Chains of 2, 4, 8, 16, 40: the levels crowd, the edges do not move

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.

Three 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 solid is a molecule that did not stop

Chains of two, four, eight, sixteen and forty sites, every level drawn at its computed energy on a single axis. The band edges are in the same place in all five columns. What grows is the number of levels between them, not the interval they occupy — which is the whole difference between a bigger molecule and a solid.

The sequence at four intermediate lengths, which is where the essay’s claim lives. Six carbons is a molecule anybody would call one and forty-eight is a piece of a polymer, and there is no column in between at which the description changes — only columns at which one quantity or another has finished converging.

The same sizes as rings rather than chains. The levels come in degenerate pairs because a ring can be traversed two ways, and the band edges are again unmoved. Whichever boundary condition is imposed, the interval is the same — which is the first hint that the interval belongs to the neighbour interaction rather than to the shape.

A band with no structure in it

Four chains, all of their levels drawn. Every energy in this picture is computed and correct, and there is no horizontal axis: nothing here says which state is which, only how many lie at each height. That missing axis is the whole subject of this essay.

Four chains an octave apart in length, with every level of each drawn. The band edges do not move and the levels between them crowd, so a gap is the one feature of this picture that can be asked about without a wavevector — does a space between two levels survive the crowding, or does it close like every other spacing? That question is answered by diagonalisation and needs no label.

Rings of eight, sixteen and forty. Every level except the top and bottom is doubly degenerate, and the degeneracy is the two directions of travel round the ring — which is the wavevector label showing itself in a spectrum. The corresponding chains have no such pairing, because a chain’s states cannot travel.

What one pair can hold together

Ring levels from two sites to forty. The lowest level sits at exactly 2β in every column — which is the fact this essay is built on, visible as a flat line across the top of a figure whose other levels are all moving.