Acoustic Möbius Insulators from Projective Symmetry

Non-Abelian Braiding of Topological Edge Bands

Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding structures, as demonstrated recently in a type of topological insulators known as Möbius insulators, whose topological edge states form two braided bands exhibiting a Möbius twist. While the formation of Möbius twist is inspiring, it belongs to the simple Abelian braid group B_{2}. The most fascinating features about topological braids rely on the non-Abelianness in the higher-order braid group B_{N} (N≥3), which necessitates multiple edge bands, but so far it has not been discussed. Here, based on the gauge enriched symmetry, we develop a scheme to realize non-Abelian braiding of multiple topological edge bands. We propose tight-binding models of topological insulators that are able to generate topological edge states forming non-Abelian braiding structures. Experimental demonstrations are conducted in two acoustic crystals, which carry three and four braided acoustic edge bands, respectively. The observed braiding structure can correspond to the topological winding in the complex eigenvalue space of projective translation operator, akin to the previously established point-gap winding topology in the bulk of the Hatano-Nelson model. Our Letter also constitutes the realization of non-Abelian braiding topology on an actual crystal platform, but not based on the "virtual" synthetic dimensions.

Higher-Order Topological Insulators via Momentum-Space Nonsymmorphic Symmetries

We theoretically construct a higher-order topological insulator (HOTI) on a Brillouin real projective plane enabled by momentum-space nonsymmorphic (k-NS) symmetries from synthetic gauge fields. Two anicommutative k-NS glide reflections appear in a checkerboard Z_{2} flux model, impose nonsymmorphic constraints on Berry curvature, and quantize bulk and Wannier-sector polarization nonlocally across different momenta. The model's bulk exhibits an isotropic quadrupole phase diagram, where the transition appears intrinsically from bulk gap closure. The model hosts the simultaneous presence of intrinsic and extrinsic HOTI features: in a ribbon geometry where one pair of boundaries gets open, the edge termination can induce boundary-obstructed topological phase within the symmetry-protected topological phase due to the breaking of k-NS symmetry. At last, we present a concrete design for the real projective plane quadrupole insulator and show how to measure the momentum glide reflection based on acoustic resonator arrays. Our results shed light on HOTIs on deformed Brillouin manifolds via k-NS symmetries.

Demonstration of Acoustic Higher-Order Topological Stiefel-Whitney Semimetal

The higher-order topological phases have attracted intense attention in the past years, which reveals various intriguing topological properties. Meanwhile, the enrichment of group symmetries with projective symmetry algebras redefines the fundamentals of topological matter and makes Stiefel-Whitney (SW) classes in classical wave systems possible. Here, we report the experimental realization of higher-order topological nodal loop semimetal in an acoustic system and obtain the inherent SW topological invariants. In stark contrast to higher-order topological semimetals relating to complex vector bundles, the hinge and surface states in the SW topological phase are protected by two distinctive SW topological charges relevant to real vector bundles. Our findings push forward the studies of SW class topology in classical wave systems, which also show possibilities in robust high-Q-resonance-based sensing and energy harvesting.

Non-Abelian physics in light and sound

Non-Abelian phenomena arise when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions. These manipulations enable the creation of a great variety of Hamiltonians, through which rich non-Abelian phenomena can be explored and observed. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics. These fundamental endeavors could enable photonic and acoustic devices with multiplexing functionalities. Our review aims to provide a timely and comprehensive account of this emerging topic. Starting from the foundation of matrix-valued geometric phases, we address non-Abelian topological charges, non-Abelian gauge fields, non-Abelian braiding, non-Hermitian non-Abelian phenomena, and their realizations with photonics and acoustics and conclude with future prospects.

