Abstract
Topological metals are conducting materials with gapless band structures and nontrivial edgelocalized resonances. Their discovery has proven elusive because traditional topological classification methods require band gaps to define topological robustness. Inspired by recent theoretical developments that leverage techniques from the field of C^{∗}algebras to identify topological metals, here, we directly observe topological phenomena in gapless acoustic crystals and realize a general experimental technique to demonstrate their topology. Specifically, we not only observe robust boundarylocalized states in a topological acoustic metal, but also reinterpret a composite operator—mathematically derived from the Ktheory of the problem—as a new Hamiltonian whose physical implementation allows us to directly observe a topological spectral flow and measure the topological invariants. Our observations and experimental protocols may offer insights for discovering topological behaviour across a wide array of artificial and natural materials that lack bulk band gaps.
Similar content being viewed by others
Introduction
Over the past two decades, immense progress has been made in predicting and observing topological phases of matter and their associated boundarylocalized states in insulators^{1,2,3,4,5} and semimetals^{6,7,8,9,10,11,12,13,14,15,16,17,18}. These developments have been predicated upon the spectral isolation of the topological phenomena in these classes of systems; although materials such as Dirac semimetals^{6,7,8}, Weyl semimetals^{9,10,11,12,13,14,15}, and nodal line/ring semimetals^{16,17,18} generally do not possess complete gaps in their band structures, the topological phenomena that manifest in these systems nevertheless appear within incomplete band gaps, allowing their boundarylocalized states to be uniquely identified at some energy and quasimomentum. In contrast, the lack of band gaps (or, more generally, mobility gaps) in metals and other types of gapless systems has made their topological analysis extremely challenging and presently there are concentrated efforts in this direction^{4}. While previous works have studied the bulk properties of topological metals, such as their topological responses and their relation to the geometry and topology of the Fermi surface^{19,20,21,22,23}, the focus of our work is on topological bulkboundary correspondence in metals. As such, for the purposes of this study, we are using ‘topological metal’ to specifically refer to systems that exhibit a bulkboundary correspondence that can be predicted using an invariant determined in the system’s bulk, but whose topologically protected boundarylocalized states or resonances are always degenerate in both energy and wave vector with bulk states. (In contrast, topological states in insulators and semimetals exhibit a range of energies and wave vectors where no bulk states exist). From a theoretical perspective, the absence of spectral or dynamical gaps essentially precludes the use of topological band theory to identify the invariants of these systems, and prohibits such theories from predicting a measure of topological protection. Moreover, any boundarylocalized phenomena in gapless systems will generally hybridize with the degenerate bulk states to create boundarylocalized resonances, which complicates their experimental observation. Thus, despite the enormous advances that have been made in topological materials, the study of topological metals has remained almost entirely unexplored.
Recently, a general theoretical method for evaluating the topology of metallic and gapless systems was put forward, opening new opportunities for discovering topology in this class of systems that could not be previously explored^{24}. This theoretical framework is rooted in the system’s spectral localizer, which makes use of the system’s realspace description and yields local invariants (synonymous with local markers) that are protected by local gaps^{25,26,27}. The key concept that links traditional band theoretic approaches to this local understanding of a system’s topology stems from a dual description of atomic limits, i.e., the limit where a complete basis of spatially localized Wannier functions exists. In band theoretic approaches, a group of bands is topologically trivial if they can be continued to an atomic limit without closing the band gap or breaking a symmetry; any obstruction to this continuation manifests as a nontrivial invariant^{28,29,30,31}. From a realspace operator perspective, an atomic limit’s complete set of Wannier functions each have both a welldefined position and energy (in crystals they can be expressed as a flat band^{32}). Thus, a ddimensional atomic limit’s Hamiltonian, H^{(AL)}, commutes with its position operators, \({X}_{j}^{({{{{{{{\rm{AL}}}}}}}})}\), \([{H}^{({{{{{{{\rm{AL}}}}}}}})},{X}_{j}^{({{{{{{{\rm{AL}}}}}}}})}]=0\), for all j ∈ 1, …, d. Using this mathematical observation, the spectral localizer ascertains a system’s topology by determining whether there is an obstruction in continuing its H and X_{j} to be commuting (given similar restrictions as before), an analysis that can be performed using recent developments from the study of C^{∗}algebras^{25,33}; any obstruction to continuing the system to possess commuting matrices yields a nontrivial invariant.
If a system’s topology is instead linked to its multiple inequivalent atomic limits (e.g. as is common in systems with chiral or crystallinebased topology), a system is only considered trivial if it can be continued to a chosen trivial atomic limit. Such cases are automatically handled by the spectral localizer, where it is necessary to explicitly specify the system’s boundary and the grading operator that defines the chiral or crystalline symmetry to evaluate the possibility of continuation. This pair of choices effectively fixes which atomic limit is considered to be trivial, and the spectral localizer ascertains whether there is an obstruction to continuing a given system to this trivial atomic limit. Altogether, by recasting the determination of a system’s topology to real space, and not relying upon a bulk band gap to be the measure of topological protection, the spectral localizer is equally applicable to insulators and metals—meaningful local gaps that protect nontrivial topology can be found in both cases^{24}. However, despite the prediction that the local topological invariants of gapless systems can be robust against system perturbations, robust topological metals that possess a bulkboundary correspondence have not been previously identified in any platform.
