How long a chain has to be before its ends stop mattering

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.

Where a molecule stops being one

The energy per site of a ring minus that of a chain of the same length, with each point labelled by the difference times the length. That product settles at about 0.71β, which is what it means for the difference to fall as one over the length. The figure prints the size at which the two agree to a thousandth of a β per site.

The three rates on one pair of logarithmic axes, where a power law is a straight line and its exponent is the slope. The energy per site’s difference from a ring falls with slope −0.99, the gap at half filling with slope −1.00, and the band edge’s shortfall with slope −2.00 — the powers the argument above predicts, fitted from the last two lengths and checked against those values rather than read off. The lines are not parallel, which is the whole finding: there is no length at which all three are small together.

The two quantities side by side, which is the whole distinction. The cost of having two ends rises to 0.719β and settles; its share of the chain’s own binding falls from 1.99 per cent at twenty sites to 0.088 at six hundred and forty. Both halves are checked — the settling on the last three lengths, and the fall by more than a factor of eight — because a figure in which both curves fell would be the ordinary story rather than this one.

The end is the hardest place to bind

What an end costs in the ground state, measured as the difference between a ring and a chain of the same length. That number — about 0.71β for two ends — is the reactivity, and it is a completely different quantity from the threshold this essay measures. One is a total energy and one is a property of the spectrum.

The chain distorts hardest where it stops

The end-energy measurement: a fixed amount of energy living at the two ends, whatever the chain’s length. It is a total, and this essay is about how that total is distributed.