Windowsill Lab · Field Explainer · M10

The magnet with
no magnetism

What if a material were perfectly, rigidly ordered — and every instrument you pointed at it read exactly zero?

2.282measured néel point
2.2692exact
≤ 0.003ordinary magnetisation
A dark tiled windowsill surface that reads as one flat grey plane until a shaft of lamplight catches one corner and reveals a crisp brass checkerboard, with a small brass dial nearby reading dead centre.
Illustration (AI-painted) — a checkerboard only the right light reveals

Take the original magnet from M01 and change one sign. Instead of neighbours wanting to agree, neighbours want to disagree. Everything else stays the same: same lattice, same temperature sweep, same code, one flipped sign in one line.

Cool it down and it orders beautifully. Every site is the opposite of all four of its neighbours, so the whole sheet locks into a perfect checkerboard. It is as rigidly organised as the ferromagnet ever was. And its total magnetisation is zero — not small, not fading, but zero at every temperature, because for every up there is an adjacent down. Across the entire published sweep the ordinary magnetisation never rises above 0.003.

So an instrument that measures magnetisation reports nothing, everywhere, and looks broken. The order is there; you are asking the wrong question. Ask instead: how checkerboard-like is it? Multiply every site by +1 or −1 according to its own square colour, then average. That quantity — the staggered magnetisation — is large when cold, zero when hot, and behaves exactly like an ordinary order parameter for a transition that was invisible a moment ago.

Where is that transition? Exactly where the ferromagnet's was: 2.2692. On this lattice the antiferromagnet is the ferromagnet in disguise — flip every site on one colour of the board and one turns into the other, bond for bond.

The panel below has one toggle, and it is the whole point. Leave it off, drag the temperature, and watch grey static that never changes. Then turn it on. Nothing about the material changed. Only the question did.

Live · square-lattice antiferromagnet, J = −1 · checkerboard Metropolis SMALL LATTICE — THE REAL RUN USED L=128
T = 2.100

left · the same data, two questions. The inset is a magnified patch — that fine static is a perfect checkerboard. · right · two needles: staggered susceptibility (ember) and ordinary susceptibility (slate), both in the same units. One sees everything. One sees nothing. On this smaller sheet the ordinary |m| wanders up to a few hundredths rather than the published 0.003 — smaller lattice, bigger wobble — and it is still nothing beside the staggered signal.

settling…

2.282néel point, staggered χ peak MEASURED
2.276specific-heat peak, same run MEASURED
0.5 %relative error MEASURED
≤ 0.003ordinary magnetisation, whole sweep MEASURED
30 swall time on GPU MEASURED

The graded run — J = −1, L = 128, 25 temperatures in [2.0, 2.6], 40,000 sweeps with 8,000 burn-in, seed 42 — put the ordering point at 2.282 from the staggered susceptibility peak, 0.5 % from the exact 2.2692, and a separate specific-heat measurement from the same run landed independently at 2.276. The ordinary magnetisation across that entire sweep peaked at 0.00311. The instrument that should show the transition shows nothing, and that nothing is the best number on the page.

The finite-lattice peak sits just above the true infinite-volume value, as it does on every rung of this ladder — 2.282 and 2.276 from two separate observables against an exact 2.2692. The stronger caveat is about what this rung does and does not demonstrate: the square lattice is bipartite and unfrustrated, so this antiferromagnet is exactly the ferromagnet wearing a disguise, and recovering Onsager's temperature here is a check that the framework handles a negative coupling cleanly rather than a new result. The genuinely hard case — an antiferromagnet on a lattice where the disagreements cannot all be satisfied — is a later rung. The strongest guard on this one is a cross-check asserting that the antiferromagnet's staggered observables match a ferromagnet's ordinary ones bond for bond, which is what would catch a silent sign error quietly reverting the whole thing to the ferromagnet.
Elsewhere on the ladder: M01 — the ferromagnet it is wearing M04 — two needles, both of them seeing M05 — the lattice where frustration begins