Here, we theoretically develop and experimentally realize robust boundarylocalized states protected by a bulk topological invariant in a gapless acoustic crystal. Unlike the forms of topology that can be found in semimetals, the topological states we observe are degenerate with bulk states in both energy and wave vector. Our design is based on coupling a topologically gapped acoustic crystal to a gapless one, yielding a system that fullwave simulations show possesses a gapless resonant spectrum. Nevertheless, when domain boundaries are introduced, both simulations and experimental observations reveal that this twolayer system possesses boundary and domainlocalized states, and the topological origins of these states can be proven using the spectral localizer. To confirm the topological origin of these localized states, we develop an experimental protocol that treats the system’s spectral localizer as the Hamiltonian of a related system, enabling the direct observation of the underlying system’s spatially resolved Ktheory, i.e., its local topology. Taken together, these measurements demonstrate that we have realized a topological metal. Given the generality of our experimental methodologies, these findings open opportunities to discover gapless topological phenomena across a broad range of natural and artificial materials.
Results
Gapless topological phononic crystal
To realize our proofofprinciple topological metal, we use a phononic crystal, as this platform has been proven to be straightforward to fabricate and reconfigure (see Fig. 1a, b)^{12,13,14,15,34,35,36,37,38,39,40,41,42,43,44,45,46,47}. Heuristically, our aim for demonstrating such a phononic topological metal is to start with a topological insulator, couple it to a second lattice such that the combined system is gapless, and then probe it to observe boundarylocalized states. For our specific design, we take the initial topological insulator to be a Su–Schrieffer–Heeger (SSH) lattice^{48}, whose gapped spectrum is shown in Fig. 1c. The second lattice is chosen to be a 1D monatomic lattice with uniform couplings, whose gapless spectrum is shown in Fig. 1d. Here, we are enforcing the resonator geometry and spacing to be the same for both lattices and we are displaying the monatomic lattice’s band structure as folded in to the same Brillouin zone as that of the SSH lattice. Finally, the two lattice layers are uniformly coupled together, resulting in a system with four resonators per unit cell that fullwave simulations show to exhibit a gapless spectrum, see Fig. 1e. We refer to this twolayer lattice as an acoustic metallized SSH lattice. Note that the choice of lattice layer couplings (and identical resonator geometry) ensures that the full system respects chiral symmetry, which is necessary to allow for the possibility of topological states at midspectrum that are associated with this local symmetry classification. (As our 1D lattice has real coupling coefficients, it is in class BDI of the Altland–Zirnbauer symmetry classification, which possesses an integer invariant in 1D^{32,49,50}).
Our phononic crystals consist of acoustic cavities coupled together via grooved channels in the system’s base and also via direct bridges (see Fig. 1a, b). The geometry of the cavities is designed so that their fundamental axial pressure modes are wellseparated in frequency from the rest of the resonant spectrum. The coupling strength between adjacent resonators is controlled through the width of the channels. The base of the system is lasercut, while the dimmers of bridged resonating cavities are fabricated by 3D printing UVcuring resin. In particular, a bipartite singlelayer metamaterial consisting of identical resonators and alternating coupling channel groove widths yields an accurate acoustic realization of the SSH lattice^{48} (see Supplementary Note 1). By designing a metamaterial with a domain boundary between two topologically distinct SSH insulating phases (without the added monatomic layer), and trivial outer edges, both simulation and experimental observations confirm that the system possesses a single topological resonant mode localized at the domain boundary, whose frequency is in the middle of the system’s bulk band gap (see Fig. 2). Moreover, this topological mode’s midgap frequency serves as implicit confirmation that the experimental platform can accurately reproduce chiral symmetry. To facilitate identification of the effects of chiral symmetry in our data we show squared frequencies, as these are the eigenvalues of the acoustic wave equation that are equivalent to energy in the Schrödinger equation.
The phononic topological metal is experimentally realized by adding a layer of identical acoustic resonators, with uniform intralayer coupling strength t_{M}, to the SSH layer and coupling the two layers together via uniform nearestneighbor couplings t_{c}, see Fig. 3a,b,c. In doing so, we are preserving the domainwall boundary in the SSH layer and its trivial edge terminations, but there is no alteration to the monatomic layer at the domain boundary. Even so, this still yields a domain wall in the combined system. Despite the full system’s gapless spectrum, both fullwave numerical simulations (Fig. 3d) and experimental observations (Fig. 3e,f) show that this metamaterial possesses edge and domain walllocalized resonant states, whose appearance is linked to the relative strength of the coupling coefficients. In particular, this system exhibits four boundarylocalized states in total, two that are approximately localized at the system’s domain wall, and one at each edge, see Fig. 3f, g. Moreover, we note that the phononic topological metal has many more topological states than one might initially expect—by itself, the SSH layer has topologically trivial edges (there are no edgeterminationlocalized states in Fig. 2c) and a single domainwalllocalized state.
