Topological sound in two dimensions

Topology is the branch of mathematics studying the properties of an object that are preserved under continuous deformations. Quite remarkably, the powerful theoretical tools of topology have been applied over the past few years to study the electronic band structure of crystals. Topological band theory can explain and predict topological phase transitions in a material, and the unusual robustness of certain band structure shapes, such as Dirac cones, against small perturbations. These findings have also unveiled a new phase of matter—topological insulators—whose exotic transport properties at their boundaries are topologically protected against imperfections and disorder. The fascinating features of topological boundary states have triggered the search for their analogs in classical wave physics. Here, we focus on the peculiar features of two‐dimensional topological insulators for sound and mechanical waves. Two‐dimensional Dirac cones and phononic topological insulators can emerge under certain conditions in periodic acoustic metamaterials, demonstrating great potential for acoustic and mechanical systems to demonstrate, over a tabletop platform, complex fundamental phenomena driven by topological concepts. In addition, these discoveries offer a direct path toward new technologies for enhanced sound control and manipulation.


INTRODUCTION
In recent years, condensed matter physics has experienced the breakthrough of two-dimensional (2D) quantum materials and of their topological states, whose robust electron transport is immune from disorder. Such discoveries have nurtured great excitement for the future of quantum devices and technologies. [1][2][3] Topological physics concepts have naturally expanded into the classical realm, including photonics, acoustics, and mechanics, where exciting phenomena predicted by topological band theory can be exploited to control the dynamics of classical waves and are far less challenging to experimentally concepts associated with the topological description of band structures in periodic systems. Then, we discuss the analog of Dirac cone physics in acoustic and elastic metamaterials. Once the existence of phononic Dirac points is established, we break their inherent degeneracy following different protocols, leading to a plethora of topological phases for sound and elastic waves. We can break time-reversal symmetry  to get the analogs of Chern insulators for sound, or remove inversion symmetry  to achieve the phononic valley-Hall effect, or introduce acoustic pseudo-spins to mimic the quantum spin-Hall effect (QSHE). Lastly, we show that it is possible to go beyond simply mimicking established condensed matter 2D topological concepts, and we review recent advances peculiar to 2D phononic topological systems, such as higher-order topological acoustic modes and non-Hermitian wave phenomena. In this paper, we present a broad overview of these phenomena and the corresponding platforms used for their experimental implementation, yet the interested readers are recommended to consult the references for a more in-depth discussion of these fascinating phenomena emerging in structured materials to control sound.

