[Paper Review] Comment on "Topological equivalence of crystal and quasicrystal band structures"

Yaacov E. Kraus, Zohar Ringel+1 more|arXiv (Cornell University)|Aug 11, 2013

Quasicrystal Structures and Properties3 citations

TL;DR

This comment refutes the claim that quasicrystals and crystals are topologically equivalent in one dimension by demonstrating that quasiperiodicity acts as a topological constraint analogous to symmetry, enabling nontrivial Chern numbers in quasiperiodic systems even without time-reversal symmetry. The key result is that topological invariants in quasiperiodic systems are robust against adiabatic deformations to periodic systems only if quasiperiodicity is preserved, invalidating the equivalence argument in Madsen et al. (2013).

ABSTRACT

Madsen et al. [arXiv:1307.2577] claim that one-dimensional insulating crystals and one-dimensional insulating quasicrystals are topologically equivalent and, thus, trivial. In this comment, we clarify that in topological classification of one-dimensional systems, quasiperiodicity plays a role similar to that of a symmetry. Hence, periodic systems are out of the quasiperiodic topological space, and the equivalence between quasicrystals and crystals becomes immaterial.

Motivation & Objective

• To resolve a claimed contradiction between the authors' prior work on topological quasiperiodic systems and Madsen et al.'s assertion of topological equivalence between quasicrystals and crystals.
• To clarify that quasiperiodicity, like symmetry, defines a distinct topological space where Chern numbers can be assigned to bulk gaps.
• To demonstrate that the absence of boundary states in Madsen et al.'s numerical results does not imply triviality, as sharp boundaries break quasiperiodicity.
• To show that adiabatic deformation from quasiperiodic to periodic systems does not imply topological triviality, as periodic systems are outside the quasiperiodic topological classification.
• To establish that only quasiperiodic Hamiltonians support topologically protected phenomena such as gap-filling boundary states and interface modes.

Proposed method

• The authors analyze 1D quasiperiodic tight-binding models with quasiperiodic (QP) hopping or on-site terms, defined by irrational frequencies b and phase φ.
• They show that bulk properties, including energy gaps, are independent of φ due to the invariance of QP functions under φ-shifts, enabling the assignment of integer Chern numbers to gaps.
• They contrast this with periodic systems (rational b), where Chern numbers depend on φ and require integration over φ to define, making single-φ Chern numbers ill-defined.
• They use the Thouless pump formalism to justify the topological robustness of QP systems under adiabatic deformations that preserve quasiperiodicity.
• They analyze the Aubry-André model numerically, showing that boundary state dependence on φ is due to local breaking of quasiperiodicity, not topological triviality.
• They argue that the persistence of large gaps during continuous deformation of b from 1.60 to 1.63 does not imply topological equivalence, as periodic systems are not in the same topological space.

Experimental results

Research questions

• RQ1 Is the topological equivalence between quasicrystals and crystals in one dimension valid under the standard classification of topological insulators?
• RQ2 Can Chern numbers be meaningfully assigned to individual quasiperiodic Hamiltonians with fixed φ, and how does this differ from periodic systems?
• RQ3 Why do boundary states in quasiperiodic systems depend on φ, and does their absence in numerical simulations imply triviality?
• RQ4 Does adiabatic deformation from quasiperiodic to periodic systems imply topological triviality, and what role does quasiperiodicity play in this context?
• RQ5 Can topological invariants like Chern numbers be defined for rational b systems via continuity, and what are the physical consequences of such a definition?

Key findings

• Quasiperiodicity acts as a topological constraint analogous to symmetry, enabling the assignment of integer Chern numbers to bulk gaps in quasiperiodic systems, even without additional symmetries.
• The Chern number of a gap in a quasiperiodic system is independent of the phase φ due to the invariance of QP functions under φ-shifts, allowing a well-defined topological invariant.
• In contrast, periodic systems (rational b) require integration over φ to define Chern numbers, making single-φ Chern numbers physically meaningless.
• The absence of subgap boundary states at sharp boundaries in Madsen et al.'s simulations is due to local breaking of quasiperiodicity, not topological triviality, and does not contradict the existence of topological phases.
• Adiabatic deformation from quasiperiodic to periodic systems does not imply topological equivalence, as periodic systems lie outside the quasiperiodic topological classification space.
• The persistence of large gaps during continuous variation of b from 1.60 to 1.63 implies that all quasiperiodic systems in this interval share the same Chern numbers in their largest gaps, but small gaps and their Chern numbers change significantly with b.