Dualities and non-Abelian mechanics
Dualities and non-Abelian mechanics
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DOI：
10.1038/s41586-020-1932-6
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发表时间：
2020-01-20
期刊：
影响因子：
64.8
通讯作者：
Vitelli, Vincenzo
中科院分区：
文献类型：
--
作者：
Fruchart, Michel;Zhou, Yujie;Vitelli, Vincenzo
对偶性--不同系统之间的数学映射--可以作为隐藏的对称性，使材料设计超越晶体空间群的建议。对偶性是数学映射，揭示了物理学几乎每个分支中明显不相关系统之间的联系（1-8）。通过对偶变换映射到自身上的系统被称为自对偶，并表现出显着的性质，如伊辛磁铁在临界点的标度不变性。在这里，我们将展示对偶如何增强动力学矩阵（或哈密顿量）的对称性，从而使超材料的设计具有逃脱标准群论分析的新兴特性。作为一个例子，我们考虑扭曲的kagome晶格（9-15），可重构的机械结构，通过折叠机制（9）改变形状。我们观察到，沿着该机制的成对不同配置表现出相同的振动谱和相关的弹性模量。我们表明，这些令人困惑的属性产生于一个机械临界点的两侧的配置对之间的对偶性。临界点对应于一个自对偶结构与各向同性的弹性，即使在没有空间对称性和一个两倍退化的频谱在整个布里渊区。光谱简并性起源于Kramers定理（16，17）的一个版本，其中费米子时间反演不变性被自对偶点处出现的隐藏对称性所取代。自对偶系统的正常模式表现出非阿贝尔几何相位（18，19），其影响波包（20）的半经典传播，导致非对易机械响应。我们的结果为完整计算（21）和机械自旋电子学提供了希望，因为它允许对声子携带的合成自旋进行即时操纵。
Dualities-mathematical mappings between different systems-can act as hidden symmetries that enable materials design beyond that suggested by crystallographic space groups.Dualities are mathematical mappings that reveal links between apparently unrelated systems in virtually every branch of physics(1-8). Systems mapped onto themselves by a duality transformation are called self-dual and exhibit remarkable properties, as exemplified by the scale invariance of an Ising magnet at the critical point. Here we show how dualities can enhance the symmetries of a dynamical matrix (or Hamiltonian), enabling the design of metamaterials with emergent properties that escape a standard group theory analysis. As an illustration, we consider twisted kagome lattices(9-15), reconfigurable mechanical structures that change shape by means of a collapse mechanism(9). We observe that pairs of distinct configurations along the mechanism exhibit the same vibrational spectrum and related elastic moduli. We show that these puzzling properties arise from a duality between pairs of configurations on either side of a mechanical critical point. The critical point corresponds to a self-dual structure with isotropic elasticity even in the absence of spatial symmetries and a twofold-degenerate spectrum over the entire Brillouin zone. The spectral degeneracy originates from a version of Kramers' theorem(16,17) in which fermionic time-reversal invariance is replaced by a hidden symmetry emerging at the self-dual point. The normal modes of the self-dual systems exhibit non-Abelian geometric phases(18,19) that affect the semiclassical propagation of wavepackets(20), leading to non-commuting mechanical responses. Our results hold promise for holonomic computation(21) and mechanical spintronics by allowing on-the-fly manipulation of synthetic spins carried by phonons.