Two bands, and the shape of each

Cumulants of a hypercubic band found a closed form where none was expected. A wrapped hypercubic structure’s levels are a sum of d independent one-dimensional levels, so its density of states is the distribution of a sum of d independent copies of one variable; cumulants add; and the two standard moment ratios follow with nothing fitted at all. It is the same construction the width of a band is a count of neighbours established for the second moment, taken two orders further.

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

Both hold to eight decimal places from one dimension to six. A band becomes a bell curve as its dimension rises, and the dimension is the sample size.

Two questions follow, and both are about what happens when the factorisation is disturbed. This essay answers them, and the second answer is the better one.

The trap that had to be avoided first #

Everything below is a statement about the shape of a sub-band, and the usual moments are about a band centred at zero.

That was not a convention. Every structure here has been bipartite with all site energies equal, so its levels are symmetric about zero, and the raw moments — the mean of x², the mean of x⁴ — are the central ones. Put a band at −Δ/2 and they are not, and a raw fourth moment there measures the offset.

The size of the mistake is worth printing, because it is exactly the kind that never announces itself. For a square net’s band displaced by a gap of eight, the raw ratio is 1.6900; at a gap of twenty it is 1.1498; at sixty it is 1.0176. The true shape is 2.2500. The wrong number is smooth, stable, of the right order and tends to one — which is what a fourth moment does when a distribution is far from the origin — and there is nothing about it that looks wrong. A density of states is not a spectrum warned about reading one of these pictures as a measurement; this is the same warning applied to a number taken off one.

So the offset is removed first, everywhere below, and the ratios are central moments. That is one function and it is new to this file, which is the honest measure of how much of this collection’s arithmetic has quietly assumed a symmetric band.

Two orbitals a site, and a coupling that can be switched off #

The first question, in full: 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 construction is a square net with two orbitals on every site, one at +Δ/2 and one at −Δ/2. Each hops to its own kind on a neighbouring site, with its own strength; the two kinds hop to each other across a bond with a strength that can be set to zero. The inter-orbital hop is taken symmetric so the matrix stays real, which makes the two orbitals of the same parity rather than an s and a p — a restriction on what the model is a model of, and not on the question being asked of it.

With the coupling off the answer is immediate and exact. The two bands are two copies of the one-band problem shifted apart, so each returns the one-band kurtosis of 2.2500 to nine decimal places. Nothing is lost by having a neighbour; the neighbour has only moved the levels.

Switch the coupling on and the answer is the one perturbation theory predicts, which is worth having as a measurement rather than as an expectation. Fitting the logarithm of the upper band’s departure against the logarithm of the coupling over the range 0.05 to 0.2 gives an exponent of 2.2007. Second order in the coupling, first order in nothing.

The lower band does something the argument did not anticipate. Its departure is negative at small coupling — the band is made slightly flatter-tailed — reaches a minimum near a coupling of 0.3, and comes back through zero somewhere between 0.4 and 0.6 before turning strongly positive. The two effects are the broadening, which pushes one way, and the repulsion from the band above, which pushes the other, and they cancel at one point. A shape measured there is indistinguishable from a shape measured at no coupling at all.

The better question: a structure that nearly factorises #

The second question is the one worth the essay.

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.

The construction is a three-dimensional torus with the hops in one direction multiplied by λ. It still factorises exactly — the graph is a product whatever the weights are — so its levels are still a sumset,

$$x = 2\cos k_1 + 2\cos k_2 + 2\lambda \cos k_3,$$

and the same cumulant argument applies without modification. Two of the three variables have variance 2 and the third has variance 2λ²; the fourth cumulant of 2cos θ is −6 for all of them; and adding cumulants gives

$$\frac{\mu_4}{\mu_2^2} = 3 - \frac{3(2 + \lambda^4)}{2(2 + \lambda^2)^2}.$$

At λ = 1 that is 3 − 3/6 = 2.5, the three-dimensional value. At λ = 0 it is 3 − 3/4 = 2.25, the two-dimensional one. Between them it is a smooth curve with nothing fitted anywhere in it.

Every point agrees with the expression to nine decimal places, which is the precision the sumset construction carries and not a tolerance.

