Quasicrystals’ Hidden Rules Unlocked, Paving the Way for New Materials Design

Scientists investigate the topological invariants governing one-dimensional quasicrystals, revealing crucial insights into condensed matter physics. Anuradha Jagannathan from Laboratoire de Physique des Solides, Université Paris-Saclay, alongside colleagues, detail a complete set of invariants for a family of these structures, including the well-known Fibonacci chain and related ‘metallic mean’ chains. This research establishes a gap labelling scheme, extended to the quasiperiodic limit, and demonstrates its accuracy through numerical analysis of edge state winding numbers. By exploring a simplified Hofstadter butterfly diagram, the team illuminate the emergence of Landau levels, significantly advancing our understanding of electronic behaviour in these aperiodic systems.

Topological invariants and winding number calculations in metallic mean quasicrystals

Scientists have uncovered a complete set of topological invariants for a family of one-dimensional quasicrystals, extending beyond the well-studied Fibonacci chain to encompass silver, bronze, and collectively termed “metallic mean” chains. This work establishes a unified framework for understanding the topological properties of these quasiperiodic structures, offering insights relevant to materials science and condensed matter physics.

The research demonstrates a novel approach to calculating these invariants by linking the quasicrystals to two-dimensional Quantum Hall problems, thereby leveraging established theoretical tools. By considering rational approximations and utilising the connection to these Hall problems, researchers developed a gap labelling scheme applicable to both finite and infinite quasicrystalline systems.

Numerical computations performed on open chains confirm that this scheme accurately predicts the winding numbers of edge states within each energy gap across all metallic mean quasicrystals. The study reveals a simplified Hofstadter “butterfly” diagram in the strict one-dimensional limit, exhibiting analogues of Landau levels as the system approaches its asymptotic state.

This diagram arises from the mapping of the quasicrystals to a series of two-dimensional models, each corresponding to a periodic approximant. The approach defines a 2D Quantum Hall problem for each finite periodic approximation of these quasicrystals, creating a subset of the Hofstadter butterfly diagram.

The complete set of these 2D problems serves as the basis for a topological indexing ansatz, a proposed method for determining the topological invariants of the entire family of quasiperiodic systems. To validate this ansatz, the winding of edge states within each gap was examined for finite chains with open boundaries.

The research demonstrates that as the size of the approximant increases, the system converges towards the true quasiperiodic limit, confirming the validity of the proposed scheme for all metallic mean chains. This work provides a global description of topological invariants for a family of quasiperiodic structures, advancing the understanding of these less-studied systems compared to their crystalline counterparts.

Fibonacci chain construction and edge state winding number calculations

Rational approximants and a connection to two-dimensional Hall problems underpin the investigation of one-dimensional quasicrystals within this work. The study begins by constructing finite systems and extending the gap labelling scheme to the quasiperiodic limit, focusing on the Fibonacci chain and its metallic mean counterparts, silver, bronze, and others.

Numerical calculations performed on open chains demonstrate the accuracy of the proposed scheme in determining the winding numbers of edge states present in each gap across all investigated quasicrystals. To facilitate the 2D modelling, the metallic mean chains are generated using the cut-and-project method, beginning with a two-dimensional square lattice.

An infinite strip of rational slope P(k−2)n /P(k−1)n is drawn, defining a unit cell of Q(k)n sites, and the chain is obtained by projecting points within the strip onto a line. Shifting the selection strip yields different finite chains for finite samples, a factor considered when examining edge states with open boundary conditions.

The P and Q series, related by Q(k)n = P(k)n + P(k−1)n, define the number of letters in the kth approximant chain of the nth quasicrystal. Tight-binding Hamiltonians are then employed to model the systems in both one and two dimensions. The 1D Hamiltonian, H1D(n, k), incorporates hopping amplitudes tA or tB, determined by the sequence of A and B elements within a given approximant Cn k.

