Experimental characterization of fragile topology in an acoustic metamaterial

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Science  14 Feb 2020:
Vol. 367, Issue 6479, pp. 797-800
DOI: 10.1126/science.aaz7654

Understanding fragile topology

Exploiting topological features in materials is being pursued as a route to build in robustness of particular properties. Stemming from crystalline symmetries, such topological protection renders the properties robust against defects and provides a platform of rich physics to be studied. Recent developments have revealed the existence of so-called fragile topological phases, where the means of classification due to symmetry is unclear. Z.-D. Song et al. and Peri et al. present a combined theoretical and experimental approach to identify, classify, and measure the properties of fragile topological phases. By invoking twisted boundary conditions, they are able to describe the properties of fragile topological states and verify the expected experimental signature in an acoustic crystal. Understanding how fragile topology arises could be used to develop new materials with exotic properties.

Science, this issue p. 794, p. 797


Symmetries crucially underlie the classification of topological phases of matter. Most materials, both natural as well as architectured, possess crystalline symmetries. Recent theoretical works unveiled that these crystalline symmetries can stabilize fragile Bloch bands that challenge our very notion of topology: Although answering to the most basic definition of topology, one can trivialize these bands through the addition of trivial Bloch bands. Here, we fully characterize the symmetry properties of the response of an acoustic metamaterial to establish the fragile nature of the low-lying Bloch bands. Additionally, we present a spectral signature in the form of spectral flow under twisted boundary conditions.

Although topological properties of phases of matter seem to be an omnipresent theme in contemporary condensed matter research, there is no single defining property of what a “topological” system is (1). For strongly interacting phases, one might use as a definition the existence of fractionalized excitations or the presence of long-range entanglement (2). For noninteracting systems, bulk-boundary correspondences can often be captured by topological indices such as Chern or winding numbers (3). Despite the vast differences between the various instances of topological matter, all these phases have one common denominator: One cannot smoothly transform the system to an “atomic limit” of disconnected elementary blocks separated in space.

For the classification of such topological systems, symmetries play an essential role. The path of smooth transformations to an atomic limit can be obstructed by symmetry constraints (4). Prime examples are the table of noninteracting topological insulators (58), or the Affleck-Kennedy-Lieb-Tasaki (9) spin-1 state as a member of the family of symmetry-protected interacting phases in one dimension (4). Lately, crystalline symmetries have been identified as an extremely rich source of topological band structures for electrons (1014). This applies not only to electronic bands but to any periodic linear system, both quantum and classical.

It has been realized that crystalline topological insulators can be divided into two main classes on the basis of their stability under the addition of other bands (10, 15). Stable topological insulators can only be trivialized by another set of topological bands and can be classified by using K-theory (5). Contrarily, there are topological bands that can be trivialized by a set of bands arising from an atomic limit. In this case, our conventional notion of topological robustness is challenged, and one needs to introduce the idea of fragile topology (1522). To understand how this fragility arises, one needs to classify the bands emerging from the 230 crystalline space groups.

Recent approaches to this challenge attempt this classification by “inverting” the idea of the atomic limit (10, 23); starting from a set of isolated orbitals at high-symmetry points in real space, one constructs all possible bands that can be induced from these orbitals. These are called elementary band representations (EBRs) (10, 24, 25). Such an EBR is characterized by the irreducible representations of the symmetry realized at high-symmetry locations in the Brillouin zone. Once all such EBRs are constructed, one can compare them with a concrete set of bands obtained from an experiment or numerical calculations. If the bands under investigation can be written as a combination of bands induced from isolated orbitals, one can perform the atomic limit by construction. In essence, the classification task is turned into a simple matching exercise between the possible EBRs and bands realized in a material.

If this matching exercise fails, two distinct ways of how a set of bands can be topological arise. First, the physical bands might realize only a fraction of the representations of an EBR. In other words, the EBR induced from a given position splits into disconnected parts. Once we take only “half” the EBR, the atomic limit is obstructed, and we deal with a topological system (10, 26). This corresponds to the familiar case of stable topology.

In the second option, the multiplicities of the representations can be reproduced by combining different EBRs with positive and negative coefficients. This means that the physical bands lack a number of high-symmetry-point representations to be compatible with an atomic limit. However, the lacking representations correspond to an EBR. In other words, an atomic limit is possible if one adds the bands of this “missing” EBR. This matches the definition of fragile topology given above.

Is fragile topology an exotic curiosity, and moreover, does it have any experimental signatures in spectral or transport measurements? The first question can be answered with a definitive “no.” Fragile band structures are abundant both in electronic materials (16), such as the flat bands of magic-angle twisted bilayer graphene (2731), as well as for classical systems in photonics and phononics (18, 32, 33). The second question was answered in a recent complementary work (34). When the system with fragile bands is slowly disconnected into several parts while preserving some of the space group symmetries, spectral flow is occurring: A number of states determined by the topology of the involved bands is flowing through the bulk gap (34). This work presents an experimental characterization of this spectral flow, turning the abstract concept of fragile topology into a measurable effect.