However, it is not possible to use topological band theories to predict the appearance of the four boundarylocalized states seen in Fig. 3f,g. Attempts to define a winding number (or another similar integer invariant for 1D chiralsymmetric systems) based on this crystal’s bulk structure cannot work; the lack of a bulk band gap would require a path of matrix determinants that tried to characterize this winding to intersect the origin, making the winding number undefined^{51}. Moreover, the topology of the observed boundarylocalized resonances in the finite system cannot be understood from the presence of lowdimensional degeneracies in the periodic gapless system’s spectrum, as is the case for topological semimetals. For a ddimensional semimetal, its topology is connected to features in its band structure with dimension ≤ d − 2, and results in states on its (d − 1)dimensional surfaces that can be identified within an incomplete band gap. In contrast, our 1D periodic phononic topological metal (Fig. 1e) cannot possess such lowdimensional band features, and upon introducing an edge or domain wall completely lacks any such kind of incomplete band gap (Fig. 3c,d). Likewise, as the observed boundarylocalized states are at midspectrum, we seek a topological classification that predicts and protects this spectral location, which precludes crystalline invariants^{28}.
Altogether, standard theories of topology are unable to distinguish between the two bulk phases in this system and identify whether these localized states are of topological origin, or provide a measure of topological protection.
Theory of the spectral localizer
Instead, to prove that the observed localized states are of topological origin and that their existence can be tied to a bulkboundary correspondence, we use the spectral localizer as this approach can be applied to systems that lack a bulk band gap^{24}. In general, a system’s spectral localizer combines its Hamiltonian and position operators using a nontrivial Clifford representation. However, as our acoustic metamaterial is a 1D chiral symmetric system in which all of the couplings are real (i.e., its effective Hamiltonian is realsymmetric), its spectral localizer can be written in a reduced form as (see Supplementary Note 2)
Here, the spectral localizer can be evaluated at any choice of parameters \(x,E\in {\mathbb{R}}\) (inside or outside of the system’s spatial and spectral extent), κ is a tuning parameter that also ensures that the terms have compatible units, Π is the system’s chiral operator, HΠ = − ΠH, and I is the identity matrix. Although the spectral localizer is basis independent, if H is written in a tightbinding basis, X is simply a diagonal matrix whose entries correspond to the coordinates of each lattice site.
The spectral localizer (of appropriate dimension) can be used to both construct the relevant local topological invariant for a system in any symmetry class, as well as define the associated local gap. For the twolayer phononic topological metal considered here (or any other 1D system in class BDI), its local invariant is given by^{25},
where sig is the signature of a matrix, i.e., its number of positive eigenvalues minus its number of negative ones. Note, ν_{L}(x) is only defined for E = 0, which reflects the fact that chiral symmetry can only protect states at the middle of the system’s spectrum. Similarly, the local gap μ_{(x, E)} is given by the smallest singular value of \({\tilde{L}}_{(x,E)}\),
More generally, μ_{(x, E)} is used to define the Clifford pseudospectrum of (X, H)^{52}.
Together, ν_{L}(x) and μ_{(x, 0)} yield a complete picture of a system’s topology. Rigorously, ν_{L}(x) is ascertaining whether the matrices H and X − xI can be continued to the chosen trivial atomic limit while preserving both operators’ realsymmetric form and chiral symmetry, and without closing the associated local gap, i.e., μ_{(x, 0)} > 0 during the entire continuation process; if ν_{L}(x) = 0, such a continuation is possible. (Here, the limit that is considered trivial is specified by the choice of Π in Eq. (1)). Moreover, this picture of topology is entirely local, different choices of x can yield different invariants—for x sufficiently far outside of the system’s spatial extent, one expects to see a system with trivial local topology, while x in the bulk of the system may reveal nontrivial local topology. At the domain boundary between these two regions, x_{0}, where ν_{L}(x_{0}) changes, the local gap must close \({\mu }_{({x}_{0},0)}=0\), which is a direct manifestation of bulkboundary correspondence^{33} and is an indication that there are eigenstates or resonances of H near x_{0} at E = 0^{52}.
When the Hamiltonian itself displays a global spectral gap, the topological invariant supplied by the spectral localizer coincides with the traditional 1D winding number^{51} (or, more generally, even/odd Chern numbers or \({{\mathbb{Z}}}_{2}\) invariants, depending on the dimensionality and symmetry of the system^{26,27}). In the absence of such a gap, ν_{L}(x) has no analogue in the traditional way of applying Ktheory: The spectral localizer simply pushes the applicability of the Ktheoretic methods to previously unclassifiable systems and, in the present context, provides the means to formulate a bulkboundary correspondence principle involving interface resonances as opposed to infinitely lived bound states. By a mechanism somewhat similar to one in complex scaling^{53}, the spectral localizer pushes away the continuum spectrum of H by opening a gap at locations away from the position being probed x, allowing for the study of spectral flows and their associated topology.
