Projectively Enriched Symmetry and Topology in Acoustic Crystals

Xue, Haoran; Wang, Zihao; Huang, Yue-Xin; Cheng, Zheyu; Yu, Letian; Foo, Y.X.; Zhao, Y. X.; Yang, Shengyuan A.; Zhang, Baile

doi:10.1103/physrevlett.128.116802

Cited by 61 publications

(25 citation statements)

References 36 publications

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“…Since glide reflection and screw rotations are the most elementary nonsymmorphic symmetries, all nonsymmorphic space groups may be realized on the reciprocal lattices by certain gauge flux configurations, which are mathematically dictated by the second cohomology groups of the space groups. Since gauge fluxes can be engineered in artificial crystals for realizing projective symmetries 23 , 24 , our work opens the door toward a fertile ground for exploring novel momentum-space symmetries and topologies of artificial crystals beyond the scope of topological quantum materials.…”

Section: Resultsmentioning

confidence: 99%

“…In particular, the Z 2 gauge field , i.e., hopping amplitudes allowed to take phases ±1, can be readily realized in these systems and have already been demonstrated in many experiments [7][8][9][10][11][12][13][14][15][16][17] . A crucial but so far less appreciated point is that under gauge fields, symmetries of the system would satisfy projective algebras [18][19][20][21] beyond the textbook group theory for crystal symmetry 22 , which has recently been experimentally demonstrated by acoustic crystals 23,24 . Then, what is the physical consequence of the projective symmetry algebra?…”

Section: Introductionmentioning

confidence: 99%

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Brillouin Klein bottle from artificial gauge fields

2022

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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 $${{\mathbb{Z}}}_{2}$$ Z 2 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 $${{\mathbb{Z}}}_{2}$$ Z 2 invariant, in contrast to the Chern number valued in $${\mathbb{Z}}$$ 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.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Brillouin Klein bottle from artificial gauge fields

2022

Self Cite

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“…177 New exciting phenomena based on projective symmetry, such as Möbius-twisted topological phases, have been implemented in coupled acoustic resonators using couplings of both positive and negative signs. 178,179 Besides, macroscopic phononic media allow to finely tune the couplings' orientation and range, leading to remarkable phenomena, such as non-Abelian phenomena 54,180,181 and nonlinearly induced topological interface modes. 182 While precisely controlling loss and gain within a solid-state material can be very challenging, phononic systems at the macroscopic scale allow easy control of loss and gain channels in the medium, offering new avenues to the experimental realization of topological phenomena of non-Hermitian nature.…”

Section: Beyond Phononic 2d Topological Insulatorsmentioning

confidence: 99%

Topological sound in two dimensions

Yves

Alù

2022

Annals of the New York Academy of Sciences

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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.

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“…In real space, a Möbius strip with a single boundary has been visualized in nanocrystal/DNA bands 17 , 18 . Möbius insulators have been proposed and realized in the reciprocal spaces of electronic 19 – 21 and acoustic crystals 22 , 23 . Under proper nonsymmorphic symmetry or projective translation symmetry, the Bloch wavefunctions of its boundary states have 4 periodicity instead of 2 , which mimics characteristics of a Möbius strip.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

et al. 2022

Nat Commun

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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.

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Projectively Enriched Symmetry and Topology in Acoustic Crystals

Cited by 61 publications

References 36 publications

Brillouin Klein bottle from artificial gauge fields

Brillouin Klein bottle from artificial gauge fields

Topological sound in two dimensions

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

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