A band becomes a bell curve

Compare a chain, a square net and a cubic structure with the band width divided out, and the shapes differ by a great deal: a chain puts more than half its levels in the outer thirds of its band and a cubic structure puts more than half in the middle third, and the binding that follows varies by nearly a factor of two.

It attributed the difference to the dimension, and it was right. It also had no way of knowing it was right, because nothing in that comparison separated the dimension from the coordination: a chain has two neighbours, a square net four, a cubic structure six, and two, four, six would have explained the same three pictures exactly as well.

Separating them shows the two behaving in completely different ways — one of them because of a theorem. The comparison was not wrong; its reason was one of two available, and it happened to be the right one.

Why a hypercubic band has a closed-form shape #

A wrapped hypercubic structure’s levels are the sum of $d$ independent one-dimensional ones. That is not an approximation: the adjacency matrix of a product graph is a Kronecker sum, so its eigenvalues are sums of the factors’ eigenvalues, and this collection computes them that way already.

A sum of $d$ independent copies of one random variable is what the central limit theorem is about. Its normalised cumulants fall as powers of $d$, and both of the shape measures used here follow with nothing fitted:

$$\frac{\mu_4}{\mu_2^2} = 3 - \frac{3}{2d}, \qquad \frac{\mu_6}{\mu_2^3} = 15 - \frac{22.5}{d} + \frac{10}{d^2}.$$

The values are 1.5, 2.25, 2.5, 2.625, 2.7 and 2.75 for one to six dimensions, and the sixth moments are 2.5, 6.25, 8.611, 10, 10.9 and 11.528 against a Gaussian’s 15. Both agree with the closed forms to $10^{-8}$, which is the eigenvalue arithmetic rather than a fit.

So the dimension in that comparison is the sample size of a limit theorem. The van Hove singularities — the divergence at a chain’s band edges, the spike at the centre of a square net’s — are what remains of a distribution that has not yet been averaged enough times, and they wash out as $d$ rises for the same reason a sample mean’s distribution smooths.

Coordination held, dimension moved #

The other half of the separation needs structures in which the two do not move together, and they are easy to build. A chain reaching to its second neighbours is one-dimensional and four-connected; a square net is two-dimensional and four-connected.

Both have second moment exactly 4, since the second moment of an adjacency spectrum counts closed two-step walks and is the coordination arithmetic. Their normalised fourth moments are both 2.250, to nine figures.

That is not the agreement it looks like. Their sixth moments are 6.719 and 6.250, their binding per unit width is 0.2546 and 0.2010 — a difference of twenty-seven per cent — and their shapes are visibly different: the chain puts 16.7 per cent of its levels in the outer thirds and the net 21.4.

So a fourth moment does not determine a band, which is worth stating because the moment expansion is often used as though it did. What determines a band is the whole sequence of moments, and two structures agreeing to the fourth and differing at the sixth is a case met before — there for a binding at fixed coordination, here for a shape at fixed coordination and different dimension.

Dimension held, coordination moved #

Three two-dimensional nets with three, four and six neighbours — honeycomb, square and triangular — give fourth moments of 1.667, 2.250 and 2.500.

That range, 1.667 to 2.500, is most of the range the dimensions one to three cover, which is 1.5 to 2.5. So at fixed dimension the coordination moves the shape nearly as much as the dimension does at fixed coordination, and neither variable can be called the one that decides.

The binding follows the same pattern. Per unit band width the three two-dimensional nets give 0.2626, 0.2010 and 0.2200, which is not even monotonic in the coordination — the honeycomb binds most and the square net least, with the triangular net between them. So the quantity a cohesive energy is does not order by neighbours at fixed dimension any more than the shape does, and the observation that shape matters more than width survives every rearrangement of the variables tried here.

The overshoot, which is the result #

The comparison that separates the two variables cleanly is a chain reaching to its third neighbour. It has six neighbours, like a cubic structure. Its fourth moment is 3.389 against the cubic 2.500 — and 3.389 is on the far side of a Gaussian’s 3.

No hypercubic structure of any dimension can be there. The closed form is $3 - 3/2d$, which is below 3 for every finite $d$ and approaches it from below.