Acoustic Higher-Order Topological Insulators Induced by Orbital-Interactions

The discovery of higher-order topological insulator metamaterials, in analogy with their condensed-matter counterparts, has enabled various breakthroughs in photonics, mechanics, and acoustics. A common way of inducing higher-order topological wave phenomena is through pseudo-spins, which mimic the electron spins as a symmetry-breaking degree of freedom. Here, this work exploits degenerate orbitals in acoustic resonant cavities to demonstrate versatile, orbital-selective, higher-order topological corner states. Type-II corner states are theoretically investigated and experimentally demonstrated based on tailored orbital interactions, without the need for long-range hoppings that has so far served as a key ingredient for Type-II corner states in single-orbital systems. Due to the orthogonal nature of the degenerate p orbitals, this work also introduces a universal strategy to realize orbital-dependent edge modes, featuring high-Q edge states identified in bulk bands. These findings provide an understanding of the interplay between acoustic orbitals and topology, shedding light on orbital-related topological wave physics, as well as its applications for acoustic sensing and trapping.

Stiefel-Whitney topological charges in a three-dimensional acoustic nodal-line crystal

Band topology of materials describes the extent Bloch wavefunctions are twisted in momentum space. Such descriptions rely on a set of topological invariants, generally referred to as topological charges, which form a characteristic class in the mathematical structure of fiber bundles associated with the Bloch wavefunctions. For example, the celebrated Chern number and its variants belong to the Chern class, characterizing topological charges for complex Bloch wavefunctions. Nevertheless, under the space-time inversion symmetry, Bloch wavefunctions can be purely real in the entire momentum space; consequently, their topological classification does not fall into the Chern class, but requires another characteristic class known as the Stiefel-Whitney class. Here, in a three-dimensional acoustic crystal, we demonstrate a topological nodal-line semimetal that is characterized by a doublet of topological charges, the first and second Stiefel-Whitney numbers, simultaneously. Such a doubly charged nodal line gives rise to a doubled bulk-boundary correspondence-while the first Stiefel-Whitney number induces ordinary drumhead states of the nodal line, the second Stiefel-Whitney number supports hinge Fermi arc states at odd inversion-related pairs of hinges. These results experimentally validate the two Stiefel-Whitney topological charges and demonstrate their unique bulk-boundary correspondence in a physical system.

General Theory of Momentum-Space Nonsymmorphic Symmetry

As a fundamental concept of all crystals, space groups are partitioned into symmorphic groups and nonsymmorphic groups. Each nonsymmorphic group contains glide reflections or screw rotations with fractional lattice translations, which are absent in symmorphic groups. Although nonsymmorphic groups ubiquitously exist on real-space lattices, on the reciprocal lattices in momentum space, the ordinary theory only allows symmorphic groups. In this work, we develop a novel theory for momentum-space nonsymmorphic space groups (k-NSGs), utilizing the projective representations of space groups. The theory is quite general: Given any k-NSGs in any dimensions, it can identify the real-space symmorphic space groups (r-SSGs) and construct the corresponding projective representation of the r-SSG that leads to the k-NSG. To demonstrate the broad applicability of our theory, we show these projective representations and therefore all k-NSGs can be realized by gauge fluxes over real-space lattices. Our work fundamentally extends the framework of crystal symmetry, and therefore can accordingly extend any theory based on crystal symmetry, for instance, the classification crystalline topological phases.

Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries

Adding magnetic flux to a band structure breaks Bloch's theorem by realizing a projective representation of the translation group. The resulting Hofstadter spectrum encodes the nonperturbative response of the bands to flux. Depending on their topology, adding flux can enforce a bulk gap closing (a Hofstadter semimetal) or boundary state pumping (a Hofstadter topological insulator). In this Letter, we present a real space classification of these Hofstadter phases. We give topological indices in terms of symmetry-protected real space invariants, which reveal the bulk and boundary responses of fragile topological states to flux. In fact, we find that the flux periodicity in tight-binding models causes the symmetries which are broken by the magnetic field to reenter at strong flux where they form projective point group representations. We completely classify the reentrant projective point groups and find that the Schur multipliers which define them are Arahanov-Bohm phases calculated along the bonds of the crystal. We find that a nontrivial Schur multiplier is enough to predict and protect the Hofstadter response with only zero-flux topology.