We constructed an acoustic sample in the wallpaper group p4mm to experimentally establish the spectral signatures of fragile bands. The symmetries and the high-symmetry points (maximal Wyckoff positions) in the unit cell of p4mm are explained in Fig. 1B. Our sample is made of two layers, in which the acoustic cavities reside at the maximal Wyckoff positions 1b and 2c. Chiral coupling channels connect the 2c cavities in different layers (Fig. 1F).

Fig. 1 Atomic limit and topological bands.

(A) Localized orbitals at the 2c maximal Wyckoff positions in the unit cell of a p4mm-symmetric system. (Top) A1 orbitals. (Bottom) A2 orbitals (Fig. 2B and table S3) (35). (B) Schematic representation of the wallpaper group p4mm. The locations and labels of the maximal Wyckoff positions are in red. Dotted black lines indicate the relevant mirror planes with respective labels, and solid lines show the action of C4 and C2 symmetry operators. (C) Sketch of the bands at high-symmetry points induced by the localized orbitals of (A) (10, 42, 43). The drawings show example orbitals that transform according to the realized irreducible representations. (D) Labels and locations of the high-symmetry points in the Brillouin zone for the p4mm wallpaper group. In parentheses are the little group realized at each high-symmetry point. (E) Bands obtained from finite-element simulations of our acoustic crystal. The irreducible representations at high-symmetry points are represented by example orbitals. (F) (Top) A rendering of the air structure of the acoustic crystal unit cell with p4mm symmetry. The labels of the maximal Wyckoff positions are in red. (Bottom) A lattice representation of the acoustic structure. (G) Photo of the experimental sample, with the soft cut indicated with the yellow dashed line. (Inset, top) A detail of the obstructions realizing the cut. (Inset, bottom) A zoom-in of the unit cell. (H) Schematic of the flow induced between fragile bands under twisted boundary conditions.

When considering the low-frequency Bloch bands, one can expect the nodeless modes of the 2c cavities to be relevant because they have a larger volume. Specifically, we presume the orbitals A1 and A2 at 2c to induce the lowest four bands (Fig. 1A). Comparing the bands induced from these orbitals in Fig. 1C with the result of the finite-element simulation of the acoustic field (35) in Fig. 1E, one observes that the representation at the high-symmetry points of the two lowest bands cannot be written by a combination of the EBRs induced from the expected orbitals. This points toward topological bands.

We now turn to the experimental validation. The acoustic crystal in Fig. 1G was fabricated by means of 3D-printing acoustically hard walls around the air-volume depicted in Fig. 1F. We excited the air at a fixed cavity in the middle of the sample with a speaker (36). By measuring at the center of each cavity, we obtained the Greens function Gα,βi,j=ψα*(i)ψβ(j), where ψβ(j) denotes the acoustic field at the speaker in unit cell j and Wyckoff position β, and likewise for the measured field ψα(i) (36). We Fourier-transformed the result to obtain the spectral information shown in Fig. 2A. Evaluating the relative weight and phase at different Wyckoff positions inside the unit cell allowed us to extract the symmetry of the Bloch wave functions (figs. S4 and S5) (35). In Fig. 2A, we label the high-symmetry points according to the irreducible representation of the respective little group (37). The names and example orbitals of the representations are shown in Fig. 2B.

Fig. 2 Irreducible representations at high-symmetry points.

(A) Measured spectrum of the acoustic crystal along high-symmetry lines. The fit to the local maxima has been overlaid at each point in momentum space, and the vertical error bars are the full width at half maxima of the fitted Lorentzians. Labels indicate the irreducible representations of the little groups GK realized at high-symmetry points K according to the names in (B) (GΓ = GMC4v and GXC2v). (Right) The integrated density of states. The frequency range of the bulk bands is shaded in gray. (B) Tables for the irreducible representations of C4v and its relevant subgroups, C2v and C2. The left column provides the standard names according to (44), the middle column provides the labels we gave the high-symmetry points in the Brillouin zone (for example, K1 → Γ1 at the Γ point), and the right column depicts an example orbital in the respective irreducible representation.

The measured symmetries confirm the expectations from the finite-element simulations. We found the following decomposition (37):Bands 1 and 2:(A1)1b(A2)2c(B2)1a(1)Bands 3 and 4:(B2)1a(A1)2c(A1)1b(2)This establishes that the lowest two sets of bands are fragile according to experimental data alone. The bands 1 and 2 have the missing EBR induced from Wyckoff position 1a, which does not host an acoustic cavity. In this case, fragility has a spectral consequence in the form of spectral flow. We quickly review how to establish this. Details can be found in the companion paper (34).

