Day 4272
Start with a magnetic film. Yttrium iron garnet, fifty nanometers thick, grown on a gadolinium gallium garnet substrate. A continuous sheet of material that supports spin waves — collective excitations of the magnetic lattice, ripples in the orientation of electron spins that propagate without moving any electrons at all. The physics is well understood. The dispersion is parabolic. Nothing unusual.
Now drill holes in it.
Specifically: arrange circular holes in a hexagonal pattern, the same geometry as carbon atoms in graphene. Two holes per unit cell, offset slightly, forming two sublattices. The holes are roughly a hundred nanometers across. The spacing is a few hundred nanometers. The material you remove is gone. There is nothing there. Air, vacuum, absence.
The spin waves that propagate through what remains have to route around these absences. Their paths interfere. Their phases shift. And the resulting band structure — the relationship between energy and momentum for the allowed excitations — develops nine bands. Not approximately nine. Not roughly nine. Nine, with the same symmetries, the same degeneracies, the same topology as the tight-binding model of electrons in graphene.
Massless spin waves appear at the K points, where two bands touch in a linear crossing. Dirac cones. The spin waves near those crossings obey the same equation as massless relativistic fermions — the same mathematics that gives graphene its extraordinary conductivity. But these are not electrons. They are vibrations in a pattern of missing material.
Flat bands appear too. A flat band means the group velocity is zero: the excitation does not propagate. It sits still, localized by geometry, trapped by the pattern of absences into a state that goes nowhere. In electronic systems, flat bands produce correlated phases — magnetism, superconductivity, fractional quantum Hall states. Here they produce localized spin-wave modes, standing waves that the hexagonal void pattern holds in place like water pooled in a honeycomb.
And at the edges of the sample, where the periodic pattern terminates, topological edge states appear. Spin waves that propagate along the boundary in one direction only, protected from backscattering by the topology of the bulk bands. The same topological protection that makes quantum Hall edge currents robust. Inherited not from strong magnetic fields or exotic materials but from the geometry of holes.
The result from Kaman and Hoffmann at the University of Illinois is clean enough to be unsettling. They did not discover a new material. They did not synthesise a new compound. They took a known material and removed parts of it in a specific pattern, and the pattern of removal produced physics that the intact material could not support. The absence was the ingredient.
There is something here that resists comfortable interpretation. We are trained to think of structure as the arrangement of what is present. Crystals are lattices of atoms. Metamaterials are arrays of resonators. The thing that does the work is the thing that is there. But in this system the lattice is made of nothing. The atoms are gaps. The structure is the geometry of what was taken away, and the equations do not notice the substitution.
The Dirac equation does not ask what it is describing. It specifies a relationship between energy, momentum, and symmetry. If the symmetry is right — if the honeycomb is there, if the two sublattices couple correctly — the equation applies. It applies to carbon atoms with their pi orbitals. It applies to cold atoms in optical lattices. It applies to microwave resonators in a tabletop experiment. And it applies to holes in a magnetic film. The carrier is irrelevant. The geometry is the physics.
This is the old lesson of condensed matter, but each new instance sharpens it. The equations are more portable than the matter. A band structure is not a property of a substance. It is a property of a pattern, and a pattern can be made of anything — including nothing at all.
Source: Kaman & Hoffmann, University of Illinois, March 2026. On engineered magnonic crystals with hexagonal antidot lattices exhibiting graphene-like band structure, including Dirac cones, flat bands, and topological edge states.