TOPOLOGICAL BAND THEORY
Topology is a universal concept in mathematics dealing with the relation between components of space (geometry). As such, it does not deal with the details of a specific geometrical shape, but it rather introduces global features that can describe widely different geometries under the same principles. Topological invariants-powerful tools introduced by topology-are quantized values describing these global features, which do not change under continuous deformations of the underlying geometrical shape. One canonical example is the features of doughnuts, mugs, and pretzels: the first two belong to the same topological family since they are characterized by a single hole. A pretzel, however, has three holes, which puts it into another topological family.
Objects belonging to the same topological class, like a mug and a doughnut, share profound similarities that are described by their topological invariant.
In physics, particularly in condensed matter, this concept has been extended to define interesting features of the electronic energy bands of quantum materials, 1,14 resulting in the tantalizing discovery of new phases of matter known as topological insulators. 15 The underlying theory, known as topological band theory, 16 predicts the emergence of exotic electron conduction states at arbitrarily shaped boundaries of these topological insulators, associated with the topological features of the bulk properties, described by a topological invariant for their energy bands. Hence, studying the topological features of the infinite medium determines how the electrons behave when a finite volume is arbitrarily shaped.
Graphene-a 2D hexagonal lattice made out of carbon atoms 17 -is a good starting example to unveil the unusual physics of topological insulators, as it allows to elegantly illustrate how topological band theory can be powerfully employed to describe its extraordinary electronic properties. The electronic π-orbital band structure of graphene, with its hexagonal unit cell shown in the inset of Figure 1A, is well known for hosting low-energy excitations associated with the two-fold degeneracy at a singular point in the energy bands 18,19 known as the Dirac cone. Remarkably, this singular point mimics the properties of massless fermions in quantum electrodynamic systems. Symmetries ensure that two-fold Dirac cones cannot exist in odd numbers in a 2D system, due to the constraints associated with time-reversal symmetry ( symmetry), which flips the time flow from the future to the past, and inversion symmetry ( symmetry), which switches the signs of spatial coordinates. Therefore, Dirac cones must come in pairs and are commonly located at the border of the Brillouin zone (BZ), the primitive cell in the reciprocal space of the lattice. In the case of graphene, this BZ takes the shape of a hexagon and Dirac cones exist at its corners, namely the K and Kʹ points ( Figure 1A) Dirac cones carry topological charges, known as Berry phases, which can be evaluated as γ n = ∮ Γ A n (k).dk following a closed path Γ in the BZ, with A n (k) being the Berry connection defined as A n (k) = −i⟨u n,k |∇ k |u n,k ⟩, where u n,k is the eigenstate for the nth energy band and k is the electron wave vector in the BZ. Remarkably, this quantity predicts the localization of the electron wave function at the edge of a finite size sample, depending on its orientation with regard to the lattice symmetry. 22,23 For example, when the graphene boundary is cut in a zigzag shape, zero-energy edge states arise localized at this boundary and their bands connect the two valleys, as shown by gray lines in Figure 1A.
The features described above make Dirac cones the mother states of 2D topological materials, whose tantalizing topological properties emerge from lifting their degeneracy by either breaking  or  symmetry in the material. The first canonical model of a 2D topological material based on breaking  symmetry was introduced by Haldane in 1988. 24 Here, the hoppings between second-neighbor sites in graphene are considered such that a periodic local magnetic flux within the honeycomb lattice is induced. In turn, this distribution of magnetic flux opens a bandgap around the Dirac point, changing graphene into an insulator ( Figure 1B Figure 1B: the associated chiral edge states propagate unidirectionally, circumventing any defect, and continue propagating in the same direction since no states exist that can carry electronic currents in the backscattering direction at a given boundary (lower panel in Figure 1B).
Conversely, a different form of topological phase can emerge when  symmetry is broken in graphene, for example, when two atoms in each unit cell of the hexagonal lattice are no longer identical, which lifts the point degeneracy of the Dirac cone in Figure 1C. This broken symmetry results in two symmetric bands separated by a bandgap.
Although being topologically trivial on average when their Berry curvature is integrated over the entire BZ, these bandgaps are characterized by local topological features-a nonzero Berry phase around the gapped Dirac cones, leading to what is known as the valley Hall effect. 28 In this scenario, we can define a local topological invariant, the valley Chern number C K∕K ′ = ±1∕2 for each valley, whose sign flips as we analyze one or the other bandgap. The bulk-edge correspondence predicts that an interface between topologically distinct domains with opposite valley Chern numbers, as shown in the lower panel of Figure 1C, supports edge modes whose propagation direction is locked to their valley DOF, as predicted by the dispersion of valley-polarized edge states in the upper panel of Figure 1C.
Since electrons are fermions, their time-reversal symmetry operator obeys T 2 = −1 and thus allows for Kramers degeneracy of the energy bands. In this fermionic case, all eigenstates of a system with  symmetry are at least two-fold degenerate, leading to the formation of two overlapping copies of the Dirac cone, each associated with a single electronic spin. This four-fold degeneracy relaxes the constraints dictated by symmetries on the necessity of an even number of Dirac cones, hence a four-fold Dirac cone can exist alone in the BZ, as demonstrated in a semiconductor quantum well 29 ( Figure 1D)

. Such four-fold
Dirac cones open the door to another topological phase of matter: the QSHE, governed by spin-orbital interactions, a process in which the intrinsic spin and angular momentum of an electron are not independent any longer. 30 We illustrate the QSHE with the model of a four-fold Dirac cone and study the corresponding spin-orbit coupling in the system. Under this scenario, a bandgap opens with topological features characterized by a Z 2 invariant taking either value 0 or 1, or an integerquantized spin Chern number C ↑,↓ = ±1, as seen in Figure 1E. 31  Another surprising feature of acoustic Dirac cones is Klein tunneling.
The acoustic version of this intriguing phenomenon has been demonstrated using three sonic crystal slabs with carefully designed Dirac cone frequencies, 41 as shown in Figure 2D. nonreciprocal device made of a trimer of cavities whose density is modulated in time with a rotating phase pattern. 73 Arranging these circulators in a honeycomb pattern generates a so-called Floquet topological phase, whose bandgaps host one-way topological edge modes not only robust to defects but also to phase distribution disorder at the scale of the bulk ( Figure 3F). If the time modulation of acoustic properties is achievable using mechanical modulation 74 Figure 3G). In addition, taking advantage of the opto-mechanical interaction at the nanoscale, several authors have studied nonreciprocal topological phases for phonons mediated by light. [77][78][79][80][81] Finally, there is a direct mapping between Floquet Hamiltonians and scattering network systems, 82 Figure 4F).

