How many sites a state occupies, and whether that depends on the ring

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.

Four 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.

The third way to be an insulator

The mean level spacing at the middle of the band, against the disorder, for three ring sizes. Every column falls by about the ratio of the ring sizes. The level-spacing criterion for telling a metal from an insulator — do the levels crowd together as the system grows — is satisfied all the way across this figure.

The participation ratio of the states at the middle of the band, against the disorder, for the same three sizes. At the left the three curves are three different numbers, each two thirds of its own ring. At the right they have converged on each other: 7.8 sites on a ring of fifty, 10.0 on a ring of two hundred.

One state from each case, drawn: the amplitude on every site of a clean ring of a hundred and twenty and of a disordered one, both at the middle of the band. The first is a wave across the whole ring. The second is indistinguishable from zero over most of it, and where it is not is decided by which sites happened to be low.

The arrangement a count cannot pick

All 1,820 ways of raising four of sixteen sites on a wrapped square net, at a contrast of four, each placed by its count of unlike bonds against the binding it gives. The count is a real predictor — the cloud rises — and it is not the right one: the arrangement that wins has twelve unlike bonds where sixteen are available.

The same 1,820 arrangements at a contrast of one instead of four. Here the count does pick the winner, and the cloud is much narrower at each count — which is the first sign of what changes at a larger contrast: the spread within a count is what has to grow before the count can be beaten.

The winner’s unlike-bond count against the contrast, with the largest available count drawn above it. Three raised sites of sixteen: the two coincide up to δ = 1.527 and separate above it, and the crossing is sharp because the winner changes from one arrangement to another rather than moving continuously.

The carriers a distortion was hiding

A ring of forty allowed to relax against its own free energy at each temperature. The falling curve is the alternation it settles at; the rising one is the carriers it has; the dashed one is the carriers counted on the same ring held flat.

The cold alternation against the stiffness, for a ring of forty and one of forty-two. One curve falls towards zero and does not reach it at any stiffness; the other reaches it between 1.6 and 3.

How many sites a state occupies once the ring is disordered rather than merely distorted. The states near the Fermi energy shrink onto a handful of sites, and a carrier that lives on a handful of sites is not a carrier in the sense counted on the rigid ring — which is the alternative this essay had to rule out before its own answer could stand.

The exponent was the window's

The slope between each neighbouring pair of points, as the transition is approached. It is still climbing where the original fit stopped, and it arrives at a half.

The same curve fitted over five windows. The numbers run in order from 0.40 to 0.50 with no plateau, so the fit is reporting where it was taken rather than what it was taken of.

Five combinations of ring size and stiffness, scaled by the cold alternation. One curve, to a few per cent — and the two largest rings to a part in two thousand.