Where a band stops being three-dimensional #

The expression turns a question of judgement into a number. Halfway between 2.25 and 2.5 is 2.375, and the anisotropy at which the closed form takes that value is

$$\lambda^* = 0.46518.$$

That is a much larger interlayer coupling than intuition suggests. A material whose out-of-plane hopping is a fifth of its in-plane hopping — which a chemist would call strongly layered — has a band shape of 2.278, only eleven per cent of the way from the two-dimensional value to the three-dimensional one. At λ = 0.3 it is 2.310, a quarter of the way.

The reason is the fourth power. The interlayer direction enters the numerator as λ⁴ and the denominator as λ², so at small λ the departure from two dimensions goes as λ² and is very slow. A band is two-dimensional for much longer than the structure is.

That is a statement about a shape, and it is worth saying what it is not a statement about. The band width is 4 + 4λ and changes linearly, so a fifth of the hopping out of plane widens the band by ten per cent — an easily visible amount. The shape is the quantity that is slow, and it is slow because it is a fourth cumulant divided by the square of a second.

The check that the product form is a matrix #

The closed form is a derivation, and a derivation of that shape is exactly the kind that is right about the levels and wrong about how many times each occurs. So the sumset is checked against a diagonalisation on a structure small enough to solve: a 4×4×4 layered torus at λ = 0.3 and at λ = 1, sorted spectra compared entry for entry.

Every level agrees to better than 10⁻⁸, multiplicities included. Comparing band edges or second moments would not have caught a multiplicity error; comparing sorted lists does.

The one-dimensional density this whole family is built from is the arcsine law, with a divergence at each band edge, and where the states pile up draws it. Everything above is a statement about what happens when several of those are added together, and the reason a sum of them becomes a bell curve is the reason any sum of independent variables does.

What this says about the closed form #

The hypercubic sequence 3 − 3/(2d) is the special case λ = 1 of the expression above, and it is worth noticing that the layered form is not a smooth interpolation of it. Setting d = 2 + λ² in 3 − 3/(2d) gives 2.3182 at λ = 0.5, where the correct value is 2.3889.

So an “effective dimension” read off the coordination is not the dimension the shape corresponds to. A band with no structure in it is what the high-dimensional limit looks like, and the layered case is the approach to it. There are two different numbers a reader might call the dimensionality of a layered band — the one that matches its second moment and the one that matches its fourth — and they disagree by an amount that grows to seven per cent in the middle of the range. That is the same shape of trouble as two counts of how many centres a pair holds: a word with two computable meanings, quoted as though it had one.

The two-band gap, and where it comes from #

One more number falls out of the two-band construction without being asked for. The gap between the two bands is 1.6 at zero coupling — Δ minus twice the sum of the two hoppings, which is the arithmetic a full band’s insulating character rests on — and it opens as the coupling is turned up: 1.6104, 1.6415, 1.7647, 1.9653, 2.2374, 2.9666, 3.8944.

That is second-order repulsion between the two bands, seen from the outside. The upper band is pushed up and the lower down, so the gap grows quadratically at first, and by a coupling of 0.8 it has more than doubled. A gap read as a measure of the site-energy difference Δ would be wrong by a factor of two and a half there, with nothing in the picture to say so.

What a shape is for #

A fourth moment is an odd thing to spend an essay on unless it decides something, and it decides two things that are already in use.

The first is binding. Where the states pile up measured that two structures of the same coordination — and therefore the same band width and the same second moment — bind by amounts differing by a factor of nearly two at half filling, and the quantity that separated them was where their levels sat within the band. That is a shape. A layered structure at λ = 0.2 has the band width of a three-dimensional one and very nearly the level distribution of a two-dimensional one, so its binding per unit width follows the plane rather than the stack.

Where all of this started is a level diagram rather than a curve: a solid is a molecule that did not stop draws a chain’s levels at two, four, eight, sixteen and forty sites and a band is what that becomes. Every moment above is a statement about that picture in the limit, and every one is computed from a finite matrix with no periodicity assumed.

The second is the gap. Two bands of a given separation overlap or do not according to their widths, and whether a full band is an insulator turns on exactly that arithmetic. A shape adds the next term: two bands that do not quite overlap have tails that reach towards one another, and how much weight is in a tail is a fourth moment rather than an edge.

