Experiment 1: find Tc from the Binder cumulant
Wolff-cluster Monte-Carlo for the current order type. U4 = 1 - <m4>/(3<m2>2) curves for different L cross at Tc. For the ferromagnet and antiferromagnet the exact value 2/ln(1+√2) = 2.2692 is known; the altermagnet toy model has no closed form.
The science
exact model Ising spins on a square lattice. Each site carries s = ±1 with H = -∑<ij> Jij si sj - h ∑i si, units |J| = kB = 1. J > 0 (ferromagnet) rewards parallel neighbours, J < 0 (antiferromagnet) rewards anti-parallel ones. The lattice is sampled by Metropolis Monte-Carlo: pick a site, flip it with probability min(1, e-ΔE/T).
Order parameters. Net magnetisation M = (1/N)∑si; staggered (Neel) magnetisation N = (1/N)∑(-1)x+y si. Ferromagnet: M ≠ 0 below Tc. Antiferromagnet: N ≠ 0 below the Neel temperature but M = 0, so it has no stray field and does not respond strongly to a field.
exact Onsager (1944). For the 2-D NN Ising model Tc = 2J/(kB ln(1+√2)) = 2.2692 J/kB and the spontaneous order is m = [1 - sinh-4(2J/kBT)]1/8. The same curve holds for N in the antiferromagnet (flip every second spin: J → -J maps one onto the other). Experiment 2 overlays it. Experiment 1 finds Tc numerically: the Binder cumulant U4 = 1 - <m4>/(3<m2>2) is size-independent at Tc, so curves for L = 8, 16, 32 cross there (sampled with the Wolff cluster algorithm to beat critical slowing down).
toy model Altermagnet. In an altermagnet the two opposite-spin sublattices are NOT related by a translation or inversion (as in a normal antiferromagnet) but by a rotation (or mirror/screw): the non-magnetic atoms around sublattice A are an elongated cage pointing one way, around B the same cage turned 90°. Here this is modelled by extra ferromagnetic next-nearest (diagonal) couplings J2 = 0.25(1 ± a) whose strong axis is turned by 90° between A and B. The ordering is still collinear Neel order with M = 0; only the thermodynamic TN shifts. "Rotate crystal 90°" swaps the roles of the sublattices.
schematic Spin-split bands. The Fermi surfaces are NOT computed from a real material. They come from a hand-made one-band tight-binding cartoon εσ(k) = -2(cos kx + cos ky) + 1.5 + [−σ U M] + [σ Δ0 a N (cos kx − cos ky)] with M and N taken live from the Monte-Carlo. Ferromagnet: the term in M shifts spin-up and spin-down surfaces apart uniformly (s-wave splitting). Antiferromagnet: neither term, bands are spin-degenerate. Altermagnet: the d-wave term is non-zero only where |kx| ≠ |ky|, flips sign between the x and y directions, vanishes along the diagonals, needs the Neel order N, and averages to zero net magnetisation. Reversing all spins flips its sign; rotating the crystal by 90° does too.
Why people care (reported, not a product). Altermagnetism was identified as a third collinear magnetic class by Šmejkal, Sinova and Jungwirth (Phys. Rev. X 12, 2022; building on earlier symmetry work such as Hayami et al. 2019 and Ahn et al. 2019). Angle-resolved photoemission on MnTe was reported to show the predicted spin-split bands (Krempaský et al., Nature 2024; Lee et al., Phys. Rev. Lett. 2024), and further candidates, RuO2-type compounds among them, are under study in 2024-2026 reports; the status of individual materials (RuO2 especially) is still debated. Research motivation: spin currents and fast memory without stray fields. Nothing here is a device claim.
Simplifications: Ising (up/down) spins instead of Heisenberg vectors; 2-D square lattice, not a real 3-D crystal such as hexagonal NiAs-type MnTe; J2 values chosen for display, not fitted; the arrow wobble is cosmetic (Ising spins only flip, they do not tilt); Fermi surfaces are a cartoon with arbitrary energy units; the "Crystal cartoon" is a layered MnTe-like sketch, not to scale.
Controls
3D: drag to orbit, wheel / pinch to zoom, buttons switch between the spin lattice and a layered crystal cartoon. On a phone use the tabs above the stage to flip between the 3D lattice and the schematic bands. Try: preset 3, then "Reverse all spins" and watch the orange and blue Fermi pockets swap.