A range of electron energies so narrow that the electrons barely move on their own. With their motion frozen out, the way they repel each other takes over – as in a packed train carriage, where nobody can walk anywhere and everything depends on how people get on with their neighbours. That is where unusual states such as and magnetism can appear.
Going deeper
The width of a band measures how easily electrons move through the crystal. When that width falls below the energy it costs two electrons to sit near each other, motion stops mattering and interactions take over – which is what happens in twisted bilayer graphene near 1.1°.
When kinetic energy stops winning
The width of a band comes from the overlap between neighbouring orbitals: the more easily an electron hops, the wider the band and the more its kinetic energy dominates. Electrons also repel each other, with a characteristic energy U for two of them on the same site. In ordinary metals the bandwidth W far exceeds U, and the electrons behave nearly independently.
A flat band inverts that ratio. With W ≪ U, kinetic energy no longer sets the ground state and interactions decide what happens – magnetism, charge ordering, superconductivity or states at particular fillings. Because the states are nearly degenerate, small perturbations produce large effects, which is why flat-band systems are both interesting and hard to predict.
Making a band flat
Several routes exist. Some lattices, such as the and Lieb lattices, have flat bands by geometry: interference between hopping paths cancels motion exactly. Strong magnetic fields flatten bands into . Heavy-element compounds with narrow f bands have long been studied for the same reason.
In the route is the . Stacking two layers with a small twist makes a superlattice with a period of tens of nanometres, and the resulting mini-bands are far narrower than the original ones. At the near 1.1° in twisted bilayer graphene the lowest bands become almost flat, and states appear at partial filling of the superlattice, with superconductivity beside them. Other moirés – twisted , aligned trilayer graphene on hBN – give flat bands too, with their own orderings.
What flatness alone does not tell you
Bandwidth is only half the story. The states also have a geometry – how the wavefunction changes across momentum space – captured by the and the . Two bands can be equally flat yet behave differently: one may host a , the other nothing, depending on that geometry. This is why the field talks about the quantum metric alongside the bandwidth.
Experimentally, flat-band physics is fragile. Twist angle varies across a sample, relaxes the moiré, and smears the filling; many early results were hard to reproduce until sample preparation improved. Reports of correlated states are now expected to come with a measured twist angle and evidence about its uniformity.
For specialists
A band whose kinetic-energy width is small compared with the interaction energy, so electron–electron interactions dominate the physics.