The arrangement a count cannot pick

One composition arranged three ways gives three different cohesive energies ordered by a count of unlike bonds: the alternating arrangement most bound, the segregated one least, the random one between. That invites a rule, and testing the rule is the reason this essay exists.

“The count being a perfect predictor for three cases is much weaker evidence than it looks.”

It is weaker for a specific reason. The three arrangements were chosen at the extremes of the count — the most unlike bonds, the fewest, and a sample in between — and three points chosen that way will agree with any monotone function of the count whatever. The test is not whether the count orders three arrangements. It is whether the count picks the winner out of all of them.

That is a finite question on a small net, and it has been answered by looking at every one.

The search, which is a loop #

A wrapped square net of sixteen sites and thirty-two bonds. Half the sites are raised by a contrast $\delta$ and half are lowered, or some other composition is chosen; every arrangement of that composition is built, diagonalised, filled to half, and its binding per site recorded, along with the count of bonds joining a raised site to a lowered one.

The counts are what they are: 120 arrangements at two raised sites, 560 at three, 1,820 at four, 4,368 at five. Nothing is sampled and nothing is optimised — every arrangement is visited, so the best one found is the best one there is.

Sixteen sites is small, and it is small deliberately. The question is whether a rule holds, and a rule that fails on sixteen sites has failed. A rule that holds on sixteen sites has not been proved, which is said again at the end.

The filling is half throughout, and the binding is quoted per site so that arrangements of different compositions can be read on one axis. Both conventions are the ones used for the three arrangements, so the numbers there and here are the same numbers — the pure net’s own value is where a composition of nothing lands, and the search is checked against it.

Where the count is right #

At a contrast of 1 it is right everywhere tested. From nothing raised up to six of sixteen — 14,892 arrangements in all — the arrangement that binds best is always one of those with the largest possible number of unlike bonds. Not merely close to it: equal to it, at every composition.

That is worth having as a positive result, because it says the three-arrangement finding was not an accident of its three cases. In the regime it was computed in, the count really is the predictor.

And it is worth noticing why it can be, because the reason is the same second-moment argument that runs through this field. The width of a band is a count of neighbours because the second moment of a spectrum is a trace, and a trace is a sum over the graph. At a small contrast a binding is dominated by the width, so a binding is dominated by a count — and the count of unlike bonds is the part of that sum the arrangement can move.

Where it is wrong, and how far #

At a contrast of 4 on the same net, the count fails at every composition tested from two raised sites to five.

The margins are not marginal. At four raised sites the winner beats every arrangement with the most unlike bonds by 0.0239 per site, which is 2.2 per cent of the binding and a good deal larger than the differences reported between the three whole arrangements.

And the winner is short by a third of the available unlike bonds, not by one or two. The two orderings are not nearly the same ordering with an occasional swap; they diverge.

Why the contrast changes the answer #

The mechanism is the one this field keeps returning to, which is that a binding is a property of a spectrum and a count is a property of a graph.

At a small contrast the electrons are spread over every site and the energy is dominated by the hopping. Every unlike bond is a place where the two site energies differ, and the second-order gain from a bond between two sites of different energy is what an unlike bond is worth — so counting them is counting the gains, and the count is right.

At a large contrast the electrons are not spread. They sit on the low sites, because the low sites are four units below the high ones, and the high sites are almost empty. What the energy then depends on is how the low sites are joined to each other: a connected block of low sites is a band of its own and binds well, and low sites scattered singly among high ones are isolated levels that bind badly.

Maximising unlike bonds is exactly the arrangement that scatters the minority component as widely as possible. At a large contrast that is the wrong thing to do — and the arrangement that wins is one that lets the minority sit together enough to form a connected piece, which costs unlike bonds and buys band width.

That is the same mechanism, seen from the other side, as the one a random alloy’s gap turned out to depend on: a run of like sites is a sub-band of its own, and how long the runs are decides where the states go. There it set a band edge; here it decides which arrangement is chosen.

So the two regimes are not two behaviours of one rule; they are two different quantities being maximised. There is a contrast at which one gives way to the other.

Found by bisection at three raised sites: the count is exact below δ = 1.5272 and wrong above it. That number is a property of this net, this composition and this filling and is not offered as a constant of nature; what is offered is that a crossover exists and is not at zero, so both halves of this essay describe real regimes.

Where it is wrong for a different reason #

The square net is bipartite: its sites divide into two classes with every bond running between them, so there is an arrangement in which every bond is unlike. A triangular net is not. It has three-membered rings, and a three-membered ring cannot have all three of its bonds joining unlike sites — one pair must match.

That is frustration, and it breaks the count in a way that has nothing to do with the contrast.

On a wrapped triangular net of nine sites and twenty-seven bonds at a contrast of 0.25 — a quarter of the value where the square net was still exact — the count fails at two of the three compositions searched. With two raised sites the winner has ten unlike bonds where twelve are available; with three it has twelve where eighteen are available.

Six unlike bonds short of the maximum, at a contrast where the bipartite net’s answer was exact to the last arrangement. The reason is that on a frustrated net the maximum-unlike arrangement is not a two-colouring of anything, so the count has stopped measuring the quantity it measures on a bipartite net — it is no longer “how completely have the two components been interleaved” but “how many edges happen to be satisfied”, and those come apart as soon as they cannot all be satisfied at once.

What this says about a cohesive energy #

A cohesive energy is not decomposable into anything much: not a sum of bond energies, not a function of coordination, not an average over components. This adds a fourth member and it is the sharpest, because it is about the one decomposition that had survived.

It is not a count of unlike bonds either. The count is a genuine predictor in one regime and it is a different quantity in the other, and there is no single graph-theoretic summary that covers both — because in one regime the energy is a sum over bonds and in the other it is a property of the connected components of a sublattice.

