• Papilla Hofstadter's Butterfly pixel-art physics T-shirt, white tee, front view

IndexKuantumTee

Papilla

Hofstadter's Butterfly Physics

\[\psi_{n+1} + \psi_{n-1} + 2\cos(2\pi n\alpha - \nu)\,\psi_n = E\,\psi_n\]
$29.99 USD
$29.99 USD
İndirim Tükendi
Size
Color

Size guide S–5XL · inches

Papilla's wings are made of allowed energies. Turn up the magnetic field and her pattern repeats inside itself, forever.

The Physics

Put an electron on a 2-D lattice in a magnetic field and its allowed energies depend on the flux through each cell. When the flux ratio is a fraction p/q, the band splits into q subbands; when it is irrational, the spectrum becomes a Cantor set. Plotted against the flux, the energies form a self-similar butterfly.

ψ_{n+1} + ψ_{n−1} + 2 cos(2πnα − ν) ψ_n = E ψ_n, α = Φ/Φ₀

  • ψ_n: wavefunction amplitude on row n (dimensionless)
  • α: flux per cell in flux quanta (dimensionless)
  • Φ: magnetic flux through one cell (Wb)
  • Φ₀: flux quantum h/e (Wb)
  • ν: phase (rad)
  • E: energy in units of the hopping energy t (dimensionless)

Douglas Hofstadter computed the spectrum in 1976; in 2013 graphene moiré superlattices showed its fingerprints in experiments.

Honest caveat: Those experiments saw butterfly features, not the whole fractal.

On the shirt

Front: Papilla, the pixel-art character, with the title. Back: Harper's equation, the p/q rule, the Cantor-set limit and the 1976 and 2013 sources. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Ölçek için: M beden 20 in (50.80 cm) genişliğindedir, yaklaşık 3.143 × 1034 Planck uzunluğu (lP = 1.616255 × 10−35 m). Mezura daha hızlıdır.

Measurements in inches, ±1.5 in tolerance.

ShippingBy destination

Shipping calculated at checkout by destination. Printed to order for you.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

My wings are made of allowed energies. Put an electron on a square lattice, thread a magnetic field through it, and the energies it may have depend on α, the flux through each cell measured in flux quanta.

When α is a fraction p/q, the band splits into q subbands. When α is irrational, the spectrum becomes a Cantor set. Plot every allowed energy against α and you get me: a butterfly that contains smaller copies of itself, all the way down. Douglas Hofstadter computed it in 1976 as a graduate student.

For decades the fields needed were far too strong for ordinary crystals. In 2013, graphene stacked on boron nitride made a moiré pattern about ten nanometres across, large enough to reach the right flux, and three groups saw my fingerprints in quantum Hall measurements. They saw features of the butterfly, not the whole fractal.

Equations

\[\psi_{n+1} + \psi_{n-1} + 2\cos(2\pi n\alpha - \nu)\,\psi_n = E\,\psi_n\]
\[\alpha = \frac{\Phi}{\Phi_0}, \qquad \Phi_0 = \frac{h}{e}\]
SymbolMeaningUnit
\(\psi_n\) wavefunction amplitude on row n \(\mathrm{1}\)
\(\alpha\) magnetic flux per lattice cell in flux quanta \(\mathrm{1}\)
\(\Phi\) magnetic flux through one lattice cell \(\mathrm{Wb}\)
\(\Phi_0\) flux quantum h/e \(\mathrm{Wb}\)
\(\nu\) phase \(\mathrm{rad}\)
\(E\) energy in units of the hopping energy t \(\mathrm{1}\)

Sources

  1. Hofstadter (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239
  2. Dean et al. (2013) Hofstadter's butterfly and the fractal quantum Hall effect in moiré superlattices, Nature 497, 598
  3. Ponomarenko et al. (2013) Cloning of Dirac fermions in graphene superlattices, Nature 497, 594
  4. Hunt et al. (2013) Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure, Science 340, 1427

Field notes

Tümünü görüntüle
1 3
Tümünü görüntüle