⟨x²⟩ is the average number of neighbours

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.

Six 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.

The width of a band is a count of neighbours

Six structures, each with its mean coordination counted directly off the list of bonds and its ⟨x²⟩ measured off the computed eigenvalues. The difference is at the level of arithmetic noise in every row. The full band width is printed beside them and obeys no comparable rule — the chain and the ring of the same length differ in width and agree exactly here.

The identity at five sizes from four sites to eighty, which is the range that shows it is not a limit. A chain of four has a mean coordination of 1.5 and a measured second moment of 1.5, agreeing to two parts in 10¹⁵; a chain of eighty has 1.975 and agrees to six parts in 10¹⁴. The two columns are equal at every size and the difference column is arithmetic noise growing slowly with the matrix. What does change with size is the full width beside them — 3.236β at four sites, 3.997β at eighty — which is the quantity that has a limit and is not the one the identity is about.

The product structures the formula is right about, in one, two and three dimensions: a wrapped chain of sixty-four, a wrapped 16×16 net, a wrapped 8×8×8 structure. The coordination counted off the edge list is 2, 4 and 6, the second moment measured off the eigenvalues is 2, 4 and 6, and the two agree to between 10⁻¹⁴ and 10⁻¹² — an identity rather than a limit, since the agreement does not improve with size and does not need to. The last column is the energy each bond supplies and it obeys no such rule: 0.6361β, 0.4027β, 0.3298β, falling as neighbours are added. That is the quantity a chemist would have wanted and the one the identity was never about.

What holds a solid together

The consequence of the exponential decay: a band’s scale is exactly the mean number of nearest neighbours, with no contribution from anything further away. That identity would be false for any of the other three interactions, all of which reach past the first shell.

The bond that weakens as neighbours multiply

Three wrapped structures with two, four and six neighbours per site. The mean of x² over the levels is the coordination exactly, to twelve decimal places. The binding each bond supplies is in the last column and falls by half across the same range.

The same neighbours, and a fifth of the binding between them. Two structures with identical coordination give quite different binding per bond, because the binding depends on the shape of the density of states and the coordination fixes only its width. That is the whole of the essay’s claim, measured on a pair of structures that agree about everything a bond count can see.

The identity in the simplest case, where the coordination is two in every row and the structures differ only in size and in whether they close. The identity holds to twelve decimal places there too, and says nothing about coordination because there is only one value of it.

Where the states pile up

The measurement this essay explains. Three wrapped structures with two, four and six neighbours per site: the second moment is the coordination exactly, the width is twice it, and the binding per site is neither — it rises more slowly than the coordination and faster than its square root. The exponent is the quantity the square-root rule gets wrong, and the shapes above are why.

Two structures with the same neighbours

The five structures, their first three moments, and the binding each gives at half filling. Every second moment is 4.000, which is what the coordination identity requires. The bindings are not equal: they span from 1.2756 to 1.6363 per site, a spread of twenty-two per cent, and the least bound is the one with the most four-step walks.

The same five structures’ level distributions, with the occupied half shaded. The second moments are identical, so the distributions have the same spread; what differs is where the weight sits within it, and the binding is an integral over the shaded part.

The rule itself: wrapped structures of two, four and six neighbours, with the second moment and the binding at half filling. The moment follows the coordination exactly and the binding follows its square root approximately, and the approximation is what the present essay is measuring the size of.

Two bands, and the shape of each

The second moment of a band against the coordination of the structure it belongs to, which is the exact identity everything here rests on: a wrapped graph with unit hops has a second moment equal to its coordination, arithmetically. In a layered structure the coordination is 4 + 2λ², which is why λ enters the denominator as a square while it enters the numerator as a fourth power.