The constant that belonged to one net

Two ways of being second order took two numbers that are both second order in the same coupling — how far a band’s shape has been pushed from its uncoupled value, and how much wider the gap is than the four band edges alone would put it — and showed that dividing one by the other leaves the coupling behind. What is left goes as the separation of the two bands and nothing else, one constant across a factor of three in that separation, to better than a per cent.

That is a result about a map from parameters to observables, and it was measured with one graph. Both sets of orbitals hopped on a square net; both bands were therefore the same band scaled by their own hopping, with the same width in units of that hopping and the same shape. A real pair of bands need not be. A σ set and a π set on the same atoms reach different neighbours, so they have different coordinations, different second moments and different shapes before anything mixes at all.

So the question is whether the quotient still cancels the coupling when the two bands are two different structures. It does. And that turns out not to be the interesting half, because the other thing the quotient had — its proportionality to the gap — comes apart, and it comes apart for a reason that has nothing to do with there being two graphs.

Two graphs, one set of sites #

The construction is the one-graph construction with one restriction lifted. Every site carries two orbitals, the lower at −Δ/2 and the upper at +Δ/2; the lower ones hop along one graph and the upper ones along another; and the two are joined across the lower graph’s bonds with a strength that is the coupling.

That third choice has to be made and cannot be derived. There is no reason a coupling between two sets of orbitals should follow either graph in particular, and following the lower one is stated rather than argued. It has the one property the comparison needs: with the two graphs equal it is the one-graph system exactly, so what follows is a generalisation of that result rather than a different calculation placed beside it.

The two graphs have to be on the same sites in the same order, and that is not bookkeeping. A wrapped square net and a wrapped triangular net at the same size index their sites identically, which means they have the same eigenvectors — and the whole two-band problem then comes apart into one two-by-two matrix per mode, each of which can be solved by hand.

The agreement is $3.5 \times 10^{-13}$ at a coupling of 0.3, which is five times larger than any coupling the exponents below are measured at. That matters more than it looks. Everything downstream is read off an expansion of the two-by-two eigenvalue in the gap, and an expansion is only worth reading if the thing being expanded is the thing being measured. Here it demonstrably is, to eleven figures, without any appeal to perturbation theory at all — the closed form is exact and the expansion is an approximation to it, checked against a matrix that knows nothing about either.

The coupling still cancels #

The first question has a clean answer. Both departures remain second order in the coupling at every pairing of graphs tested.

The fitted exponents are 1.971, 2.000 and 2.000 — the lower band’s shape, the upper band’s shape, and the excess gap. Across all four pairings measured here they run from 1.971 to 2.044, which is the spread a least-squares slope on five points has rather than a spread in the physics.

So the coupling divides out of any quotient of these numbers, on any pair of graphs. That is the part of the one-graph argument that generalises, and it generalises for the reason it was given: at second order the coupling enters every one of these quantities as its square, and a square divided by a square is one.

The gap does not #

The second half is where it goes.

With one graph, the exponents are $-2.011$ and $-2.012$ for the two shape departures and $-1.011$ for the excess gap, which is the received answer: the shape sees the ratio of coupling to gap and nothing else, so it goes as the inverse square; the excess gap sees the coupling squared over the gap, so it goes as the inverse first power. Their quotient is therefore the gap, and dividing by the gap leaves a constant. Across a factor of four in the separation that constant varies by 0.00 per cent.

With a square net below and a triangular one above, the exponents are $-1.894$, $-0.985$ and $-1.017$. The lower band keeps its order. The upper band does not: its shape departure has become inverse first order in the gap, which is the same order as the excess gap — so the quotient of those two contains no gap at all.

Measured the original way, the quotient varies by 134 per cent over the range where it varied by a tenth of one. The lower band’s quotient survives in the sense that it is still a decreasing function of the same thing, but it drifts by 17.6 per cent, which is twenty times the drift of the one-graph case and far too much to call a constant.

The reason is not that there are two graphs #

The obvious reading is that a quotient built to cancel one coupling between two identical bands cannot be expected to cancel it between two different ones, and that the one-graph result is the special case of matched structures. That reading is wrong, and the calculation that refutes it is one line different from the one that suggests it: put the triangular net on both sides.

