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Ising Model Hoodie

EVERY SCALE AT ONCE | Pixelated Physics

\[H = -J\sum_{\langle ij\rangle} s_is_j,\qquad s_i = \pm1\]
$54.99 USD
$54.99 USD
Obral Habis
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Size guide S–5XL · inches

A grid of tiny magnets, each pointing up or down, each preferring to agree with its neighbours. Heat tries to scramble them. A 2D Ising magnet simulated at its exact critical temperature, kTc = 2J/ln(1+√2): up and down domains of every size at once. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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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

In the Ising model each spin points up or down and prefers to match its neighbours, while heat tries to scramble them. Onsager solved the two-dimensional case exactly, finding a sharp critical temperature below which the spins line up into a magnet.

A grid of tiny magnets, each pointing up or down, each preferring to agree with its neighbours. Heat tries to scramble them.

Below a critical temperature they line up into a magnet. Above it they don’t. Lars Onsager solved the two-dimensional case exactly in 1944: the transition sits at kTc = 2J/ln(1 + √2).

Exactly at Tc there is no typical domain size. Islands of up spins contain lakes of down spins that contain islands again, at every scale. The art is a 144 × 144 lattice equilibrated at Tc with Wolff cluster updates, then coarse-grained once by majority vote over 2 × 2 blocks, Kadanoff’s renormalization step. At Tc the coarse picture looks statistically like the fine one.

Out of the equilibrium snapshots, the one closest to zero net magnetization was chosen.

Every scale at once.

Equations

\[H = -J\sum_{\langle ij\rangle} s_is_j,\qquad s_i = \pm1\]
\[k_BT_c = \frac{2J}{\ln(1+\sqrt2)} \approx 2.269\,J\]
SymbolMeaningUnit
\(H\) energy of a spin configuration \(\mathrm{J}\)
\(J\) coupling between neighbouring spins \(\mathrm{J}\)
\(s_i\) spin at site i, up or down \(\mathrm{1}\)
\(\langle ij\rangle\) sum over nearest-neighbour pairs \(\mathrm{1}\)
\(k_B\) Boltzmann constant \(\mathrm{J/K}\)
\(T_c\) critical temperature \(\mathrm{K}\)

Sources

  1. Ising (1925) Beitrag zur Theorie des Ferromagnetismus, Z. Phys. 31, 253
  2. Onsager (1944) Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117
  3. Wolff (1989) Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62, 361
  4. Kadanoff (1966) Scaling laws for Ising models near Tc, Physics 2, 263

Source Onsager, L. (1944). Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Physical Review 65, 117–149.doi.org/10.1103/PhysRev.65.117

Field notes

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