Topological connections between the 2D Quantum Hall problem and the 1D quasicrystal

TL;DR

The paper addresses how the topology of the $2$D Quantum Hall problem is inherited by one-dimensional Fibonacci quasicrystals. It introduces the Fibonacci-Hall model based on the cut-and-project construction and a geometric flux to interpolate between $2$D and $1$D, enabling adiabatic continuity of band topology and linking the $2$D Chern numbers to the $1$D gap labels. The authors compute the band Chern numbers via Bott indices and show a bulk-edge correspondence in open chains, with edge-state crossings controlled by the phason angle and yielding topological pumping observable in experiments. They discuss extensions to other $1$D quasiperiodic models and higher-dimensional CP tilings, and outline experimental platforms (e.g., cold atoms, photonics) to realize and measure the predicted topological responses.

Abstract

1D quasicrystals such as the Fibonacci chain have been said to ``inherit" their topological properties from the 2D Quantum Hall problem. Yet, a direct way to see the connection was lacking until a common ancestor, the Fibonacci-Hall model, was introduced recently \cite{aj2025}. This 2D ancestor model relates the role of the external magnetic flux in the Hall problem and that of a geometric flux which describes the winding of the quasicrystal in 2D, in the cut-and-project method. Doing this enables us to extend the notion of Chern numbers as defined in 2D, to the energy bands of the 1D chain by adiabatic continuity. The older notion of gap labels in the 1D system are now seen to be derivable from the Chern numbers of the 2D bands. The Fibonacci-Hall model thus provides an important link between physics of two paradigmatic models, the Fibonacci quasicrystal and the quantum Hall insulator. The generalization to other 1D quasiperiodic models is expected to be relatively straightforward. The extension to 2D cut-and-project tilings is left for future studies.

Figures (5)

• Figure 1: (Left) Schema for the Fibonacci hopping Hamiltonian with two different hopping amplitudes $t_a$ and $t_b$. (Right) Schema for the Quantum Hall problem, showing a portion of a square lattice. The magnetic flux per plaquette of $\phi$ is shown, and hopping amplitudes $t_a$ and $t_b$ along the $x$ and $y$ directions respectively are indicated.
• Figure 2: (Left) A portion of the 2D plane illustrating the F-H Hamiltonian for approximant $k=4$ having 5 sites per unit cell. The black zigzag lines join points belonging to different parallel approximant chains. Intra-chain hoppings take place on the thick black bonds while the inter-chain bonds are shown in red. A structural unit cell is outlined in red. (Right) A single unit square with edges identified to form a torus, showing the winding of the blue line of slope $2/3$ representing the strip S for the $k=4$ approximant.
• Figure 3: (Left) The blue line joins points which lie within the CP selection strip shown with its window, $W_1$ (phase angle $\theta_1$). Shifting the window results in discrete changes of the blue zigzag line, which recovers its original shape (upto a shift) when the window reaches the position $W_2$ (phase angle $\theta_1+2\pi$) shown in red. The displacement corresponds to a complete cycle of the phase angle $\theta$. (Right) As the position of the selection window is varied and $\theta$ cycles through $2\pi$, the periodic approximant chain undergoes discontinuous changes. Successive changes are shown here for the case $N=5$. In each case only two unit cells of the infinite chains are shown. The original chain, shown at the bottom of the figure, changes discontinuously and cycles through 5 different shapes at $\theta=\theta_j$ ($j=1,...,5$), returning to its original shape at the end of the cycle.
• Figure 4: Plot of energy bands and energies of edge states as a function $\phi$ for the $k=5$ approximant ($N=13$)
• Figure 5: Plots of wavefunction amplitudes versus site number for a chain of 91 sites (7 unit cells of the $k=5$ approximant) showing the evolution of edge states as the phason angle $\phi$ is varied between 0 and $2\pi$. The edge states shown correspond to the energy gap of label $q=-3$.