Applying the spectral localizer to a tightbinding approximation of the acoustic metallized SSH model proves that the localized states observed in this lattice are connected to a bulk topological invariant (see Supplementary Note 6). In particular, we numerically observe the local invariant ν_{L}(x) in this system to change a few times, both at the system’s boundaries and twice at the domain wall within the lattice’s interior. Moreover, despite the fact that this lattice does not possess a bulk band gap, for most values of x we find that the localizer \({\tilde{L}}_{(x,E)}(X,H)\) does have a reasonable spectral gap at E = 0 that protects the bulk topological invariant.
Using the localizer as a system Hamiltonian
Beyond numerically calculating the phononic topological metal’s local invariant and associated strength of protection, the form of the spectral localizer at E = 0 also inspires an experimental approach to verify the system’s topology directly. In particular, because the reduced spectral localizer Eq. (1) is Hermitian at E = 0, it can be reinterpreted as a set of Hamiltonians itself, with
in which κ(X − xI)Π is now an onsite potential with a sign that is sublatticedependent (i.e., a modification of the central frequencies of each resonator), and the choice of x recenters this potential at a different lattice site (or anywhere in between lattice sites). Thus, by simulating and observing the spectrum of \({\tilde{H}}_{x}\), we are directly measuring the spectrum of \({\tilde{L}}_{(x,0)}\), which, through Eqs. (2) and (3), determines the topology and associated protection of the underlying system described by H at x.
In practice, this reinterpretation presents a challenge, as ∥X − xI∥ can become arbitrarily large as the lattice’s size increases, but it is not possible to alter a resonator’s geometry to yield arbitrarily large or small resonance frequencies. Instead, we can circumvent this challenge by using the substitution
such that
As this bounded operator is linear in the vicinity of x = 0 (the range of approximate linearity is set by α), one can prove that it preserves the necessary information for determining the system’s topology using Eq. (2) (see Supplementary Note 3). Moreover, this choice of alteration to the system’s resonators can be experimentally realized for lattices of any size (see Fig. 4).
We directly confirm the topological behavior of the phononic topological metal described by the Hamiltonian H by numerically and experimentally observing the properties of its spectrally localized counterpart given by \({\tilde{H}}_{x}\) at many different choices of x (a realization for a single x is shown in Fig. 5a). In particular, the sublatticedependent onsite potential is realized by modifying all of the resonator heights by up to 2 mm, which preserves their wellseparated fundamental axial mode but yields a shift their frequency (i.e., a different onsite potential). First, fullwave simulations of the localized metamaterial, \({\tilde{H}}_{x}\), demonstrate that there are four x locations in the underlying system where the local invariant changes and the local gap closes (Fig. 5b,c,d), two locations at the outer edges of the system, and two next to the domain boundary. As locations where μ_{(x, 0)} = 0 predict the presence of states of H, the localized states seen in the original phononic topological metal (Fig. 3f,g) are a direct manifestation of bulkboundary correspondence and are necessarily of topological origin. Thus, despite the absence of a bulk band gap in the system, these topological states are protected by the nonzero local gap μ_{(x, 0)} ≈ 0.1 kHz^{2} surrounding the locations where these states appear. In other words, chiralpreserving perturbations to the system H → H + δH cannot alter the local topology at x so long as ∥δH∥ < μ_{(x, 0)}. Furthermore, we note that these states possess additional protection due to a relatively large secondary gap ≈ 0.4 kHz^{2}, see and Fig. 5e and Supplementary Note 4. Heuristically, every gap in the localizer can be associated with an element in a Ktheory group, and thus can provide some form of topological protection^{54}. Although most such gaps are too small for the resulting topology to be physically robust, for this particular system the secondary gap is relatively large and thus provides strong protection for one of the two states at the domain boundary at the system’s center.
We experimentally realize the localized metamaterial, and directly characterize the changes in the underlying system’s local Ktheory (Fig. 5f, g, h). In particular, the simulated resonance spectrum is reproduced with high fidelity by experimental measurements, confirming the topological properties of our phononic topological metal. Likewise, the two central bands (Fig. 5f) are observed to be well separated from the remainder of the spectrally localized system’s bulk bands (Fig. 5g). Although different choices of x in \({\tilde{H}}_{x}\) yield distinct physical systems (see Fig. 4b), our acoustic metamaterial is reconfigurable, and thus we do not need to fabricate a new system for each choice of x shown in Fig. 5. The discrepancies between simulation and experiment are likely the result of fabrication imperfections and variations, as well as measurement errors. However, the discrepancy observed in Fig. 5 is still of a similar magnitude to those observed both Figs. 2 and 3. In particular, the observed discrepancy in the SSH lattice (Fig. 2) shows that these differences are standard to the acoustic metamaterial platform, and are not substantially larger or smaller for our metallic (Fig. 3) or spectrally localized systems (Fig. 5).