The simplest approach is given by a real-space picture (34). One can characterize all states below the spectral gap of a finite sample by their transformation properties under the C2 symmetry around the central 1a position. This allows us to write a real-space index (RSI) (34)δ = m(B) – m(A)(3)which counts the imbalance between C2-odd and C2-even states (Fig. 2B). One can now smoothly disconnect the sample in a C2-symmetric way (Fig. 1G) and reconnect it with the opposite hopping sign. The initial and final dynamics are related by a gauge transformation, in which we multiply all degrees of freedom in one half of the sample by minus one. However, this gauge transformation turns C2-even into C2-odd states. Hence, we expect δ states flowing through the gap during this cutting and reconnecting procedure.

The remaining task is to bind this RSI δ to the decomposition in Eq. 1. This can be achieved with an exhaustive analysis of all EBRs in a given space group, akin to the matching exercise to establish fragility (34). For our situation, we found (34)δ = 1(4)To implement this soft cut in our sample, we inserted obstructions of growing sizes into the channels along the yellow dashed line in Fig. 1G. Without using higher-order resonances in the connecting channels (38), we could only perform half of the twisting cycle: from the original sample (λ = 1) to a total disconnect (λ = 0), where λ is the hopping multiplication factor across the cut. However, the spectrum at λ and – λ is related by a gauge transformation. Hence, the path λ = 0 → −1 does not provide further information (35).

The measured spectral flow is shown in Fig. 3, which depicts how a state is flowing from the upper bulk bands to the lower bulk bands in the course of the cutting procedure. We obtained these results from the local density of states measured at the symmetry center of the sample. This measurement gives a clean and robust spectral signature associated to fragile topology. The opposite flow of a C2-odd state from the lower bands to upper ones occurs for λ < 0 as ensured by the symmetry λ → –λ (34, 35).

Fig. 3 Spectral flow between fragile bands.

Measured local density of state at the symmetry center of the acoustic crystal for different values λ of the hopping multiplication factor across the cut. The overlaid dots indicate the fit to the local maxima, and the vertical error bars correspond to the full width at half maxima of the fitted Lorentzians. The frequency range of the bulk bands has been shaded in gray. The data at each λ have been individually normalized with respect to their maximum value.

Through the results presented here, we achieved two goals. First, we established the presence of fragile bands through full band tomography. Second, we demonstrate that fragile topology, a concept that challenges our understanding of topology, can yield a clean and simple experimental observable. In the context of metamaterials design, such fine control of well-localized states is an important building block for mechanical logic and other wave-control applications (39). Moreover, many classical topological metamaterials rely on crystalline symmetries (40). This attributes fragility a more prominent role than anticipated (18, 32, 35). Last, the expectation that fragility plays a role in the reported strongly correlated superconductivity in twisted bilayer graphene (2731, 41) raises the natural question of how classical nonlinearities are influenced by these intricate band effects.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S13

Tables S1 to S7

References (4663)

References and Notes

  1. Supplementary text is available as supplementary materials.
  2. Materials and methods are available as supplementary materials.
  3. Bands that are classified as trivial according to their crystalline symmetry eigenvalues could still have some nontrivial topology captured by Berry phases.
Acknowledgments: We thank G. Blatter, T. Neupert, and V. Vitelli for insightful discussions. Funding: S.D.H., V.P., M.S.-G., and P.E. acknowledge support from the Swiss National Science Foundation, the Swiss National Center of Competence in Research QSIT, and the European Research Council under grant agreement 771503 (TopMechMat). Z.-D.S. and B.A.B. are supported by the Department of Energy grant DE-SC0016239, the National Science Foundation (NSF) EAGER grant DMR1643312, Simons Investigator Grant 404513, Office of Naval Research N00014-14-1-0330, NSF–Materials Research Science and Engineering Center DMR-142051, the Packard Foundation, and the Schmidt Fund for Innovative Research. B.A.B. is also supported by a Guggenheim Fellowship from the John Simon Guggenheim Memorial Foundation. Part of the group theory analysis of this work was supported by U.S. Department of Energy grant DE-SC0016239.Author contributions: S.D.H., V.P., Z.-D.S., R.Q., and B.A.B. performed the theoretical part of this work. S.D.H., V.P., X.H., W.D., and Z.L. designed the samples. V.P., M.S.-G., and P.E. conducted the experiment. All authors contributed to the writing of the manuscript. Competing interests: The authors declare no competing interests. Data and materials availability: The data shown in this work are available at (45).

Correction (23 March 2020): U.S. Department of Energy funding information was added to the Acknowledgments.

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