One graph, twice, exactly as in the original construction. The exponents come out at $-1.105$, $-1.105$ and $-1.094$, and both quotients drift by 128 per cent. Every symptom of the two-graph case, with no second graph anywhere in the system.

So the thing that made the quotient a constant was never the matching. It was the net.

What was actually doing the work #

Expanding a mode’s exact eigenvalue in the separation puts one extra term on each band’s dispersion at first order in $1/\Delta$: the upper band picks up $+\text{mix}^2\varepsilon_c(k)^2/\Delta$, where $\varepsilon_c$ is the structure factor of the graph the coupling runs along. Whether that term can move the band’s shape is a question about a single derivative — how fast the fourth-moment ratio changes as a multiple of $\varepsilon_c^2$ is added to the dispersion — and that derivative can be computed from the closed form alone, with no two-band matrix anywhere in it.

If the derivative is zero the first-order term cannot move the shape, the leading effect is whatever survives at the next order, and the departure is inverse square in the gap. If it is not zero, the departure is inverse first order. That is a prediction, and it is made before the system is built.

For each pairing, the sensitivity of a band’s shape to the first-order term can be predicted from the moments before the matrix is built, and the gap exponent measured afterwards from the diagonalisation. The prediction is right in all eight cases, which is what turns the observation into a rule rather than a table.

The sensitivities that vanish do so as arithmetic and not as small numbers: the largest of them is $1.1 \times 10^{-12}$, against 8.889, 9.999, 11.9998 and 19.9992 for the ones that do not. And the pattern of which vanish is the answer to the whole question. A sensitivity is zero exactly when the band’s coupling graph is the band’s own graph and that graph’s spectrum is symmetric about its centre.

The underlying asymmetry is visible in the bare spectra: a square net’s band is symmetric about zero and every odd moment of it vanishes, while a triangular net’s is not and its third moment does not. That single difference is what the constant is sensitive to, and it is a property of the graph rather than of the coupling.

A graph with no closed walk of odd length has a spectrum symmetric about zero, because every walk that contributes to an odd moment can be paired with one of the opposite sign. The square net has that property and the triangular net does not — the triangle is a closed walk of length three. Adding a multiple of $\varepsilon^2$ to a dispersion $t\varepsilon$ changes the fourth-moment ratio at first order through $\langle\varepsilon^3\rangle$ and $\langle\varepsilon^5\rangle$, and on a symmetric spectrum both of those are zero.

So the constant was a statement about closed walks of odd length. Nothing in the argument that produced it mentioned the lattice at all, and its whole content was that one particular lattice happened to be bipartite.

What it costs an experiment #

The one-graph result was not a curiosity about exponents; it was the claim that two measurements of second-order quantities fix two parameters, which is what makes a set of band measurements worth inverting. That claim can be priced.

With one graph, the upper band’s shape and the excess gap invert at a condition number of 6.74, and one per cent measurements of both fix the coupling to 1.16 per cent and the separation to 1.49 per cent. With a triangular net above a square one, the same pair inverts at 160.5, and the same measurements fix them to 22.7 and 45.4 per cent. The pair has not become singular, but it has stopped being useful, and it has done so silently — every observable is still measurable, still second order in the coupling, still a smooth function of both parameters.

The lower band’s shape, paired with the same excess gap, still inverts at 6.75. So the repair is available and it is a repair about which band to measure, which is not a choice anybody would have known they were making.

What was computed, and how #

Every level quoted here comes from diagonalising a real symmetric matrix of $2n$ sites with an eigensolver checked against closed forms, and every eigenpair is required to satisfy $Av = xv$ before anything is read off it. The band shapes are fourth central moments divided by the square of the second, computed over the levels of each band separately after the level list is cut at its largest interior gap.

The excess gap is measured against a written-down bare gap — the four band edges, each scaled by its own hopping — and not against a second calculation with the coupling switched off. The reason is sharper here than before: with two graphs the two bands have different half widths, so a subtraction of two numerical answers would hide exactly the arithmetic that has to be visible.