Thus, altogether, our experimental results, coupled with our fullwave simulations, prove that our underlying phononic metamaterial (Fig. 3) is a gapless topological material.
The underlying Ktheory
The relative simplicity of the equations of the spectral localizer and the local topological invariants it provides tends to obscure the Ktheory that it rests on. Thus, to show how our experimental protocol for altering a system to directly observe its topology stems from Ktheory, we provide a brief discussion for an interested reader aimed at illuminating the spectral localizer’s mathematical foundation. A traditional form of Ktheory, topological Ktheory^{55,56}, works with continuous functions that map from a given topological space to a space of structured matrices. These spaces of structured matrices are called classifying spaces, as they can be used to compute all 10 Ktheory groups associated to the original given space. (Classifying spaces for classes of real or complex vector bundles would be special cases associated with this mapping.) A newer, more powerful form of Ktheory is the Ktheory of C^{∗}algebras^{57}, which still applies when one has momentum space but also applies when momentum space is lost. Here, we work with modified forms of C^{∗}algebra Ktheory^{25,58,59,60,61,62,63,64} that are more directly applicable to finite systems. Moreover, these newer forms of Ktheory lead to efficient numerical algorithms and are adaptable to different symmetry classes. The main speedup of these approaches comes from avoiding spectral flattening, as numerically, operations like projecting into an occupied subspace tend to produce dense matrices even when the original system can be described with sparse matrices.
Following Ref. ^{32}, we briefly review how classical topological Ktheory arises in the case of a periodic 1D system in class BDI. The position operator is used to define momentum space, which is a copy of the circle \({{\mathbb{T}}}^{1}\). The chirality of the Hamiltonian means
If we have taken the optional step of spectrally flattening H, we find that U(k) is unitary. Thus, the topology in class BDI arises from attempting to classify the ways in which continuous functions can map from the circle to the classical groups of unitary matrices; that is, homotopy classes of elements of \(C({{\mathbb{T}}}^{1},{{{{{{{\mathcal{U}}}}}}}}(n))\). Timereversal symmetry manifests itself as U^{∗}(k) = U(−k). As Kitaev explains^{32}, homotopy classes in \(C({{\mathbb{T}}}^{1},{{{{{{{\mathcal{U}}}}}}}}(n))\) can be used to form a group, one of the classic groups in topological Ktheory^{56}. Finally, the Ktheory group element determined by H(k) can be calculated using a winding number.
For nonperiodic, finite 1D systems in class BDI, we use the form of Ktheory that associates groups to certain algebras. The relevant algebra is \({{{{{{{\mathcal{A}}}}}}}}={{{{{{{{\boldsymbol{M}}}}}}}}}_{2n}({\mathbb{C}})\), treated as a graded, real C^{∗}algebra, and 2n is the number of sites in the system. The grading is determined by Π and we use the standard reality structure (real matrix means real entries). The zeroth group of Ktheory for this algebra \({K}_{0}({{{{{{{\mathcal{A}}}}}}}})\) is built out of out of homotopy classes of 2nby2n unitary matrices U that also satisfy ΠU Π = U^{†} and U^{⊤} = U^{60}. However, if we want to avoid spectral flattening, we can instead look at homotopy classes of invertible matrices M such that ΠM Π = M^{†} and M^{⊤} = M. Finally, the element of \({K}_{0}({{{{{{{\mathcal{A}}}}}}}})\) determined by M is calculated using half the signature of MΠ.
To apply this general discussion to the spectral localizer, consider the full 1D spectral localizer at E = 0
which at most positions x will contain two invertible matrices, M_{x} = κ(X − xI) + iH and its adjoint. Notice that M_{x}Π is Hermitian (but not real), and thus we can determine the underling physical system’s topology by calculating sig(M_{x}Π). Moreover, it has been established^{25} that in the BDI symmetry class M_{x}Π is unitarily equivalent to the real symmetric matrix κ(X − xI)Π + H (i.e., Eq. (4)), and thus \({{{{{{{\rm{sig}}}}}}}}({M}_{x}{{\Pi }})={{{{{{{\rm{sig}}}}}}}}\left({\tilde{H}}_{x}\right)\), which is what is used in Eq. (2).
Altogether, a more standard approach to topological materials would consider the unitary U_{x} that is derived from M_{x} by spectral flattening (that is, U_{x} is the unitary polar factor of M_{x}). Then one can apply the graded trace, which is more familiar in pure math than the graded signature. However, since
we have many mathematically equivalent formulas to choose from to determine the underlying system’s topology. In particular, the latter two of these formula can be immediately recognized from the invertible matrix form of C^{∗}algebra Ktheory that underpin the spectral localizer. From the perspective of numerical efficiency, the formula involving U_{x} would be the slowest, assuming one actually performs the spectral flattening. The formula involving \({\tilde{H}}_{x}\) will be the fastest, as this matrix will be real, symmetric, and (usually) sparse.
