Quantum many-body topology of quasicrystals

TL;DR

This paper explores the topological phases of quasicrystals, revealing richer structures than crystals, including new phases and a classification framework called the Quasicrystalline Equivalence Principle.

Contribution

It introduces a topological classification of quasicrystals, uncovering intrinsically quasicrystalline phases and extending crystalline topological phase theories.

Findings

Identification of non-trivial topological terms in quasicrystals

Discovery of intrinsically quasicrystalline phases with no crystalline analogues

Development of the Quasicrystalline Equivalence Principle

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 $\mathrm{U}(1)$ 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.