The reason is the reason the closed form exists. A hypercubic spectrum is a sum of independent contributions, one per direction, so the central limit theorem applies and drives the shape toward a Gaussian. A chain’s longer hops are not independent of anything: every one of them is a function of the same angle, $2\cos\theta + 2\cos 2\theta + 2\cos 3\theta$, so there is no sum of independent variables anywhere in it and no theorem pushing it anywhere. It is the same distinction that separates a band from a set of levels: what makes a band is not how many states there are but how they combine.

So the two variables are not two names for one thing, and one of them has a theorem behind it and the other does not.

What the shapes look like #

The moments are a summary and the histograms are the thing summarised, and it is worth looking at the two four-connected structures together, because their agreement to the fourth moment is not visible in them.

The chain of reach two puts 16.7 per cent of its levels in the outer thirds of its band and 41.4 in the middle third. The square net puts 21.4 and 41.8. So they agree closely on the middle and differ by a quarter on the edges — which is exactly where a fourth moment is least sensitive, since it weights the tails by the fourth power of a quantity that has been divided by the width.

This is the general reason a low moment is a weak summary, and it is worth carrying: the moments weight the tails, and the features that distinguish real densities of states from one another — van Hove singularities, gaps, spikes — are mostly not in the tails.

What this says about the comparison by dimension #

The comparison’s conclusion survives and its reason is replaced.

It concluded that the shape of a density of states is a more useful summary of a band than the width, on the evidence that the width was held fixed and the binding still varied by nearly a factor of two. That is still true, and the seven structures here extend it: at equal width the binding per unit width runs from 0.1667 for the cubic structure to 0.3183 for the chain, a factor of 1.91 — a spread comparable to the one a set of structures at fixed coordination showed for a different reason.

What has changed is the account of why the shapes differ. It is not that a higher-dimensional structure has more neighbours, and it is not that it is higher-dimensional as such. It is that a hypercubic structure of dimension $d$ is a sum of $d$ independent things, and the number of independent things a spectrum is a sum of is the quantity that governs the shape. For a hypercubic lattice that number happens to be the dimension.

Why the closed forms have the coefficients they do #

The two expressions are not fitted and they are not deep, and it is worth showing where the numbers come from, because a closed form nobody can derive is a curve fit with airs.

For a sum of $d$ independent copies of a variable, the cumulants add: $\kappa_n^{\text{sum}} = d\,\kappa_n$. Normalising by the variance raised to the right power divides by $(d\kappa_2)^{n/2}$, so the normalised cumulants fall as $d^{1-n/2}$. The fourth normalised cumulant — the excess kurtosis — therefore falls as $1/d$, and the whole shape is

$$\frac{\mu_4}{\mu_2^2} = 3 + \frac{\gamma_2}{d},$$

where $\gamma_2$ is one chain’s own excess kurtosis. A chain’s is $1.5 - 3 = -1.5$, so the coefficient is $-1.5$ and the expression is $3 - 3/2d$. Nothing else is involved.

The sixth follows the same way with two terms, because the sixth normalised cumulant falls as $1/d^2$ while the square of the fourth falls as $1/d$ and both feed into the moment. That is where the $-22.5/d$ and the $+10/d^2$ come from, and it is why the sixth needs two terms and the fourth needs one.

So the arcsine law’s own shape is the only chemical input, and everything else is arithmetic about sums. That is also the sharpest form of the essay’s point: a hypercubic band’s shape is a property of a chain’s shape and of a counting number, and the counting number is the number of factors rather than the number of neighbours.

A third route to many neighbours, and a third limit #

Two ways of raising the coordination have been tried here: add dimensions, which drives the shape toward a Gaussian, and reach further along one line, which drives it past a Gaussian and keeps going. There is a third, and it lands somewhere neither of them does.

Branch. A structure in which every site has $z$ neighbours and there are no closed loops at all — the tree the width of a band met as the case that refuses the $2z\beta$ formula — has a fourth moment with its own closed form, and it comes from the same walk-counting the second moment uses.

Count the closed four-step walks from a site of a tree. There are two kinds and no others: out to a neighbour and back, then out to a neighbour and back, which is $z^2$ ways; and out, out, back, back, which is $z(z-1)$, since the second step may go anywhere except straight home. A third kind would have to be a walk round a four-membered ring, and a tree has none. So

$$\frac{\mu_4}{\mu_2^2} = \frac{2z^2 - z}{z^2} = 2 - \frac{1}{z},$$

against the hypercubic sequence, which in the same variable is $3 - 3/z$.

