Vortia & Antivortia

Vortia and Antivortia spin opposite ways, bound together in the cold, until the heat sets them free.

The Physics

In 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.

E = πJ ln(R/a), S = 2k_B ln(R/a); ħ² ρ_s / (m² k_B T_BKT) = 2/π

• E: energy of one isolated vortex (J)
• J: spin stiffness of the 2D XY model (J)
• R: system size (m)
• a: core size or lattice spacing (m)
• S: entropy (J/K)
• k_B: Boltzmann constant (J/K)
• ρ_s: areal superfluid mass density just below T_BKT (kg/m²)
• m: mass of one helium-4 atom (kg)
• T_BKT: transition temperature (K)

Berezinskii (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.

Honest caveat: 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.

On the shirt

Front: Vortia & 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.