{"product_id":"vortia-antivortia-the-bkt-transition-physics-t-shirt","title":"Vortia \u0026 Antivortia – The BKT Transition Physics T-Shirt","description":"\u003cp\u003eVortia and Antivortia spin opposite ways, bound together in the cold, until the heat sets them free.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn a flat 2D world, spins can't line up across all space, yet a phase transition still happens: vortex–antivortex pairs unbind. A lone vortex costs energy that grows with the log of the system size, but its entropy grows the same way. When temperature makes the free energy E − TS change sign, free vortices proliferate. At the transition, the superfluid stiffness jumps by a universal amount.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE = πJ ln(R\/a),  S = 2k_B ln(R\/a);   ħ² ρ_s \/ (m² k_B T_BKT) = 2\/π\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eE: energy of one isolated vortex (J)\u003c\/li\u003e\n\u003cli\u003eJ: spin stiffness of the 2D XY model (J)\u003c\/li\u003e\n\u003cli\u003eR: system size (m)\u003c\/li\u003e\n\u003cli\u003ea: core size or lattice spacing (m)\u003c\/li\u003e\n\u003cli\u003eS: entropy (J\/K)\u003c\/li\u003e\n\u003cli\u003ek_B: Boltzmann constant (J\/K)\u003c\/li\u003e\n\u003cli\u003eρ_s: areal superfluid mass density just below T_BKT (kg\/m²)\u003c\/li\u003e\n\u003cli\u003em: mass of one helium-4 atom (kg)\u003c\/li\u003e\n\u003cli\u003eT_BKT: transition temperature (K)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eBerezinskii (1971) and Kosterlitz and Thouless (1973) found it; Bishop and Reppy saw the jump in helium films in 1978. Kosterlitz and Thouless shared the 2016 Nobel Prize.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The simple estimate k_B T = πJ\/2 is a bare value; the renormalised transition temperature is lower. 2D order isn't absent, it is quasi-long-range.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Vortia \u0026amp; Antivortia, the pixel-art character, with the title. Back: the vortex energy and entropy, the bare transition estimate, the universal jump, and the 1971\/1973\/1978\/2016 history. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960824434791,"sku":"28156819958764855060","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960824467559,"sku":"17698890735234799546","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960824500327,"sku":"24986943417801669810","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960824533095,"sku":"23566283020927263749","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960824565863,"sku":"23152816362326898804","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960824598631,"sku":"27913446324559551087","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960824631399,"sku":"19832609618286057308","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960824664167,"sku":"42082332405081277035","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960824696935,"sku":"15660087523906892576","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960824729703,"sku":"11690198199465569797","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960824762471,"sku":"20619284614492748524","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960824795239,"sku":"66543661093001735231","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960824828007,"sku":"30517303145086066810","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960824860775,"sku":"20011867660906096625","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960824893543,"sku":"29469019029609635429","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960824926311,"sku":"31988724245615027941","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/bkt-transition_front_01_white.png?v=1793628542","url":"https:\/\/pixelatedphysics.com\/ko\/products\/vortia-antivortia-the-bkt-transition-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}