Kohmoto Model Indices & Butterfly Coloring

• Kohmoto Model Indices are integer invariants that classify spectral gaps and characterize the fractal structure of one-dimensional quasiperiodic Hamiltonians.
• They are computed via a novel recursive combinatorial framework that resolves the discontinuity issues preventing the use of conventional Chern numbers.
• This method uniquely colors the Kohmoto butterfly, offering a rigorous classification tool for theoretical studies and applications in quasicrystals and photonic systems.

Kohmoto Model Indices are integer-valued invariants that classify the spectral gaps and topological structure of one-dimensional quasiperiodic Hamiltonians, particularly exemplified by the Kohmoto model and related Sturmian systems. Unlike standard Chern numbers used in the Hofstadter model and other smooth (differentiable) quasiperiodic systems, the discontinuity in the Kohmoto potential prohibits direct application of conventional topological invariants. Instead, Kohmoto Model Indices are constructed using a new combinatorial and recursive framework that resolves the modulo ambiguity inherent in the Diophantine labeling of periodic approximants, yielding a complete classification and “coloring” of the Kohmoto butterfly—a fractal spectral phase diagram characteristic of these models.

1. Spectral Problem and the Challenge of Discontinuous Potentials

The Kohmoto Hamiltonian is a tight-binding operator of the form

$$(H_{α, V}\psi)(n) = \psi(n + 1) + \psi(n - 1) + V_α(n)\psi(n),$$

with the potential defined as

$V_α(n) = λ \; χ_{[1-α,\,1)}(nα \bmod 1),$

where α ∈ [0,1] is an irrational frequency, λ is the coupling, and $χ_{[1-α,\,1)}$ is the characteristic function of the indicated interval. For periodic approximants (α = p/q), the spectrum consists of exactly q bands (intervals), separated by gaps. The integrated density of states (IDS) N_α(E) in the nth gap traditionally satisfies a Diophantine equation

$N_α(E) = \frac{n}{q} = c_α \bmod 1,$

where c_α is an integer-valued index invariant, modulo q. In the Hofstadter problem with a smooth potential, this ambiguity is resolved by relating c_α to a Chern number; however, the discontinuity in the Kohmoto model prevents a direct topological identification, necessitating a new combinatorial solution (Band et al., 28 Sep 2025).

2. Spectral Tree Graph and Periodic Approximants

To surmount the absence of differentiability, the Kohmoto butterfly is encoded using a spectral tree construction that systematically organizes the spectra of all periodic approximants:

• Any irrational α is represented as a continued fraction expansion

$$α = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \ldots}}.$$

• Truncated expansions α_k = p_k / q_k yield rational approximants with spectra of q_k bands.
• The spectral tree (“α-tree”) is then constructed with:
  • Root node (level k = -1)
  • Recursive connections to band (A), gap (G), and intermediate (B) vertices, dictated by the continued fraction digits {a_k}:
  • From an A (or B) vertex at level k, depending on a_{k+1}, there are 2M+1 child vertices in level k+1 (M = a_{k+1} - 1 for A, M = a_{k+1} for B), alternating between G and A.
  • Gaps propagate as G-nodes, bands as A-nodes; every G-node at level k connects to one B-node at level k+1.
  • This graph encodes all periodic spectra and the parentage of each gap through all scales.

3. Index Assignment: Matrix-Based Formula and Centered Modulo

Each node v (gap) in the spectral tree at a given level k is assigned a matrix

$$Q_k(v) = \begin{bmatrix} N_A^{(k)} & A(v) \ N_B^{(k)} & B(v) \end{bmatrix},$$

where N_A^{(k)} (N_B^{(k)}) is the total number of A (B) nodes at level k, and A(v) (B(v)) is the number of A (B) nodes left of v. The raw index is then

$i_k(v) = (-1)^k \cdot \mathrm{det} Q_k(v).$

To resolve the periodicity ambiguity, a centered modulo operation is applied: $c_k(v) = i_k(v)\bmod^*q_k,$ where x mod^* q is the unique representative in [–q/2, q/2). This makes the index sensitive to sign and parity and avoids the ambiguities present in the original Diophantine equation. Thus, c_k(v) provides a unique integer label for every gap in every periodic approximant.

4. Conservation of Indices and the Quasiperiodic Limit

A crucial property of the construction is conservation of the index along certain infinite paths γ = (v_0, v_2, ..., v_{2m}, ...) in the tree. The path is chosen by parity/sign conventions such that for relevant vertices,

$c_k(v) = c_{k+2m}(v_{2m})\quad\forall m \geq 0,$

for each even-indexed vertex v_{2m} descended from v. This ensures that the index, defined at a periodic level, stabilizes and remains well-defined in the quasiperiodic (irrational α) limit, aligning with the requirements of the gap-labelling theorem: $N_α(E) = c\,α \mod 1,\quad c \in \mathbb{Z},$ with every possible integer c appearing in the full quasiperiodic spectral labeling.

5. Full Coloring of the Kohmoto Butterfly

With the unique index assignment, the Kohmoto butterfly—the fractal union of all σ(H_{p/q, V}) plotted against α—can be completely and uniquely “colored.” Every gap, at every scale, receives its natural integer label, and the coloring is unambiguous even in regions where conventional topological invariants fail. The approach is robust to the presence of spectral defects (localized impurities) that occur when irrational α are approximated by rationals: the indices remain conserved across spectral branches, and impurities appear as discrete gap states indexed by the same formalism (Beckus et al., 23 Oct 2024).

6. Comparison to Hofstadter Butterfly and General Implications

While the “gap-labeling theorem” for the Hofstadter model (and Almost–Mathieu operator) resolves modulo ambiguities via nontrivial Chern numbers—reflecting the differentiable structure of the potential—the Kohmoto model requires the combinatorial tree construction. Nevertheless, both systems show similarities in their fractal structure, recursive band and gap formation, and scaling of the spectral Hausdorff distance with suitable metrics (Farey metric for Kohmoto, continued fraction/Hölder for Hofstadter), although the coloring and topological invariants fundamentally differ due to underlying differences in potential regularity (Beckus et al., 23 Oct 2024).

This integer-valued spectral labeling rooted in the continued fraction structure and recursive tree is unique to models with discontinuous potentials and provides both a rigorous mathematical invariant and a physical classification of spectral gaps, accessible to analysis in concrete experimental and theoretical contexts involving quasicrystals, photonic waveguides, and related systems. The solution extends to a broad class of aperiodic models where standard topological tools are not directly applicable, supplying a paradigm for “coloring” spectral butterflies in quasiperiodic systems.