Two impurities at 2 to 14 sites 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.

Nine 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 defect is a level in the gap

The threshold measured at the end of the chain and in its middle, against the length. The middle’s number halves with every doubling and is going to zero; the end’s converges on exactly one β and stays. Two sites in the same chain, one binding for free and one demanding a whole β, which is the end is the hardest place to bind’s subject.

Two defects, and the level between them

The splitting between the two levels a pair of impurities pulls out of a chain, against their separation, on a logarithmic scale. It falls by a constant factor per site, and that factor is computed from the isolated bound state’s energy rather than fitted to the points.

The same measurement at h = −1.2β, barely past the binding threshold. The isolated level is shallow, so its tail decays slowly, so the splitting survives out to separations where the h = −2 pair has nothing left — the whole curve is lifted and its slope on a logarithmic axis is shallower. Weakly bound states are the ones that interact at a distance, which is the opposite of the intuition that a strong impurity is an influential one.

At the threshold itself the state is on the point of dissolving into the band and its participation diverges with the chain length, which is the sense in which a marginally bound state is not really bound at all.

The same measurement for stronger impurities, at h = −3β. The isolated level is deeper — −3.606β against −2.828 — and the decay is faster, so the splitting falls more steeply. The predicted κ and the measured ratio agree here too.

A vacancy is not an impurity

And two impurities, at nine separations, with the splitting between their levels measured. It falls by a factor of 0.41421 for every site of separation — a number predicted from the isolated level alone. This is the behaviour a vacancy pair does not have.

One defect is a level, many are a band

The level a single impurity pulls out of the band, at six depths, against the closed form. The diagonalisation of a ring of a hundred and sixty sites agrees with −√(h² + 4) to better than a millionth at every depth. The number above each point is how many sites the state occupies: 3.73 for the shallowest impurity here and 1.24 for the deepest.

Two impurities at seven separations. The splitting is 0.6 in these units when they are two sites apart and indistinguishable from zero by fourteen, and the decay is the state’s own decay length seen twice over.

The impurity levels of a ring of a hundred and sixty, drawn as a bar from the lowest to the highest, against the fraction of sites that are impurities. At the lowest concentrations the bar has no height at all — every level is at the closed form’s answer. By thirty per cent it spans 2.41 and has closed to within 0.25 of the host band, which is shaded.

A particle in a box the alloy made

The same physics on purpose rather than by accident: two impurities in a chain, and the splitting between the symmetric and antisymmetric combinations against how far apart they are. A pair split by a thousandth is a pair about nine sites apart.

The length at which levels become a band

The two quantities that depend on the chain length, for runs of six. They run against each other and cross once.

The splitting against separation, with the fitted exponential. Two decades of decay over ten sites.

The two chains the splitting is measured on. One run gives one level near the box energy; two give two.

A band that is a hundred and seventy decades of nothing

The splitting between two runs one site apart — the largest any chain can produce — and the splitting at the separation two runs typically have, for run lengths four to eight. The axis spans a hundred and seventy-five decades.

The measured splittings between two runs of six, and the fitted exponential continued to the two hundred and fifty sites they typically sit apart. Everything the measurements cover is in the top few per cent of the axis.

Each run length’s level, with a bar showing the widest that level’s band can ever become. The dashed line at 2 is where an infinitely long run’s level would sit.

A count rather than an average

The fraction of runs whose nearest neighbour of the same length is coupled more strongly than a threshold, for five run lengths.

For each threshold: the separation it corresponds to, the fraction the closed form predicts, and the fraction generated chains actually have.

The closest pair, the mean separation and the median separation, with the coupling at each and the share of runs at least that strongly coupled.

The share that was read as a line

The share of runs of six in a resonant pair against the chain length at five concentrations, with straight lines carried on from the first nine decades.

The exact per-decade rise of the share against the chain length at five concentrations, with the linear rise marked at the left.

The chain length at which half of the runs of six are resonant, from the exact share and from the straight line through its first nine decades, across concentrations.