The two are the same shape of expression and they converge to different numbers. Raising the coordination on a hypercubic lattice takes the shape to 3, which is a Gaussian; raising it on a tree takes the shape to 2, which is a semicircle. Both limits are approached as one over the coordination, both are exact, and neither is reachable from the other.

Three routes, three destinations: 3 by adding dimensions, 2 by branching, and past 3 without limit by reaching further along one line. Coordination alone decides nothing, and the essay’s headline — a band becomes a bell curve — is a statement about one of the three.

What separates them is visible in the walk count itself, which is the satisfying part. On a hypercubic lattice the dominant four-walks go out and back along two different axes, and those axes are independent, which is exactly the condition the central limit theorem needs. On a tree there are no genuinely distinct pairs of directions to combine, because every pair of edges from a site leads into a branch that never rejoins, so only the degenerate walks survive and the shape saturates lower. On a long-reach chain every neighbour lies along one line, so the contributions are functions of the same variable rather than of independent ones, they reinforce, and the shape overshoots.

So the variable that governs the shape is not how many neighbours a site has but how independent they are, and dimension is a good proxy for that only because perpendicular axes are independent by construction. A tree has as many neighbours as anything and no independence at all; a long-reach chain has many neighbours and perfect dependence. Both are visible in one count of four-step walks.

What this cannot say #

There is no wavevector anywhere in it. Every number above is a moment of a list of eigenvalues of a finite matrix, and a density of states is a histogram of that list. What that costs and why it is a decision is the standing note of this field, and it applies with its usual force: the closed forms here are correct as statements about a distribution and carry none of the information a band structure carries.

The structures are all one orbital per site, one hopping, no repulsion. A real solid has three things a hypercubic lattice does not, and only one of them is addressed here. Several orbitals per site and unequal hoppings are still undone, and the second of them is what a chain that cannot stay even is about.

The sizes are finite. The closed-form check uses the sumset expression, which costs nothing, so those are at tens of thousands of sites. The seven nets are diagonalised outright at around two hundred, which is enough for a fourth moment and is not enough for a fine histogram; the moments quoted for them are exact for the finite structure and are not the infinite-size limits.

Nothing here is about a real material. These are graphs with unit hoppings, and their moments are moments of an adjacency spectrum. A real band’s shape is set by overlaps that fall with distance, several orbitals per site, and a lattice — so what has been separated here is two variables of a model, and the separation is what makes the model’s own statements checkable.

And the cumulant argument is about independence, not about space. A structure that is a product of two identical chains is a sum of two independent variables whether or not it is drawn in a plane. The theorem does not know about dimension; it knows about factorisation, and for these structures the two coincide.

Every structure in this essay is at the far end of a process drawn elsewhere: a ring’s discrete levels closing up into a band as it grows. Large enough that the histogram is smooth is the condition under which the moments quoted here mean anything, and all of them are past it.

What was checked #

Both closed forms to $10^{-8}$, at six dimensions apiece, which is twelve independent agreements between an eigenvalue computation and an expression derived from cumulant additivity.

The sequence increasing and staying below three, which is the direction the theorem requires and would catch a sign error in the derivation.

The second moment equal to the coordination, exactly, in all seven nets — the arithmetic check that the graphs are what they are supposed to be.

Two four-connected structures of different dimension agreeing on the fourth moment to nine figures and disagreeing on the sixth and on the binding.

Three two-dimensional nets with three different shapes.

And the overshoot, in both directions: the chain of reach three above 3 and the cubic structure below it, with the cubic structure’s value required to be exactly the closed form’s — which is the check that ties the two halves of the essay together, since it makes the same number a member of the theorem’s sequence and a point in the comparison.

Still open: several orbitals per site #

The obvious open question is the second of those three: several orbitals per site. A chain with two orbitals per site and different site energies has a two-band density of states with a gap in it, which no single-orbital structure of any dimension has, and the moments of the whole thing are then a mixture rather than a limit. Whether the closed form survives in each band separately is a question with an answer.

The nearer question is about the independence. The theorem here applies because a hypercubic structure factorises exactly. A structure that factorises approximately — a layered one, say, with strong coupling in a plane and weak coupling between planes — should approach the two-dimensional shape as the interlayer coupling falls, and the rate at which it does is a measurable statement about how anisotropic a material has to be before its band is effectively two-dimensional. Nothing here computes it yet, and nothing new is needed to do so — the same wrapped-graph construction, with one hopping made small.