## 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.

## Abstract

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 (*5*–*8*), 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 (*10*–*14*). 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 (*15*–*22*). 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 (*27*–*31*), 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 *p*4*mm* to experimentally establish the spectral signatures of fragile bands. The symmetries and the high-symmetry points (maximal Wyckoff positions) in the unit cell of *p*4*mm* are explained in Fig. 1B. Our sample is made of two layers, in which the acoustic cavities reside at the maximal Wyckoff positions 1*b* and 2*c*. Chiral coupling channels connect the 2*c* cavities in different layers (Fig. 1F).

When considering the low-frequency Bloch bands, one can expect the nodeless modes of the 2*c* cavities to be relevant because they have a larger volume. Specifically, we presume the orbitals *A*_{1} and *A*_{2} at 2*c* 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 _{β}(*j*) denotes the acoustic field at the speaker in unit cell *j* and Wyckoff position β, and likewise for the measured field ψ_{α}(*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.

The measured symmetries confirm the expectations from the finite-element simulations. We found the following decomposition (*37*):*a*, 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 *C*_{2} symmetry around the central 1*a* position. This allows us to write a real-space index (RSI) (*34*)δ = *m*(*B*) – *m*(*A*)(3)which counts the imbalance between *C*_{2}-odd and *C*_{2}-even states (Fig. 2B). One can now smoothly disconnect the sample in a *C*_{2}-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 *C*_{2}-even into *C*_{2}-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 *C*_{2}-odd state from the lower bands to upper ones occurs for λ < 0 as ensured by the symmetry λ → –λ (*34*, *35*).

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 (*27*–*31*, *41*) raises the natural question of how classical nonlinearities are influenced by these intricate band effects.

## Supplementary Materials

science.sciencemag.org/content/367/6479/797/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S13

Tables S1 to S7

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

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- Bands that are classified as trivial according to their crystalline symmetry eigenvalues could still have some nontrivial topology captured by Berry phases.
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**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*).