Thus, as \({\tilde{H}}_{x}\) is the Hamiltonian for the localized system (Fig. 5) that we realize to observe the topology of the underlying phononic topological metal (Fig. 3), our experimentally methodology is inextricably linked to the Ktheory of C^{∗}algebras.
Discussion
In conclusion, we have demonstrated a topological metal in an acoustic metamaterial and directly observed its boundarylocalized states despite its lack of a bulk band gap. To do so, we have used the spectral localizer, a local theory of topological materials that is able to predict topological phenomena, and a measure of topological protection, even in the absence of a bulk band gap. Moreover, we have introduced an experimental protocol that uses a system’s spectral localizer as its Hamiltonian, providing a direct probe of the underlying system’s local Ktheory. Here, it is worth emphasizing that this protocol can be applied to any topological system. Although our specific demonstration has leveraged the system’s symmetries to yield a realsymmetric spectral localizer, the spectral localizer for any system is, by definition, Hermitian, and as such, it can always be adapted to be an observable system. Thus, the overall methodology that we have introduced may enable the prediction and observation of topological metals across a broad array of systems, including materials that exhibit higherorder topology and those whose topology is determined by its crystalline symmetries. Finally, as the spectral localizer takes an operatorbased, rather than eigenstatebased, approach to topology, it is potentially broadly applicable to interacting systems, a class of systems that traditional band theories of topology have had difficulty gaining traction with.
Methods
Fabrication
Our fabrication process is modular and the acoustic crystals are assembled from parts that are independently manufactured with different automated process. This approach enables a high throughput of acoustic crystals, which can be disassembled and stored after use.
One leg of the process is the manufacturing of the supporting bases, which consist of two layers of 3mm thick black acrylic plates (Fig. 3b) and one top layer of 1.5mm thick transparent acrylic plates (Fig. 3a,b), of which the top transparent layer and middle black layer have through holes, lasercut at specific geometries with the Boss Laser1630 Laser Engraver. The middle layer with 3mm deep through channels provides coupling channels between the dimers. The top transparent layer with through square holes holds the dimers in place.
The resonators were manufactured using an Anycubic Photon 3D printer, which uses UV resin and has 47μm XYresolution and 10μm Zresolution. The thickness of their walls is 2 mm, to ensure a good quality factor and to justify rigid boundaries in our numerical simulations. The inner dimensions of the resonators are supplied in Fig. 1a,b. For the metallizedSSH system, the resonators are 3D printed as dimers with identical narrow channels connecting the resonators. To implement the spectral localizer, the resonators were printed with different heights according to the algorithm Eq. (6) and were made ready for the assembling.
The resonators were mounted and coupled through the channels grooved in the acrylic plates of the base. Let us specify that the resonators are interchangeable so that they can be move around and acoustic crystals with different probe positions can be generated, as described in the main text. Finally, we note that although the coloration of the resonators is not uniform in either of the structures shown in Fig. 3a,b or 5a, this is simply an artifact of the fabrication process, and these color differences do not impact their behavior in any way.
Experimental protocols
The protocol for the acoustic measurements reported in Fig. 3 and Fig. 5 was as follows: Sinusoidal signals of duration 1 s and amplitude of 0.5 V were produced with a Rigol DG 1022 function generator and applied on a speaker placed in a porthole opened in a resonator. A dbx RTAM Reference Microphone with a Phantom Power was inserted in a porthole opened in the same resonator where the speaker was inserted and acquired the acoustic signals. The signals were read by a custom LabVIEW code via National Instruments USB6122 data acquisition box and the data was stored on a computer for graphic renderings.
The local densities of states reported in Fig. 3d and Fig. 5f,g were obtained by integrating the local density of states acquired from resonators whose index are the same as the position of the probe. Same instrumentation was used. The measurements were repeated with moving the position of the probe. For each measurement, the frequency was scanned from 4200 Hz to 5200 Hz in 10 Hz steps.
Simulation
The simulations reported in Fig. 1, 2, 3, and 5 were performed with the COMSOL Multiphysics pressure acoustic module. The wave propagation domains shown in Fig. 1 were filled with air with a mass density 1.3 kg/m^{3} and the sound’s speed was fixed at 343 m/s, appropriate for room temperature. Because of the huge acoustic impedance mismatch compared with air, the 3D printing UV resin material was considered as hard boundary.
Data availability
All the data generated in this study have been deposited in the Zenodo database https://zenodo.org/record/7765335#.ZB3GnbMLD6.
Code availability
The computer codes used in this study have been deposited in the Zenodo database https://zenodo.org/record/7765335#.ZB3GnbMLD6.
