The idea that the laws governing electrons would look just as valid if you ran a film of them backwards. Reversing time flips every motion and every , so a material with no magnetism of its own usually keeps this symmetry, while a magnet – or a magnetic field – breaks it. Which case applies decides a lot: some protected edge currents need the symmetry, and others need it broken.
Going deeper
Run time backwards and every motion and every spin flips. Without magnetism the reversed state is just as allowed, so every electron state has a partner with opposite momentum and spin at the same energy – the pairing that protects the edge states of topological insulators.
Running the film backwards
Film a ball flying through the air and play it backwards: the reversed motion is just as possible as the original. The laws that govern electrons in a crystal have the same property as long as nothing magnetic is involved. Reversing time turns every velocity around and flips every spin, and in a non-magnetic material the result is another allowed state with the same energy.
One consequence is Kramers’ rule: in such a material every electron state has a partner with opposite spin and opposite momentum at the same energy. and applied magnetic fields break the symmetry, because reversing time would also flip the magnet’s own spins, and the flipped magnet is a different state.
Protection from symmetry
Kramers’ rule is what protects the of a . Its edges carry pairs of states moving in opposite directions with opposite spins, and as long as time-reversal symmetry holds, no ordinary impurity can scatter an electron from one to the other, so current flows along the edge without turning back. A magnetic impurity or field breaks that protection.
The symmetry also governs , which it forces to cancel when summed over the whole crystal. That is why a non-magnetic can show a valley Hall effect, with electrons of the two drifting to opposite sides, but no overall sideways voltage – and why the effect needs magnetism.
Catching it broken
Some of the most debated claims about correlated 2D and layered materials are that a phase breaks time-reversal symmetry on its own, without ordinary magnetic order – in , in twisted graphene, in certain . The evidence has to come from effects the symmetry forbids: a rotation of the polarisation of reflected light, a spontaneous with no applied field, or a tiny internal field seen by muons. Each probe has its own artefacts, and in several materials they disagree, which is why such a claim should always name the method behind it.
For specialists
The antiunitary operation t → −t, which reverses momenta and spins; for spin-½ electrons T2 = −1, giving Kramers degeneracy of every state at time-reversal-invariant momenta. It is broken by magnetic order or applied fields. Its presence protects the helical edge states of quantum spin Hall insulators and makes Berry curvature odd in k, so valley Hall but not anomalous Hall effects survive; its breaking allows Chern insulators, the anomalous Hall effect and , the standard probes of spontaneous breaking in .