Both exponents are least-squares slopes on logarithms over five or six points spanning a factor of four, rather than derivatives at a point. That is deliberate. A derivative cannot tell a power law from a curve, and one of the findings here is that a quantity the received argument calls a power law is not one over the range anybody would measure it in.

The sensitivities are central differences on the closed-form spectrum, each computed at two step sizes a factor of two apart, with the two required to agree — the same rule everything this collection inverts is subject to, and for the same reason: an entry wrong in its third digit is a condition number wrong in its first.

The refusal is the pairing that must not behave: with the square net on both sides the sensitivity has to vanish, and with a triangular net anywhere it has to be large. A version of this argument that reported a small sensitivity everywhere would be measuring its own step size.

Where the model stops #

Every structure here is wrapped, which is what makes the modes exact and the closed form available. A finite piece of either net with ends would have surface states, and an end changes what a band does in ways this comparison would read as a shape departure. That is a real limitation on the reading, not a caution about the arithmetic: the exponents are properties of the bulk spectrum and are measured on a system that has only bulk.

The coupling is taken symmetric and real, which makes the two sets of orbitals the same parity — an s set and another s set rather than an s set and a p set. What is being asked is whether a band’s shape survives having a differently-shaped band beside it, and that question does not turn on parity; but no figure here may be read as a statement about a σ–π mixing, which has a sign structure this model does not carry.

And there is no electron repulsion anywhere. The bands are one-electron bands, the gap is a parameter rather than a consequence, and nothing here can see the insulator a half-filled band is sometimes attached to.

The generalisation #

The shape of what happened is worth stating without the nets in it. A result was obtained by dividing one quantity by another so that a nuisance parameter cancelled, and the quotient was then found to depend on one remaining parameter in a simple way. The cancellation was real and general. The simple dependence was an accident of the example, and it was an accident of a property nobody had written down — that the example’s spectrum was symmetric.

The same shape has turned up twice before from the other end. A fitted exponent turned out to belong to the window it was fitted over rather than to the system, and a decay length turned out not to exist because the profile it was fitted through was not an exponential. Both of those are failures of a fit. This one is not: every exponent here is a good fit to a real power law over the range measured. The failure is one level up, in the claim that the exponent is a property of the arithmetic rather than of the system the arithmetic was demonstrated on.

The instrument that separates them is the same in all three cases, and it is cheap: run the argument on a second example chosen to differ in something the argument does not mention. The one-graph argument does not mention bipartiteness, which is precisely why a triangular net was the right second example.

Who found it, and when #

Bipartite graphs and the symmetry of their spectra are Coulson and Rushbrooke’s from 1940, in the form the pairing theorem takes in Hückel theory — the levels of an alternant hydrocarbon come in $\pm$ pairs, and it can be checked in both directions so that the non-alternant systems are required to fail it. That the same property controls the odd moments of a lattice band is the same statement with a larger graph in it.

The moment expansion of a density of states is Cyrot-Lackmann’s, from the late nineteen-sixties, and the observation that the low moments are counts of closed walks is what makes it usable: the second moment counts neighbours, the third counts triangles, and a lattice with no triangles has none. The connection running the other way — that a missing third moment is what makes a perturbation invisible at first order — does not appear to be a standard remark, and it is the only thing here that is not textbook.

Still open: the coupling graph, and the third moment #

The obvious open question is the coupling graph, which was fixed to the lower band’s by fiat here and then found to be the deciding object. If the sensitivity is zero exactly when the coupling graph is the band’s own and that graph is bipartite, then coupling two square-net bands along a triangular set of bonds should destroy the constant while leaving both bands untouched — two bipartite bands, one non-bipartite coupling, and an exponent of −1. That is a sharper test of the mechanism than anything here, because it moves the only object the explanation names and nothing else.

The nearer question is the third moment itself. The argument above says the first-order term is invisible because $\langle\varepsilon^3\rangle$ vanishes, which predicts that the size of the effect on a non-bipartite net should track that moment rather than the coordination or the shape. Three nets of the same coordination and different triangle counts would test that directly, and they are easy to build — two structures with the same neighbours is the pairing, and it was built to separate coordination from dimension. Asked about the skewness instead, it would say whether the third moment is the whole of the mechanism or merely the first term of it.