{"product_id":"papilla-hofstadters-butterfly-physics-t-shirt","title":"Papilla – Hofstadter's Butterfly Physics T-Shirt","description":"\u003cp\u003ePapilla's wings are made of allowed energies. Turn up the magnetic field and her pattern repeats inside itself, forever.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003ePut 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eψ_{n+1} + ψ_{n−1} + 2 cos(2πnα − ν) ψ_n = E ψ_n,   α = Φ\/Φ₀\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eψ_n: wavefunction amplitude on row n (dimensionless)\u003c\/li\u003e\n\u003cli\u003eα: flux per cell in flux quanta (dimensionless)\u003c\/li\u003e\n\u003cli\u003eΦ: magnetic flux through one cell (Wb)\u003c\/li\u003e\n\u003cli\u003eΦ₀: flux quantum h\/e (Wb)\u003c\/li\u003e\n\u003cli\u003eν: phase (rad)\u003c\/li\u003e\n\u003cli\u003eE: energy in units of the hopping energy t (dimensionless)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eDouglas Hofstadter computed the spectrum in 1976; in 2013 graphene moiré superlattices showed its fingerprints in experiments.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Those experiments saw butterfly features, not the whole fractal.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: 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.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50936423809127,"sku":"24514473121024317917","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50936423841895,"sku":"22917331663404980497","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50936423874663,"sku":"22577136792632353747","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50936423907431,"sku":"20271402960609501284","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50936423940199,"sku":"27330723619580894410","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50936423972967,"sku":"10435240210161079499","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50936424005735,"sku":"29024263304547455512","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50936424038503,"sku":"12834058414266461556","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50936424071271,"sku":"29672131250800659822","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50936424104039,"sku":"27247775413272444321","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50936424136807,"sku":"94758219692805567217","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50936424169575,"sku":"18892129280176307588","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50936424202343,"sku":"24750266961566727583","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50936424235111,"sku":"12230344499082881539","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50936424267879,"sku":"27014172608399633319","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50936424300647,"sku":"32082559289615077922","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/hofstadter-butterfly_front_02_white.png?v=1793595175","url":"https:\/\/pixelatedphysics.com\/ko\/products\/papilla-hofstadters-butterfly-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}