[论文解读] Metallic mean quasicrystals and their topological invariants

Anuradha Jagannathan|arXiv (Cornell University)|Feb 10, 2026

Quasicrystal Structures and Properties被引用 0

一句话总结

The paper derives a complete set of topological invariants for the metallic mean family of one-dimensional quasicrystals by linking their finite approximants to two-dimensional quantum Hall problems and validating with edge-state windings.

ABSTRACT

Topological invariants govern many important physical properties in condensed matter systems. In this work, we obtain the complete set of topological invariants for a family of one-dimensional quasicrystals. The first and best-studied member of the family is the Fibonacci chain, while the successive ones are known in the literature as silver, bronze... and collectively as the metallic mean chains. By considering rational approximants, and by making use of the relationship between these chains and two dimensional Quantum Hall problems, we write down a gap labeling scheme for finite systems, and extend it to the quasiperiodic limit. We show, by numerical computations on open chains, that the proposed scheme correctly yields the winding numbers of edge states in each of the gaps, in all of the quasicrystals. In the strict 1D limit, we discuss properties of a simplified Hofstadter ``butterfly" diagram, with the analogues of Landau levels appearing in the asymptotic limit.

研究动机与目标

• Provide a global, unified description of topological invariants for the metallic mean family of 1D quasicrystals.
• Generalize gap labeling from the Fibonacci (golden) case to silver, bronze, and higher metallic means.
• Establish a 2D quantum Hall framework for finite approximants and extend the labeling to the quasiperiodic limit.
• Demonstrate bulk-edge correspondence by analyzing edge-state windings under phason variations.
• Explore simplifications in the large-n limit and relate results to Hofstadter-like butterfly structures.

提出的方法

• Construct 1D tight-binding Hamiltonians for metallic mean approximants with A and B hopping sequences.
• Introduce a 2D Hofstadter-like model with a geometric flux phi^(k)_n determined by P_n^(k)/Q_n^(k) to access topological indices.
• Use adiabatic continuity from the 2D QH problem to infer 1D topological invariants for the metallic mean chains.
• Employ the gap labeling theorem to relate integrated density of states in gaps to integers p_j and q_j via I_n(j)=p_j+q_j P_n^(k)/Q_n^(k).
• Compute edge-state windings by varying the phason angle theta in finite open chains to verify the proposed labeling.
• Analyze asymptotic large-n behavior to reveal a simple, Landau-level-like structure and a simplified gap labeling as n becomes large.

实验结果

研究问题

• RQ1 Can a complete gap-labeling scheme be extended from the Fibonacci chain to the entire metallic mean quasicrystal family?
• RQ2 How do 2D quantum Hall mappings illuminate the topological invariants of 1D metallic mean chains?
• RQ3 Do edge states in open finite approximants wind with the phason angle in a manner that matches the gap labels?
• RQ4 What is the behavior of the gap labeling and edge-state structure in the large-n limit?
• RQ5 How do the spectra organize into a Hofstadter-like butterfly across the metallic mean family?

主要发现

• A Diophantine gap-labeling relation I_n(j)=p_j+q_j P_n^(k)/Q_n^(k) is established for all metallic mean approximants.
• The integers q_j (with |q_j| ≤ N/2) label the gaps and determine edge-state windings, validating bulk-edge correspondence in finite open chains.
• In the infinite quasiperiodic limit, I_n(q) tends to Mod[q ω_n/(1+ω_n),1], linking gap labels to the metallic mean ω_n.
• The 1D spectra exhibit a Hofstadter butterfly structure with Landau-like levels appearing at the corners and gaps labeled by increasing |q|.
• As n grows large, most energy levels group into n bands with a simple, consecutive labeling pattern (±1, ±2, ±3, …).
• Open-chain edge-state windings across gaps reproduce the predicted q_j values, confirming the topological labeling scheme across the metallic mean family.