Two things this essay inherits rather than establishes are worth pointing at. A gap is a statement about a sequence and not about a material — a uniform chain’s shrinks as one over its length while an alternating one’s settles, which a chain cannot stay even plots at four lengths — and everything here is about bands whose gap is the second kind. And a moment measured on a finite matrix is a moment of a band only once the histogram has stopped moving; the sizes used above, forty cubed for the layered structure and a hundred and forty-four sites for the two-orbital net, sit well inside that region.

What this cannot say #

No electron repulsion anywhere. Every band here is a one-electron band on a graph with numbers on its edges. Two bands with a gap between them is the situation in which repulsion matters most — it is where a Mott insulator lives — and none of it is in this model.

The two orbitals are of the same parity. A real second band is usually a p band, and its coupling to an s band is antisymmetric between the two directions along a bond, which needs complex phases or a doubled matrix. The symmetric coupling here has the same second-order structure and a different sign pattern, so the exponent is trustworthy and the sign of a particular band’s shift is not.

The layered structure has one weak direction. A material with two weak directions — chains packed side by side — has its own expression from the same argument, and it is not computed here.

And a shape is not a spectrum. Two moments do not determine a distribution. What they do is separate structures that differ, and everything above is a comparison rather than a reconstruction.

What the calculation requires #

The product form is a matrix: every level of a 4×4×4 layered torus, at two anisotropies, agrees with the sumset to 10⁻⁸ with multiplicities included.

The closed form holds at every anisotropy, to nine decimal places, and reproduces the two-dimensional and three-dimensional values at its two ends — which are the numbers the closed form already gave, arrived at a different way.

A band with no neighbour keeps its own shape exactly: with the coupling off, both bands return the one-band kurtosis to nine decimal places.

The departure is second order, with a fitted exponent between 1.8 and 2.3 over the small-coupling range.

The lower band’s shift changes sign, which is tested rather than remarked on, because it is the claim that a measured band shape does not measure a coupling.

And the refusal is the offset: a raw fourth moment on a band centred at −Δ/2 must differ from the central answer by more than 0.4 at three gaps, and must tend to one as the gap grows — the tripwire that says why the moments have to be taken about the band’s own centre and not about zero.

Which low dimension is easier to hold on to #

The companion expression — one strong direction and two weak ones, a bundle of chains rather than a stack of layers — was written down above and left uncomputed. It costs nothing to evaluate, and the comparison it allows is the one worth having, because it is a statement about the arithmetic rather than about any material.

Both curves start at their own low-dimensional value and end at 2.5. Read as the fraction of that journey completed:

The bundle loses its low-dimensional shape roughly twice as fast at every coupling. Its halfway point falls at $\lambda = 0.348$ against the layered structure’s $0.465$, and by $\lambda = 0.5$ — a weak direction by any structural description — a chain-like material is three quarters of the way to being three-dimensional while a layered one is barely past half.

The factor of two has an obvious source and is worth naming rather than leaving as a number. The bundle has two weak directions contributing and the layered structure has one, so at small $\lambda$ the bundle’s variance and fourth cumulant both pick up twice as much, and the ratio moves twice as fast. It is the same arithmetic that gave the layered case its slow approach, applied to twice as many terms.

Which inverts a plausible expectation. A one-dimensional material looks like the most robustly low-dimensional thing available — a chain is more obviously confined than a layer — and its band shape is the more fragile of the two. Confinement in the structure and confinement in the spectrum are counted differently: the structure counts how many directions are weak, and the spectrum counts what they contribute, which is the same number twice over.

Still open: a material’s axis, and the two-band measurement #

The comparison of the layered and bundled expressions above is made at fixed anisotropy, which is the arithmetic’s own variable and not a material’s. What a real comparison would need is the two structures at fixed ratio of bond strengths as measured, and a bond strength is not a hopping integral — it is a quantity with a length and a force constant in it. Putting the two curves on an axis a material could be placed on is therefore a question about how a hopping relates to a bond, which nothing here computes.

The nearer question is what to do with the two-band case now that its rate is known. The departure being second order in the ratio of coupling to gap means the product of the two is what a measurement constrains, so a single shape measurement fixes a hyperbola in the two-parameter plane rather than a point. What would fix a point is a second observable with a different dependence — the gap itself, which goes as the square of the coupling divided by the gap, and is drawn above. Two quantities with the same second-order origin and different combinations of the parameters is exactly the situation in which two measurements are worth four times one, and setting that up properly is a calculation of its own.