Skip to main content
Sandbox Physics

Topological Matter 04 · magnetic Bloch bands

Hofstadter Butterfly · Fractal Magnetic Spectrum

Diagonalize the Harper tight-binding Hamiltonian to build the fractal energy spectrum versus dimensionless flux. Park the probe near any rational flux, watch the magnetic unit cell expand into subbands, and label a selected gap with its Hall Chern number from the Diophantine equation.

Interactive modelHofstadter Butterfly · Fractal Magnetic Spectrum
Flux fraction α\alphaα=13\alpha=\frac13
Magnetic cell qqq=3q=3
Energy probe EE0.00 t0.00\,t
Gap Chern number CrC_rCr=+1C_r=+1

Physics tutorial

The Hofstadter butterfly: flux, magnetic cells, and fractal bands

BackgroundA two-dimensional lattice electron in perpendicular field feels both Bloch periodicity and cyclotron phase. At rational flux the periods combine into a magnetic unit cell.

Why it mattersThe Hofstadter spectrum unifies band theory, quantum Hall topology, and fractal self-similarity in one minimal model.

Start with the essentials

Focus question
Why does continuously changing flux repeatedly split an ordinary band into finer subbands and gaps?
One-sentence intuition
At plaquette flux α=p/q\alpha=p/q, magnetic translation has a period enlarged by qq, so one band splits into qq subbands. Sweeping all rationals produces the self-similar spectrum.

Core mathematical model

Harper equation

ψn+1+ψn−1+2cos⁡(2παn+ky)ψn=Eψn\psi_{n+1}+\psi_{n-1}+2\cos(2\pi\alpha n+k_y)\psi_n=E\psi_n

The Peierls phase converts the two-dimensional magnetic problem into a one-dimensional difference equation with a quasiperiodic potential.

Flux fraction

α=ΦΦ0=pq\alpha=\frac{\Phi}{\Phi_0}=\frac{p}{q}

Φ0=h/e\Phi_0=h/e is the single-electron flux quantum; the denominator sets magnetic-cell size.

Gap Diophantine equation

r=Crp+srqr=C_rp+s_rq

For gap rr, the integer solution CrC_r is the Hall Chern number of all occupied bands below it.

Common difficulties

The butterfly is not a particle path

Typical misconceptionThe wings show the shape of electron motion in real space.

Better mental modelThe horizontal axis is flux and the vertical axis is allowed energy; every point is an eigenvalue of a different Hamiltonian.

A finite plot is not the mathematical infinite fractal

Typical misconceptionA sufficiently large finite matrix displays every scale exactly.

Better mental modelEvery numerical plot has size and sampling cutoffs and can only approach the self-similar infinite-lattice spectrum layer by layer.

Run the experiment

  1. 01

    Compare simple fractions

    Select α=12\alpha=\frac12, α=13\alpha=\frac13, and α=25\alpha=\frac25 in sequence.

    What to observe: The right-hand flux slice gains structure as the magnetic-cell denominator grows.
  2. 02

    Move through the gaps

    Hold α=13\alpha=\frac13 and move the energy probe vertically.

    What to observe: The nearest gap count changes and the Diophantine Chern label jumps.
  3. 03

    Approximate irrational flux

    Choose the golden-ratio approximation and increase the Harper matrix size.

    What to observe: The spatial eigenstate and local spectrum become more intricate, while the finite system still resolves only finite hierarchy.