References
Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045 (2010).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011). publisher: American Physical Society.
Bansil, A., Lin, H. & Das, T. Colloquium: Topological band theory. Rev. Mod. Phys. 88, 021004 (2016).
Chiu, C.K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Wang, Z. et al. Dirac semimetal and topological phase transitions in A_{3}Bi (A=Na, K, Rb). Phys. Rev. B 85, 195320 (2012).
Liu, Z. K. et al. Discovery of a ThreeDimensional Topological Dirac Semimetal, Na3Bi. Science 343, 864 (2014).
Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Threedimensional Dirac semimetal and quantum transport in Cd_{3}As_{2}. Phys. Rev. B 88, 125427 (2013).
Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl Semimetal Phase in Noncentrosymmetric TransitionMetal Monophosphides. Phys. Rev. X 5, 011029 (2015).
Lv, B. Q. et al. Observation of Weyl nodes in TaAs. Nat. Phys. 11, 724 (2015).
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622 (2015).
Xiao, M., Chen, W.J., He, W.Y. & Chan, C. T. Synthetic gauge flux and Weyl points in acoustic systems. Nat. Phys. 11, 920 (2015).
Li, F., Huang, X., Lu, J., Ma, J. & Liu, Z. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30 (2018).
Xie, B. et al. Experimental Realization of TypeII Weyl Points and Fermi Arcs in Phononic Crystal. Phys. Rev. Lett. 122, 104302 (2019).
Yang, Y. et al. Topological triply degenerate point with double Fermi arcs. Nat. Phys. 15, 645 (2019).
Burkov, A., Hook, M. & Balents, L. Topological nodal semimetals. Physical Review B 84, 235126 (2011).
Xu, Q. et al. Twodimensional oxide topological insulator with ironpnictide superconductor lifeas structure. Physical Review B 92, 205310 (2015).
Fu, B.B. et al. Dirac nodal surfaces and nodal lines in zrsis. Science advances 5, eaau6459 (2019).
Hor^ava, P. Stability of Fermi Surfaces and K Theory. Phys. Rev. Lett. 95, 016405 (2005).
Alexandradinata, A., Wang, C., Duan, W. & Glazman, L. Revealing the Topology of FermiSurface Wave Functions from Magnetic Quantum Oscillations. Phys. Rev. X 8, 011027 (2018).
Sun, S., Song, Z., Weng, H. & Dai, X. Topological metals induced by the Zeeman effect. Phys. Rev. B 101, 125118 (2020).
Yuan, N. F. Q. & Fu, L. Topological metals and finitemomentum superconductors. Proceedings of the National Academy of Sciences 118, e2019063118 (2021).
Kane, C. L. Quantized nonlinear conductance in ballistic metals. Phys. Rev. Lett. 128, 076801 (2022).
Cerjan, A. & Loring, T. A. Local invariants identify topology in metals and gapless systems. Phys. Rev. B 106, 064109 (2022).
Loring, T. A. Ktheory and pseudospectra for topological insulators. Annals of Physics 356, 383 (2015).
Loring, T. A. & SchulzBaldes, H. The spectral localizer for even index pairings. J. Noncommut. Geom. 14, 1 (2020).
Loring, T. A. & SchulzBaldes, H. Finite volume calculation of Ktheory invariants. New York J. Math. 23, 1111 (2017).
Kruthoff, J., de Boer, J., van Wezel, J., Kane, C. L. & Slager, R.J. Topological Classification of Crystalline Insulators through Band Structure Combinatorics. Phys. Rev. X 7, 041069 (2017).
Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298 (2017).
Po, H. C., Vishwanath, A. & Watanabe, H. Symmetrybased indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).
Cano, J. et al. Building blocks of topological quantum chemistry: Elementary band representations. Phys. Rev. B 97, 035139 (2018).
Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conference Proceedings 1134, 22 (2009).
Hastings, M. B. & Loring, T. A. Topological insulators and C^{*}algebras: Theory and numerical practice. Ann. Phys. 326, 1699 (2011).
SerraGarcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342 (2018).
Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higherorder topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113 (2019).
Ni, X. et al. Observation of Hofstadter butterfly and topological edge states in reconfigurable quasiperiodic acoustic crystals. Commun. Phys. 2, 1 (2019).
Wen, X. et al. Acoustic Landau quantization and quantumHalllike edge states. Nat. Phys. 15, 352 (2019).
Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higherorder topological insulator on a kagome lattice. Nat. Mater. 18, 108 (2019).
Xue, H. et al. Realization of an Acoustic ThirdOrder Topological Insulator. Phys. Rev. Lett. 122, 244301 (2019).
Apigo, D. J., Cheng, W., Dobiszewski, K. F., Prodan, E. & Prodan, C. Observation of Topological Edge Modes in a Quasiperiodic Acoustic Waveguide. Phys. Rev. Lett. 122, 095501 (2019).
Peri, V. et al. Experimental characterization of fragile topology in an acoustic metamaterial. Science 367, 797 (2020).
