoustic topological insulators, where generic disorder will break the unitary symmetry that ultimately protects the transport (38). To quantify the effect of lattice scale disorder, we consider a setup with two drains, one to the clockwise and one to the counterclockwise direction, as shown in Fig. 4*B*. In the absence of disorder, the vibrations travel clockwise (in this example) and are absorbed in the right drain; only very weak residual backscattering occurs at the sharp hexagon corners. In the presence of lattice-scale disorder, a portion of the excitations will be backscattered and subsequently reach the left drain. In Fig. 4*C* we plot the portion

## Discussion

We emphasize once more that one of the great practical advantages of the pseudomagnetic-field-based design put forward here is that it relies entirely on a simple geometry. Because time-reversal symmetry is not broken, there is no need for driving. This is in contrast to the first proposal for engineered chiral sound wave transport at the nanoscale (22), where a patterned slab illuminated by a laser realizes a Chern insulator for sound. The laser drive in ref. 22 breaks the time-reversal symmetry, enabling unidirectional topologically protected transport, but it also requires a suitable engineering of the wavefront and rather strong intensities. The other existing proposal for topologically protected sound transport in a 2D nanoscale lattice can do without driving, because it is based instead on a topological insulator (23). However, it involves a geometry that seems to be hard to fabricate, in contrast to the design described here, which is based on an experimentally proven simple structure. In addition, of course, a major difference between the topological insulator and our pseudomagnetic field approach is that the present design offers access to the entire physics of the quantum Hall effect, such as spatially inhomogeneous magnetic field distributions or flat bulk Landau levels. The latter may become particularly interesting for situations with interactions between the excitations.

Because our design is scale-invariant, a variety of different implementations can be easily envisioned. At the nanoscale, the fabrication of thin-film silicon snowflake crystals and resonant cavities has already been demonstrated with optical readout and actuation (31). At the macroscale, the desired geometry could be realized using 3D laser printing and similar techniques. A remaining significant challenge relates to the selective excitation of helical sound waves and the subsequent readout. In an optomechanical setting, the helical sound waves can be launched by carefully crafting the applied radiation pressure force. For the typical dimensions of existing snowflake optomechanical devices operating in the telecom wavelength band (lattice spacing *SI Appendix*). Helical sound waves can then be launched by either directly modulating the light intensity or a photon–phonon conversion scheme, using a strong red-detuned drive, with signal photons injected at resonance. In the micron regime one can benefit from electromechanical interactions. A thin film of conducting material deposited on top of the silicon slab in combination with a thin conducting needle pointing toward the desired triangles forms a capacitor. In this setting, an a.c. voltage would induce the required driving force. The vibrations could be read out in the same setup as they are imprinted in the currents through the needles. In a different electromechanical approach, the phononic crystal could be made out of a piezoelectric material and excitation and readout occur via piezoelectric transducers (39).

We have shown how to engineer pseudomagnetic fields for sound at the nanoscale purely by geometrical means in a well-established platform. Our approach is based on the smooth breaking of the

## Acknowledgments

This work was supported by European Research Council Starting Grant OPTOMECH (to V.P., C.B., and F.M.) and by the European Marie-Curie Innovative Training Network cQOM (F.M.). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant 732894 (Hybrid Optomechanical Technologies) (to V.P. and F.M.). This work was also supported by the Air Force Office of Scientific Research–Multidisciplinary University Research Initiative (MURI) Quantum Photonic Matter, Army Research Office–MURI Quantum Opto-Mechanics with Atoms and Nanostructured Diamond Grant N00014-15- 1-2761, the Institute for Quantum Information and Matter, and NSF Physics Frontiers Center Grant PHY-1125565 with support of Gordon and Betty Moore Foundation Grant GBMF-2644 (all to O.J.P.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: christian.brendel{at}mpl.mpg.de.

Author contributions: V.P., O.J.P., and F.M. designed research; C.B., V.P., and F.M. performed research; C.B., V.P., and F.M. analyzed data; and C.B., V.P., O.J.P., and F.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1615503114/-/DCSupplemental.

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