Photonic Möbius topological insulator from projective symmetry in multiorbital waveguides

The gauge fields dramatically alter the algebraic structure of spatial symmetries and make them projectively represented, giving rise to novel topological phases. Here, we propose a photonic Möbius topological insulator enabled by projective translation symmetry in multiorbital waveguide arrays, where the artificial π gauge flux is aroused by the inter-orbital coupling between the first (s) and third (d) order modes. In the presence of π flux, the two translation symmetries of rectangular lattices anti-commute with each other. By tuning the spatial spacing between two waveguides to break the translation symmetry, a topological insulator is created with two Möbius twisted edge bands appearing in the bandgap and featuring 4π periodicity. Importantly, the Möbius twists are accompanied by discrete diffraction in beam propagation, which exhibit directional transport by tuning the initial phase of the beam envelope according to the eigenvalues of translation operators. This work manifests the significance of gauge fields in topology and provides an efficient approach to steering the direction of beam transmission.

Classification of time-reversal-invariant crystals with gauge structures

A peculiar feature of quantum states is that they may embody so-called projective representations of symmetries rather than ordinary representations. Projective representations of space groups-the defining symmetry of crystals-remain largely unexplored. Despite recent advances in artificial crystals, whose intrinsic gauge structures necessarily require a projective description, a unified theory is yet to be established. Here, we establish such a unified theory by exhaustively classifying and representing all 458 projective symmetry algebras of time-reversal-invariant crystals from 17 wallpaper groups in two dimensions-189 of which are algebraically non-equivalent. We discover three physical signatures resulting from projective symmetry algebras, including the shift of high-symmetry momenta, an enforced nontrivial Zak phase, and a spinless eight-fold nodal point. Our work offers a theoretical foundation for the field of artificial crystals and opens the door to a wealth of topological states and phenomena beyond the existing paradigms.

Unsupervised Data-Driven Classification of Topological Gapped Systems with Symmetries

A remarkable breakthrough in topological phase classification is the establishment of the topological periodic table, which is mainly based on the classifying space analysis or K theory, but not based on concrete Hamiltonians that possess finite bands or arise in a lattice. As a result, it is still difficult to identify the topological phase of an arbitrary Hamiltonian; the common practice is, instead, to check the incomplete and still growing list of topological invariants one by one, very often by trial and error. Here, we develop unsupervised classifications of topological gapped systems with symmetries, and demonstrate the data-driven construction of the topological periodic table without a priori knowledge of topological invariants. This unsupervised data-driven strategy can take into account spatial symmetries, and further classify phases that were previously classified as trivial in the past. Our Letter introduces machine learning into topological phase classification and paves the way for intelligent explorations of new phases of topological matter.

Spinful Topological Phases in Acoustic Crystals with Projective PT Symmetry

For the classification of topological phases of matter, an important consideration is whether a system is spinless or spinful, as these two classes have distinct symmetry algebra that gives rise to fundamentally different topological phases. However, only recently has it been realized theoretically that in the presence of gauge symmetry, the algebraic structure of symmetries can be projectively represented, which possibly enables the switch between spinless and spinful topological phases. Here, we report the experimental demonstration of this idea by realizing spinful topological phases in "spinless" acoustic crystals with projective space-time inversion symmetry. In particular, we realize a one-dimensional topologically gapped phase characterized by a 2Z winding number, which features double-degenerate bands in the entire Brillouin zone and two pairs of degenerate topological boundary modes. Our Letter thus overcomes a fundamental constraint on topological phases by spin classes.

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.

Acoustic Realization of Surface-Obstructed Topological Insulators

Recently, higher-order topological insulators have been attracting extensive interest. Unlike the conventional topological insulators that demand bulk gap closings at transition points, the higher-order band topology can be changed without bulk closure and exhibits as an obstruction of higher-dimensional boundary states. Here, we report the first experimental realization of three-dimensional surface-obstructed topological insulators with using acoustic crystals. Our acoustic measurements demonstrate unambiguously the emergence of one-dimensional topological hinge states in the middle of the bulk and surface band gaps, as a direct manifestation of the higher-order band topology. Together with comparative measurements for the trivial and phase-transition-point insulators, our experimental data conclusively evidence the unique bulk-boundary physics for the surface-obstructed band topology. That is, the topological phase transition is determined by the closure of the surface gap, rather than by closing the bulk gap. Our study might spur on new activities to deepen the understanding of such elusive topological phases.