Cheng, W., Prodan, E. & Prodan, C. Experimental Demonstration of Dynamic Topological Pumping across Incommensurate Bilayered Acoustic Metamaterials. Phys. Rev. Lett. 125, 224301 (2020).
Ni, X., Li, M., Weiner, M., Alù, A. & Khanikaev, A. B. Demonstration of a quantized acoustic octupole topological insulator. Nat. Commun. 11, 2108 (2020).
Chen, Z.G., Zhang, R.Y., Chan, C. T., & Ma, G. Classical nonAbelian braiding of acoustic modes, https://doi.org/10.1038/s41567021014319Nat. (2021).
Xue, H. et al. Observation of DislocationInduced Topological Modes in a ThreeDimensional Acoustic Topological Insulator. Phys. Rev. Lett. 127, 214301 (2021).
Jiang, B. et al. Experimental observation of nonAbelian topological acoustic semimetals and their phase transitions. Nat. Phys. 17, 1239 (2021).
Deng, Y. et al. Observation of Degenerate ZeroEnergy Topological States at Disclinations in an Acoustic Lattice. Phys. Rev. Lett. 128, 174301 (2022).
Su, W., Schrieffer, J. & Heeger, A. J. Solitons in polyacetylene. Physical review letters 42, 1698 (1979).
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. W. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New Journal of Physics 12, 065010 (2010).
Asbóth, J. K., Oroszlány, L. & Pályi, A. A short course on topological insulators. Lecture notes in physics 919, 166 (2016).
Cerjan, A., Loring, T. A. & Vides, F. Quadratic pseudospectrum for identifying localized states. J. Math. Phys. 64, 023501 (2023).
Simon, B. Resonances and complex scaling: a rigorous overview. International Journal of Quantum Chemistry 14, 529 (1978).
SchulzBaldes, H. & Stoiber, T. Invariants of disordered semimetals via the spectral localizer. EPL 136, 27001 (2021).
MAtiyah,M. F. Ktheory (W. A. Benjamin, Inc., New YorkAmsterdam, 1967) pp. v+166+xlix, lecture notes by D. W. Anderson.
Karoubi, M. Ktheory: An introduction, Vol. 226 (Springer Science & Business Media, 2008).
Blackadar, B. https://doi.org/10.1007/9781461395720Ktheory for operator algebras, Mathematical Sciences Research Institute Publications, Vol. 5 (SpringerVerlag, New York, 1986) pp. viii+338
Boersema, J. L. & Loring, T. A. Ktheory for real C^{*}algebras via unitary elements with symmetries. New York J. Math 22, 1139 (2016).
Hastings, M. B. & Loring, T. A. Topological insulators and C^{*}algebras: Theory and numerical practice. Annals of Physics 326, 1699 (2011).
Trout, J. On graded Ktheory, elliptic operators and the functional calculus. Illinois Journal of mathematics 44, 294 (2000).
Browne, S. L. A Bott periodicity proof for real graded C^{*}algebras. Journal of Mathematical Analysis and Applications 479, 658 (2019).
Doll, N. & SchulzBaldes, H. Skew localizer and z2flows for real index pairings. Advances in Mathematics 392, 108038 (2021).
Großmann, J. & SchulzBaldes, H. Index pairings in presence of symmetries with applications to topological insulators. Communications in Mathematical Physics 343, 477 (2016).
Loring, T. A. A Guide to the Bott Index and Localizer Index, Preprint at http://arxiv.org/abs/1907.11791 (2019).
Acknowledgements
T.L. acknowledges support from the National Science Foundation, grant DMS2110398. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. A.C. and T.L. acknowledge support from the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE’s National Nuclear Security Administration under contract DENA0003525. The views expressed in the article do not necessarily represent the views of the U.S. DOE or the United States Government. C. P. acknowledges support from the National Science Foundation, grant CMMI2131759. E.P. was supported by the U.S. National Science Foundation through the grants DMR1823800 and CMMI2131760.
Author information
Authors and Affiliations
Contributions
W.C., assisted by S.Y.C., performed the fullwave simulations, fabrication, and experimental characterization. A.C. designed the underlying physical model. A.C., E.P., and T.A.L. developed the mathematical framework for analyzing this model. All authors analyzed and interpreted the data. W.C. and A.C. wrote the manuscript with input from all authors. E.P., T.A.L., and C.P. supervised the project.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests
Peer review
Peer review information
Nature Communications thanks Yuning Guo, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Cheng, W., Cerjan, A., Chen, SY. et al. Revealing topology in metals using experimental protocols inspired by Ktheory. Nat Commun 14, 3071 (2023). https://doi.org/10.1038/s41467023388622
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467023388622
This article is cited by

Classifying topology in photonic crystal slabs with radiative environments
npj Nanophotonics (2024)

Bulklocaldensityofstate correspondence in topological insulators
Nature Communications (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.