PSEUDO-SPINS: PHONONIC SPIN-HALL INSULATORS
If phononic Chern insulators show promising transport with topologically robust properties, the fact that they rely on a strictly controlled angular momentum bias or modulation can become an issue for practical devices. Alternatively, as illustrated in Figure 1D,E, we can seek topological modes in spin systems obeying  symmetry. Unfortunately, phononic waves are bosons, meaning that in their case the time reversal operator follows T 2 = 1, thus Kramers degeneracy does not hold, preventing the existence of a natural quantum spin Hall effect for phonons and the associated topological phenomena.
Nevertheless, this challenge can be circumvented by introducing external DOFs within a tight-binding structure, a sort of pseudo-spin, by engineering couplings that mimic a spin-orbit interaction. A first example was proposed in lattices of pendula dimers coupled with designed multilayer spring arrangements 124,125 ( Figure 5A). Here, the polarizations of each dimer act as a pseudo-spin degree of freedom, and the experimental demonstration using high-speed cameras demonstrates quite remarkably the robust helical edge propagation. 124,126 A different demonstration has relied on the addition of a second layer, along with chiral interlayer couplings, 126 as shown in Figure 5B.
Remarkably to two distinct topological behaviors. An interface between the two topologically distinct media carries helical waves whose direction of propagation is locked to the sense of rotation of local acoustic vortices along the interface ( Figure 5D). However, the topological protection of these modes directly relies on the spatial symmetries of the lattice that construct the pseudo-spin states of the modes. As the interface inevitably breaks these symmetries, the topological protection is hampered, leading to a small bandgap opening between the helical edge bands, hence making these interface modes inherently less robust than their electronic counterparts or even the phononic bilayer designs mentioned above, 124,126,127 which are independent of the lattice symmetries. Nevertheless, their peculiar features allow designing topological beam splitters ruled by pseudo-spin conservation. 128 The concept of opening a nontrivial topological bandgap from two overlapping Dirac cones in the center of the BZ can also be applied within triangular lattices consisting of six "meta-atoms," which can be either shrunken or expanded. 129 In this case, the initial four-band degeneracy of the unperturbed lattice is purely due to artificial band folding.
The relative simplicity of this protocol has motivated a plethora of studies of various platforms, ranging from soda can metamaterials 130 and macroscopic phononic crystals [131][132][133][134][135][136][137][138][139][140][141][142][143] ( Figure 5E) to on-chip systems, 144-148 whose reconfigurability opens the door to practical devices. 141 tight-binding systems to the realm of phononic crystals described by Bragg interference, hence expanding the range of platforms supporting higher-order topological phases. In a related context, a second family of HOTIs is characterized by bulk polarization and it does not require negative coupling to achieve topological corner modes, since it only relies on the difference between inter-and intracell couplings. 155 This phase has also been widely investigated in phononics, [160][161][162][163][164][165][166][167][168][169][170][171][172][173] as shown in Figure 6B for the case of breathing Kagome lattices, along with lattices made of rotated scatterers [174][175][176] and combined with Floquet physics. 177 183 Notably, fine control of phase distribution within the unit cell of the lattice induces topological "audio lasing" of whispering gallery modes with a specific handedness, 184 as shown in Figure 6C. Other intriguing phenomena, such as the non-Hermitian phononic skin-effect 185-188 ( Figure 6D) or non-Hermitian higherorder acoustic phases [188][189][190] (Figure 6E), have also been investigated experimentally based on this platform. Finally, phononic metamaterials offer a straightforward platform to go beyond purely periodic topological phases and investigate the impact of the disorder on wave propagation. One example is the demonstration of phononic topological defects, which can be used as robust waveguides 191 but also to realize topological cavities induced by disclination, 192,193 as, respectively, shown in Figure 6F and G. Topological cavity modes can also be induced through a Kekulé distortion conferring on them strong robustness against spatial perturbations [194][195][196][197] ( Figure 6H). Nonperiodic phononic media have also been studied in the context of elastic Chern insulators 71 ( Figure 6I) and higher-order phases. 198

CONCLUSION AND FUTURE DIRECTIONS
In this article, we have discussed how Dirac cones in 2D electronic materials and the associated topological phases can be transposed

ACKNOWLEDGMENTS
This work was partially supported by the Simons Foundation and the Department of Defense.

AUTHOR CONTRIBUTIONS
S.Y., X.N., and A.A. contributed to the preparation of the manuscript.

COMPETING INTERESTS
The authors declare no competing interests.