A chiral symmetry present in the model results in a spectrum symmetric around E = 0. Topological numbers are computed by linking the 1D Fibonacci chain to a 2D Quantum Hall problem, describing three electrons hopping on a square lattice subject to a fictitious geometric flux φ(k)n = P(k)n Q(k)n. The 2D Hamiltonian, H2D(φ(k)n), utilizes the Landau gauge with the fictitious vector potential parallel to the y axis, and hopping amplitudes depend on position.

Adiabatic continuity is established between the isotropic 2D Quantum Hall model and the 1D quasiperiodic system, allowing topological invariants calculated in the 2D model to be transferred to the 1D metallic mean chains. The integrated density of states in each gap, In(j), is defined by the equation In(j) = pj + qj P(k)n Q(k)n, where pj and qj are integers representing gap labels. This Diophantine equation serves as a principal result, enabling the determination of the gap labels and, consequently, the edge state winding numbers, as demonstrated through energy spectra plots for gold, silver, and bronze means.

Topological invariants and edge state winding numbers in metallic mean quasicrystals

Researchers detail a comprehensive analysis of topological invariants within a family of one-dimensional quasicrystals known as metallic mean chains. The study establishes a gap labelling scheme for finite systems, successfully predicting the winding numbers of edge states across all gaps in the quasicrystals through numerical calculations on open chains.

This work extends beyond the well-studied Fibonacci chain, encompassing silver, bronze, and other metallic mean chains, providing a unified framework for understanding their topological properties. The research defines a two-dimensional quantum Hall problem for each finite periodic approximant of these quasicrystals, linking them to a subset of the Hofstadter butterfly diagram.

The substitution matrix, central to generating the approximant chains, exhibits eigenvalues governing their asymptotic growth, with the number of tiles in the kth chain increasing as ωk n, where ωn represents the largest root of the equation ω2 = nω + 1. Specifically, for the Fibonacci chain where n equals 1, the largest root is ω1 = (√5 + 1)/2.

Further analysis reveals that the ratio of B tiles to A tiles in the infinite chain approaches 1:ωn. Rational approximants of ωn are derived through continued fraction truncation, with the Pell numbers emerging for the silver mean (n=2) and their ratio tending towards ω2 = (√2 + 1)/2 as the approximation order increases.

The number of letters in the kth approximant chain, denoted by Q(k) n, is determined by a recursive relation and linked to another series, P(k) n, which satisfies the same recursion but with different initial conditions. These 2D models facilitate a topological indexing ansatz applicable to the entire family of quasiperiodic systems, offering insights into their unique electronic properties and potential applications.

Topological invariants and energy gap labelling in metallic mean quasicrystals

Scientists have established a comprehensive understanding of topological invariants within a family of one-dimensional quasicrystals known as metallic mean chains. These chains generalise the well-known Fibonacci chain and include silver and bronze structures, allowing for a unified description of their topological properties.

By relating these quasicrystals to two-dimensional Quantum Hall models, a scheme was developed to label the gaps in their energy spectra, accurately predicting the winding numbers of edge states within each gap. Numerical computations on finite chains confirmed the validity of this gap labelling scheme, demonstrating its ability to correctly identify the behaviour of edge states across all members of the metallic mean family.

Furthermore, analysis revealed a simplified Hofstadter butterfly structure within these quasicrystals, with analogues of Landau levels appearing in the asymptotic limit. This work provides a global picture of topological characteristics in quasicrystals, moving beyond studies focused on specific structures.

The authors acknowledge that their analysis primarily focused on pure hopping Hamiltonians, with the gap labelling scheme potentially applicable to a wider range of quasiperiodic Hamiltonians provided level crossings do not occur. Future research could explore the effects of coupling these metallic mean chains to phonons, potentially leading to novel charge density modulated phases and topological phase transitions. The findings have implications for experimental systems, particularly those involving edge states and phenomena like unusual Josephson effects, potentially extending beyond the Fibonacci case to the broader metallic mean family.