A ring filled to 50 per cent chooses its own period

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.

One essay draws 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 distortion the filling chooses

A ring of twenty-four sites with its bonds modulated at period three, drawn at the strength each bond actually has. Two bonds in every three are weakened by half of what the third is strengthened by, which is what keeps the elastic cost equal to the alternating pattern’s.

And the same ring modulated at period four, which is what a quarter-filled chain chooses. The pattern repeats every four bonds and the elastic cost is identical to both of the others by construction.

A ring of 120 sites with 120 electrons — one per site — and what a distortion of each available period is worth. Period two is worth 2.479 in units of β and the best of the others is worth 0.407, a margin of six to one at identical elastic cost.