In this paper, we characterize quasicrystalline interacting topological phases of matter, i.e., phases protected by some quasicrystalline structure. We show that the elasticity theory of quasicrystals, which accounts for both “phonon” and “phason” modes, admits nontrivial quantized topological terms with far richer structure than their crystalline counterparts. We show that these terms correspond to distinct phases of matter and also uncover intrinsically quasicrystalline phases, which have no crystalline analogs. For quasicrystals with internal U(1) symmetry, we discuss a number of interpretations and physical implications of the topological terms, including constraints on the mobility of dislocations in $d=2$ quasicrystals and a quasicrystalline generalization of the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem. We then extend these ideas much further and address the complete classification of quasicrystalline topological phases, including systems with point-group symmetry as well as noninvertible phases. We hence obtain the “quasicrystalline equivalence principle,” which generalizes the classification of crystalline topological phases to the quasicrystalline setting.

中文翻译：

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准晶的量子多体拓扑

在本文中，我们表征了物质的准晶相互作用拓扑相，即受某些准晶结构保护的相。我们表明，准晶体的弹性理论同时考虑了“声子”和“相子”模式，它承认具有比晶体对应物丰富得多的结构的非平凡量子化拓扑项。我们表明这些术语对应于物质的不同相，并且还揭示了本质上没有晶体类似物的准晶相。对于具有内部 U(1) 对称性的准晶体，我们讨论了拓扑项的许多解释和物理含义，包括对位错迁移率的限制$d=2$准晶体和 Lieb-Schultz-Mattis-Oshikawa-Hastings 定理的准晶体推广。然后我们进一步扩展这些想法并解决准晶拓扑相的完整分类，包括具有点群对称性和不可逆相的系统。因此，我们获得了“准晶等效原理”，它将结晶拓扑相的分类推广到准晶设置。