Physical realization of topological Roman surface by spin-induced ferroelectric polarization in cubic lattice

Topology, an important branch of mathematics, is an ideal theoretical tool to describe topological states and phase transitions. Many topological concepts have found their physical entities in real or reciprocal spaces identified by topological invariants, which are usually defined on orientable surfaces, such as torus and sphere. It is natural to investigate the possible physical realization of more intriguing non-orientable surfaces. Herein, we show that the set of spin-induced ferroelectric polarizations in cubic perovskite oxides AMn₃Cr₄O₁₂ (A = La and Tb) reside on the topological Roman surface—a non-orientable two-dimensional manifold formed by sewing a Möbius strip edge to that of a disc. The induced polarization may travel in a loop along the non-orientable Möbius strip or orientable disc, depending on the spin evolution as controlled by an external magnetic field. Experimentally, the periodicity of polarization can be the same or twice that of the rotating magnetic field, which is consistent with the orientability of the disc and the Möbius strip, respectively. This path-dependent topological magnetoelectric effect presents a way to detect the global geometry of a surface and deepens our understanding of topology in both mathematics and physics.

A non-orientable surface can mirror reflecting the man travelling on it. Realizing such topological object is fascinating. Here, the authors discover that antiferromagnetic-induced polarization in a solid can realize a non-orientable Roman surface.

Brillouin Klein bottle from artificial gauge fields

A Brillouin zone is the unit for the momentum space of a crystal. It is topologically a torus, and distinguishing whether a set of wave functions over the Brillouin torus can be smoothly deformed to another leads to the classification of various topological states of matter. Here, we show that under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}}}_{2}$$\end{document}Z2 gauge fields, i.e., hopping amplitudes with phases ±1, the fundamental domain of momentum space can assume the topology of a Klein bottle. This drastic change of the Brillouin zone theory is due to the projective symmetry algebra enforced by the gauge field. Remarkably, the non-orientability of the Brillouin Klein bottle corresponds to the topological classification by a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}}}_{2}$$\end{document}Z2 invariant, in contrast to the Chern number valued in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}$$\end{document}Z for the usual Brillouin torus. The result is a novel Klein bottle insulator featuring topological modes at two edges related by a nonlocal twist, radically distinct from all previous topological insulators. Our prediction can be readily achieved in various artificial crystals, and the discovery opens a new direction to explore topological physics by gauge-field-modified fundamental structures of physics.

Topological states are exploited based on crystalline symmetry, but under artificial gauge fields, symmetries may satisfy projective algebras, which remains less studied. Here, the authors reveal that projective symmetry algebra leads to momentum-space nonsymmorphic symmetry, resulting in new topological states over a momentum-space Klein bottle.

Projectively Enriched Symmetry and Topology in Acoustic Crystals

Symmetry plays a key role in modern physics, as manifested in the revolutionary topological classification of matter in the past decade. So far, we seem to have a complete theory of topological phases from internal symmetries as well as crystallographic symmetry groups. However, an intrinsic element, i.e., the gauge symmetry in physical systems, has been overlooked in the current framework. Here, we show that the algebraic structure of crystal symmetries can be projectively enriched due to the gauge symmetry, which subsequently gives rise to new topological physics never witnessed under ordinary symmetries. We demonstrate the idea by theoretical analysis, numerical simulation, and experimental realization of a topological acoustic lattice with projective translation symmetries under a Z_{2} gauge field, which exhibits unique features of rich topologies, including a single Dirac point, Möbius topological insulator, and graphenelike semimetal phases on a rectangular lattice. Our work reveals the impact when gauge and crystal symmetries meet together with topology and opens the door to a vast unexplored land of topological states by projective symmetries.