From Quasiperiodicity to a Complete Coloring of the Kohmoto Butterfly

TL;DR

This work resolves topological indexing for the Kohmoto model, a paradigmatic quasicrystal with a discontinuous potential that eludes standard Chern-number-based invariants. By encoding the spectra of periodic Kohmoto Hamiltonians into a spectral $\alpha$-tree built from continued-fraction data, the authors define gap indices $\mathfrak{c}_{k}(v)$ via $\mathfrak{c}_{k}(v)=(-1)^{k}\det Q_{k}(v)\pmod{*} q_{k}$, and prove Diophantine consistency and index conservation along infinite paths in the tree. They show convergence of periodic-gap labels to the quasiperiodic limit, yielding a full coloring of the Kohmoto butterfly and resolving the modulo ambiguity in a manner consistent with the gap-labelling theorem. The results provide a structural framework for other quasiperiodic models and offer precise, finite-size realizations for experiments, while drawing connections and distinctions with the Hofstadter butterfly.

Abstract

The spectra of the Kohmoto model give rise to a fractal phase diagram, known as the Kohmoto butterfly. The butterfly encapsulates the spectra of all periodic Kohmoto Hamiltonians, whose index invariants are sought after. Topological methods - such as Chern numbers - are ill defined due to the discontinuous potential, and hence fail to provide index invariants. This Letter overcomes that obstacle and provides a complete classification of the Kohmoto model indices. Our approach encodes the Kohmoto butterfly as a spectral tree graph, reflecting the quasiperiodic nature via the periodic spectra. This yields a complete coloring of the phase diagram and a new perspective on other spectral butterflies.

Figures (6)

• Figure 1: The right panel shows the Kohmoto butterfly -- the spectral bands are plotted for different rational frequencies $\alpha$. In the left panel each spectral gap for a periodic Hamiltonian is colored according to its index.
• Figure 2: An example of a spectral $\alpha$-tree is sketched if $\alpha$ has continued fraction expansion $\left(a_{k}\right)_{k=0}^{\infty}$ starting with $0,1,2,3$. The vertices of the graph are drawn as the spectral bands to which they correspond; their labels ($A/B)$ are indicated. Vertices representing gaps $G$ are indicated by circles.
• Figure 3: Illustration of the tree for a continued fraction beginning with $0,3,2,1,2$. The index is marked within the circle of the $G$-vertex. Four $G$-vertices are highlighted with the corresponding infinite paths having index $\mathfrak{c}_{k}(v)=-1$ respectively $\mathfrak{c}_{k}(v)=4$.
• Figure 4: An illustration of the conservation deduced from (\ref{['eq:=000020ind=000020relations=000020-=000020k=000020even']}) and (\ref{['eq:=000020ind=000020relations=000020-=000020k=000020odd']}) for odd $k$. For all blue vertices $i_{\ast}(\ast)$ is preserved and for all red vertices $i_{\ast}(\ast)-q_{\ast}$ is preserved. We indicate the path $\gamma$ which is the one is chosen if $\mathfrak{c}_{k}(v)<0$.
• Figure 5: Illustration of different $G$-vertices: sandwiched between an $\mathit{A}$-vertex and a $\mathit{B}$-vertex (Left) or between two $\mathit{A}$-vertices (Right).