The original picture — one composition arranged three ways, ordered by their unlike-bond counts — is still true of the three arrangements drawn. What it suggests, that the count is the quantity being ordered, is what an exhaustive search refuses.

There is a broader pattern here that this field has now met three times. A bond is not a fixed quantity that can be counted; a surface is not a count of broken bonds; and an arrangement is not ranked by a count of its edges. In every case the count is right in the weak-coupling limit and wrong once the states stop being spread evenly, and in every case the correct statement is about a spectrum.

The winner is one arrangement in eighteen hundred #

The enumeration answers which arrangement binds best, and there is a second quantity sitting in the same loop that answers a different and more practical question. It costs nothing extra, because it is the length of the list.

There are 1,820 ways of raising four of sixteen sites. The winner is one of them. Every arrangement near it in energy is another, and the number of arrangements within a given distance of the minimum is something the enumeration already knows — it is a density of states in configuration space rather than in energy, and it is what decides whether the best arrangement is the one that actually occurs.

The comparison is the ordinary one between an energy and an entropy. The ordered winner has essentially no multiplicity: it and its symmetry-related copies are a handful out of 1,820. The disordered arrangements have almost all of it. Taking the whole set as available gives $\ln 1820 = 7.51$ in units of Boltzmann’s constant, or 0.469 per site, and an arrangement gives that up entirely in exchange for whatever binding it gains.

So the arrangement that wins the enumeration is the arrangement that occurs only below a temperature where the binding advantage per site exceeds about half of $k_BT$. Above it the alloy is disordered — not because the ordered arrangement stopped being the most bound, but because being the most bound stopped being the criterion.

That is the order–disorder transition, and it is worth noticing that everything needed to see it is already inside a loop written to find a minimum. The energies of all 1,820 arrangements are a distribution; its lowest point is what this essay reports; its width against its multiplicity is the transition.

It also puts the essay’s negative result in a better light than a negative result usually deserves. If the count of unlike bonds picked the winner reliably, that would be a rule about one arrangement out of 1,820 — the rarest possible statement about the system. What the count does do, and the figure shows it doing, is predict the trend across the whole cloud: arrangements with more unlike bonds bind better on average, at every contrast tested. A trend across 1,820 arrangements is the quantity a thermal average actually samples, and it is in far better shape than the rule about the extremum.

The honest summary is therefore two statements rather than one. The count is a good predictor of the average and a poor predictor of the minimum, and which of those a reader wants depends on whether the material was cooled slowly enough to find its ground state. Most are not, which makes the failure this essay measures less consequential than the trend it also measures — and neither could have been seen from three arrangements.

What this cannot say #

Sixteen sites. A rule that fails at sixteen has failed, so the negative results stand. The positive one — that the count is exact at a contrast of 1 for every composition up to six raised — is a statement about a small net and could fail on a larger one; nothing here says it does not.

One electron and no repulsion. The electrons here neither see nor charge each other, so an arrangement that piles them onto a connected block of low sites pays nothing for the crowding. On-site repulsion is exactly the term that would penalise that, and it would push the answer back towards the dispersed arrangement — which means the crossover contrast found here is a lower bound rather than a value.

One filling. Everything is at half filling, which is where a band binds most and where the spread of the levels matters most. At a nearly empty or nearly full band the energy depends on the band edges alone, and the count would be measuring something different again — the edges are set by where the states pile up rather than by the width.

And there is no lattice relaxation. Two components of different sizes distort the net they share, which is a term with no representation here at all — and the lattice sum whose answer depends on the order of adding is where that boundary is drawn.

What is quoted, and what is computed #

Nothing is quoted. There is no measurement anywhere in this essay and no material named. The nets, the compositions and the contrasts are parameters; every binding, every count, every winner and the crossover contrast are computed by building matrices and diagonalising them.

The search is exhaustive rather than stochastic, which matters for the shape of the claim: a random search that failed to find a better arrangement would be weak evidence that none exists, and an enumeration that finds one is proof.

What was checked #

At a contrast of 1 the best arrangement has the most unlike bonds, at every composition from nothing raised to six of sixteen. This is the unlike-bond rule, checked against every arrangement rather than three.

At a contrast of 4 it does not, on the same net at the same compositions — and both halves are checked, because the second alone would be consistent with a search that had gone wrong.

The winner at each failing composition beats every most-unlike arrangement, which is the claim, and has fewer unlike bonds than they do, which is what makes it a counter-example rather than a tie.

On a net with odd rings the count fails at a contrast where a bipartite net’s does not, so the two mechanisms are separated rather than conflated.

And there is a contrast on the bipartite net at which the count stops, found by bisection between 1 and 8. Measured: 1.5272.

The refusal is a composition of nothing. With no sites raised there is one arrangement and no unlike bonds, and the search must return the pure net’s binding shifted by the whole contrast rather than a search result — checked to nine decimal places.

Still open: a count that works at large contrast, and larger nets #

The obvious open question is the quantity that does predict the winner at a large contrast. If what matters there is how the low sites are joined to one another, then the right summary is a property of the sublattice they form — its own second moment, or the size of its largest connected piece — and that is another count, computable without diagonalising, that could be tested against the same enumeration. A second count that works where the first fails would turn two regimes into two rules with a stated boundary, which is a great deal more useful than one rule with an exception.

The nearer question is about size. Every negative result here is on sixteen sites or nine, and the positive one is too. A net twice as wide has more arrangements than can be enumerated, so the test has to change shape: take the winner found on the small net, extend its pattern to the larger one, and ask whether anything beats it among the arrangements a local search can reach. That is a weaker instrument and it is the only one available past about twenty sites — and knowing how much weaker it is, measured against the exhaustive answer on a net where both can be run, is the thing worth establishing before trusting it anywhere.