One band width, three shapes

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.

Five 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 the states pile up

The density of states of a wrapped chain, a wrapped square net and a wrapped cubic structure, each with 4,096 levels and each scaled so that its band runs from −1 to +1. The three shapes have nothing in common. The chain crowds its states into the two edges, the cubic structure crowds them into the middle and thins to almost nothing at the edges, and the square net sits between the two with a spike at the centre.

The chain alone. The density rises without limit at both edges — this is a histogram, so the outermost bars are finite, but the closed form behind them is 1/(π√(4 − x²)) and it diverges. Between the edges the density is nearly flat, which is why the outer thirds hold so much.

The cubic structure alone, on the same axes. The density falls to nothing at both edges — the outermost bars hold 0.07 of the average — and rises to a broad hump either side of the middle. A level at the very bottom of this band requires every one of three directions to be at its own extreme at once, and almost nothing is.

A band becomes a bell curve

The normalised fourth moment of a hypercubic band against its dimension, with the closed form drawn through it. They agree to eight decimal places at every dimension from one to six. A chain’s 3/2 is the arcsine law, a Gaussian’s is 3, and the sequence approaches it and never arrives.

A chain’s density of states, which is the one-dimensional case: the arcsine law, with its divergences at the band edges. Every hypercubic band of higher dimension is this distribution convolved with itself, and the divergences survive only in the sense that a convolution of things with sharp features has softer ones.

Seven wrapped structures placed by how many neighbours a site has and by the shape of its band. Neither variable decides the other. Two four-connected structures of different dimension share a fourth moment; two six-connected structures of different dimension share one too; and a one-dimensional structure with six neighbours lands on the far side of a Gaussian, where no hypercubic structure can go.

Two bands, and the shape of each

Each band’s shape as the coupling between the two orbitals is turned up. With the coupling off both sit exactly on the one-band value. The upper band then departs as the square of the coupling; the lower one is pushed down, then back, and crosses its own uncoupled value between 0.4 and 0.6 — so a single measurement of a band’s shape says nothing about how mixed it is.

The shape of a layered band against the coupling between its layers, with the closed form drawn through it. The points are eigenvalues of a forty-cubed structure and the curve is arithmetic, and they agree to nine decimal places at every anisotropy. The marked point is where the expression sits halfway between the two limiting values.

The sequence the layered expression is being compared against: the normalised fourth moment of a wrapped hypercubic band against its dimension, with 3 − 3/2d drawn through it and agreeing to eight decimal places from one dimension to six. The chain is at 1.5, the square net at 2.25, the cubic structure at 2.5, and three — the Gaussian value — is approached and never reached. This is the λ = 1 line of the layered form, and reading a non-integer point off it is exactly the move the next paragraph refuses.

Two ways of being second order

The departure itself, which is the quantity the exponent above was taken of: each band’s fourth-moment ratio against the coupling between them, with the uncoupled value drawn across. The scale on which this moves is thousandths, which is why the question of what a measurement of it fixes is a real one.

The other closed form, and the reason this one is worth the trouble: where a band’s shape has an exact expression, a measurement of the shape is a measurement of something. Where it has not, it is a number.

The constant that belonged to one net

The upper band’s shape departure against the separation of the two bands, for four pairings of graphs. The square-on-square pairing — the top line — has a slope of −2.01. The others do not.