Quantum many-body topology of quasicrystals

Abstract

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 non-trivial 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 analogues. 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 non-invertible phases. We hence obtain the "Quasicrystalline Equivalence Principle," which generalizes the classification of crystalline topological phases to the quasicrystalline setting.

Effective field theories of topological crystalline insulators and topological crystals

“…Further attempts have been made in order to turn it into a concrete theoretical tool either in the field theory framework [53][54][55] or on the lattice [55]. There has been an alternative proposal which characterizes cSPT phases through their response to elastic deformations [56][57][58][59][60]. This approach is very physical and is well-defined in a continuous field theory.…”

From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal

“…Recent advancements in the field, including Else's exploration of the quantum manybody topology in quasicrystals [22], have opened new theoretical frontiers. The works of Baggioli et al [23,24] on viscoelastic models and phason dynamics have significantly advanced our understanding of quasicrystal physics.…”