Where two bands lie, as their centres are pulled apart

One of the figures on extended structures: Chains and rings taken far enough to behave like solids — bands, gaps, fillings, defects and ends — every one of them a finite matrix diagonalised, with no lattice anywhere in the argument.

Six essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

A full band is not an insulator

The intervals the two sets occupy, as the difference in site energy between the two orbitals is swept from nothing to six. Each band is centred on its own orbital energy and is four times its coupling wide, so the two are clear of one another exactly when the difference in centres exceeds twice the sum of the couplings. Below that mark the intervals share energies.

The same electron count at a weaker coupling between the two bands. The overlap shrinks and does not close, so the metal survives the weakening — and what would close it is not a smaller coupling but a larger separation between the two bands’ centres, which is a statement about the atoms rather than about the structure.

The gap above the completely filled lower set, at five ring sizes, for a separated pair of bands and an overlapping pair. Both systems have two electrons per site and both have their lower set exactly full. The separated pair gives 1.000 at every size — a property of the material. The overlapping pair gives a number that falls away as the ring grows.

Two bands, and the shape of each

Where two bands sit as a function of their own widths, from the arithmetic that decides whether they overlap. The gap in the two-orbital construction above starts here and then opens further as the two are allowed to mix, which is the second-order shift the shape measurement sees from the inside.

Two bands, if the chain is short enough

The other way two bands can fail to leave a gap, computed on the same kind of chain: two bands that overlap in energy while remaining distinct in character. A measurement that sees no gap has not distinguished that case from this essay’s, and the two have entirely different causes.

The constant that belonged to one net

Two orbitals on every site of a wrapped net of sixty-four, the lower ones hopping on the square net’s bonds and the upper ones on a triangular net’s. The two bands have different widths and different shapes before the coupling is switched on, which is the situation a one-graph construction has no way to build.

Each dot is one of the sixty-four modes: the coupling it feels across, the separation of its two levels up. Solving all sixty-four two-by-two problems reproduces every one of the hundred and twenty-eight levels the full diagonalisation returns.

The same construction with the same net above and below, which is the control the inversion needed. Two square bands coupled at the same strength and the same separation give an exponent of −2 as well — so the constant belongs to the shape of the lower band rather than to the pairing, and the one-graph result had generalised from one case.

Seven points that looked like a switch

Every wrapped net here, with its dimension, coordination, moments, and the exponent the coupled-band measurement returns. One column sorts the table.

Each net by the skewness of its own band and by the exponent it returns. The points lie on two horizontal lines rather than on a curve.

A square net and a chain reaching to its second neighbours: the same coordination, the same second moment, the same kurtosis, and opposite behaviour.

The triangles that were never in the bands

Nine combinations of two bands and a coupling, with the composite third moment split into the part inside a band and the part that crosses the gap.

A closed three-step walk in the composite either uses three band bonds or one band bond and two coupling bonds. There is no third case.