What it does
A magnon is a quantum of spin wave, a ripple in the direction of the spins. A two-sublattice antiferromagnet has two kinds. One turns the spins clockwise and the other anticlockwise: they carry opposite chirality and opposite spin (−1 and +1 in units of ħ). In an ordinary antiferromagnet the two always have the same energy, so their effects cancel.
In an altermagnet they do not. Their energies split, and the splitting changes sign around the Brillouin zone, just like the spin splitting of the electron bands (d-wave in this model). A temperature gradient along x can then carry a net spin current made of magnons, which could be useful for low-power spintronics.
This tool solves the simplest spin model that shows this, using linear spin-wave theory.
The model
Spins of length S sit on a square checkerboard with two sublattices: A (up) and B (down).
- J1 couples every atom to its four nearest neighbours on the other sublattice and favours opposite spins. It must be positive.
- Ja and Jb couple atoms of the same sublattice one cell apart and favour parallel spins. On A, Ja acts along x and Jb along y. On B it is the other way round. This is what makes the magnet an altermagnet: only a 90° rotation turns A into B. Set Ja = Jb and you have a plain antiferromagnet.
- K is an easy-axis anisotropy that holds the spins along their axis. It opens a gap at Γ.
All energies are in whatever unit you give the couplings.
What each result means
- Largest splitting ωα − ωβ: how far apart the two chiralities get, and where. For this model it is 4S|Ja − Jb|, at the X and Y points with opposite signs. Along the diagonals kx = ±ky the two stay degenerate.
- Gap at Γ: 2S√(K(4J1 + K)). Without anisotropy the magnons are gapless.
- Top of the bands: the highest magnon energy on the path.
- Net thermal spin: always 0. A 90° rotation turns one chirality into the other, so equal numbers of each are excited and the magnet stays compensated. Only a current or gradient that picks a direction shows the splitting.
- Bands chart: α (solid) and β (dashed) along Y–Γ–X–M–Γ.
- Map: where α is higher and where β is higher.
If the couplings make the up-down order unstable, for example with a negative K or same-sublattice couplings that are too antiferromagnetic, a magnon energy goes negative or imaginary. The tool then says so instead of drawing bands, because linear spin-wave theory only applies to a stable state.
Worked example
Use the default made-up couplings: J1 = 1, Ja = 0.3, Jb = 0.05, K = 0.02 and S = 1.
- Gap at Γ: 2√(0.02 × 4.02) = 0.567.
- At X: the α magnon sits at 5.24 and the β magnon at 4.24, a splitting of +1.0.
- At Y: the order reverses, so the splitting is −1.0.
- At M and along the diagonals: the two are degenerate (5.44 at M).
- Largest splitting: 4 × 1 × |0.3 − 0.05| = 1.0 in size. The tool reports it as −1 at the Y point, which is the same size as +1 at X.
Now set Jb = 0.3 so that Ja = Jb. The splitting disappears everywhere and the bands of a plain antiferromagnet come back.
FAQs
Which mode is which? α lowers the total spin by one unit and lives mostly on the spin-up sublattice. β raises it and lives mostly on the spin-down sublattice. They precess in opposite senses.
Is linear spin-wave theory exact? No. It is the first step, valid at low temperature where few magnons are excited. Interactions between magnons shift and broaden the bands at higher energy.
Does it model a real material? No. It is a toy model with made-up couplings, chosen to show the effect clearly.
Is anything sent anywhere? No. Everything runs in your browser.
Where the numbers come from
- Method: linear spin-wave theory with Holstein–Primakoff bosons and a Bogoliubov transformation for a collinear two-sublattice magnet (standard textbook treatment). The closed-form energies are worked out by fibub for this model.
- Model: a made-up square-checkerboard Heisenberg model in which same-sublattice couplings are anisotropic in opposite ways on the two sublattices, the simplest way to make an altermagnet. The idea that altermagnet magnons are chirality-split comes from the literature since 2023 and is cited for the concept only; no paper's parameters or figures are used.
- Checked in the tests against NumPy eigenvalues of the paraunitary (σ3·H) magnon matrix.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-08.