The gap follows the winner late

What holds a solid together is, in a band picture, the energy its electrons gain by spreading across neighbouring sites. An alloy of two kinds of atom on a lattice has, for each composition, many ways of arranging them — and a mixture does not bind as the average of its ends — so the one that binds best is the one the solid should adopt. In a tight-binding model with one orbital per site, the two kinds differ by a site energy of +δ or −δ, the band is half filled, and the binding of an arrangement is the energy of its filled levels. The census of arrangements finds the best ones by steepest ascent from two hundred random starts, swapping one raised site with one lowered site at a time, and records every local optimum — every basin — the starts reach.

Two results framed what followed. A count of unlike bonds — raised sites next to lowered ones — picks the winner on a square net below a contrast of 1.5272 and fails above it. Half filling turned out to be the easy composition, not the hard one. And on a triangular net, which cannot be coloured in two so that every bond is unlike, half filling was still easy: one basin, every start reaching it, at the smaller contrast. The arrangement reached made thirty unlike bonds where thirty-two were possible, so whatever made it the winner was not a count. The explanation proposed was spectral. An arrangement that opens a wide gap at the Fermi level pushes its occupied levels down and its empty ones up, and at half filling that is most of what binding is — so the winner should be the arrangement with the widest gap.

That is a claim about numbers the census already holds, and it can be checked basin by basin.

Eighteen of twenty #

For both nets, at contrasts of 1 and 4, and for every composition from four raised sites of sixteen to eight, the census’s basins were each diagonalised and the gap between the eighth and ninth levels — the highest occupied and lowest empty of a half-filled band — recorded.

In eighteen of the twenty cases the winning basin has the widest gap. Sometimes trivially, because there is only one basin; often not, as on the triangular net at a contrast of 4 with six sites raised, where four basins compete and the winner’s gap of 1.464 is the widest of them, tied with two others. The gap is a good predictor, far better than the count of unlike bonds, which failed at four of the square net’s compositions at the larger contrast. It also costs nothing extra: the levels that give an arrangement’s binding give its gap, so a search that already diagonalises every arrangement it visits has the gap in hand at every step.

The two exceptions are the square net at a contrast of 1 with six sites raised, where the winner’s gap is 0.806 and the runner-up’s 0.898, and the triangular net at half filling at a contrast of 4, where the winner’s gap is 4.947 and a runner-up’s 5.088. The second is exactly the composition the gap was proposed to explain. Whether it is a marginal disagreement or a real one needs the frustrated net’s half filling followed across the contrast rather than sampled at two values.

The square exception is of a different kind and is worth separating from it. With six of sixteen sites raised at a contrast of 1, the census finds two basins: the winner makes twenty-four unlike bonds and the runner-up sixteen, and the winner binds better by 0.030 per site with the narrower gap. There the count of unlike bonds picks the winner and the gap does not — the reverse of the frustrated half filling at low contrast, where the gap picks it and the count does not. That composition is below half filling, where the Fermi level falls inside a band rather than between two, and the gap there is a less natural measure of what the electrons gain. It is recorded as a failure of the gap and not pursued; the frustrated half filling is the case the gap was offered for.

The winner changes, and the gap does not notice at once #

At eleven contrasts from 1 to 6, the triangular net’s half-filled composition has one basin up to a contrast of 2.5, and its winner makes thirty unlike bonds. From 3 there are two basins and the winner still makes thirty. At 4, the winner makes thirty-two — an arrangement at the counting ceiling — and it keeps winning at 5 and 6.

So the winning arrangement changes somewhere between 3.5 and 4. That is not a matter for the census to resolve, because the two contenders can be held fixed: the thirty-bond arrangement that wins at 3.5 and the thirty-two-bond arrangement that wins at 4, each evaluated at any contrast without searching.

Below the crossing the thirty-bond arrangement binds better, by up to 0.011 per site at a contrast of 2; the difference narrows, changes sign and grows the other way. Bisection puts the change of winner at δ = 3.7900. Both contenders are basins of the census at 3.5 and at 4, on either side of the crossing, so near it this is an exchange between two structures that both exist, not the appearance of a new one. Below a contrast of about 2.7 the census does not find the thirty-two-bond arrangement as a basin at all; starts that pass near it climb on to the thirty-bond one, which is why a single basin was reported there. It still exists as an arrangement and can still be evaluated, which is what makes it possible to follow the pair below the contrast where the census first sees them both.

The gaps of the same two arrangements tell a different story.

The thirty-bond arrangement has the wider gap at every contrast up to 4.8487, where the two gaps cross — more than a whole unit of contrast after the binding crosses. Between 3.790 and 4.849, the thirty-two-bond arrangement binds better with the narrower gap. At a contrast of 4, inside that interval, it binds better by 0.0011 per site while its gap is narrower by 0.140. The gap follows the winner, but late.

That is why it is right eighteen times in twenty. The gap and the binding are strongly correlated across arrangements, and they rank two contenders the same way except near a change of winner, where the binding has already moved and the gap has not. A census that samples the contrast coarsely will usually sit away from such an interval and find the gap reliable; one that happens to sample inside it will find the gap wrong about the one case that matters.

Why the gap lags #

The binding at half filling is the sum of all eight occupied levels. The gap is the distance between two of them. Those are different functions of an arrangement, and nothing requires them to change their ranking of two arrangements at the same contrast.

There is a reading of why they separate here, offered as a reading. At large contrast the site energies dominate the hopping, the raised and lowered sites form two narrow sub-bands, and each unlike bond lowers the occupied levels by an amount that no longer depends much on which other bonds surround it — so the binding approaches a sum over unlike bonds, and the arrangement at the counting ceiling must eventually win. That is the regime the thirty-two-bond arrangement is heading into. The gap, by contrast, is set by the top of the lower sub-band and the bottom of the upper one, and those edges depend on how the raised and lowered sites are connected into clusters rather than on how many contacts there are in total. An arrangement can therefore have enough unlike bonds to win on binding while its clusters still leave the band edges closer together.

At small contrast, the opposite limit, the band is a single band perturbed by the site energies, most of the binding comes from levels near the middle, and the gap and the binding are governed by the same few levels. That is where they agree. The interval of disagreement lies between the two regimes, which is also where the winner changes — so the lag is not an accident of where it happens to fall.

What the numbers establish is the lag itself: where it starts, where it ends, and that inside it the gap and the binding disagree about which of two real arrangements wins. The reading explains why the thirty-two-bond arrangement should win eventually; it does not predict 3.790 or 4.849, and it is offered only as a reading.

Where a second basin appears #

The other half of the question left open was the degeneracy. At half filling the triangular net goes from one basin to several between the two contrasts sampled before, and the square net does the same. If the appearance of a second basin were the same phenomenon as the count’s failure, it would happen at the same contrast.

It does not. Bisected ten times on the census, a second basin appears on the square net at 1.237 and on the triangular net at 2.714. The count of unlike bonds stopped picking the square net’s winner at 1.5272. The square net’s second basin appears a quarter below that threshold; the triangular net’s more than a whole unit above it. And neither onset is where the triangular winner changes, at 3.790.

The square net’s second basin is rare — five starts in two hundred reach it at the onset, a thirty-two-bond winner and a twenty-four-bond runner-up — and a rarer basin at a lower contrast could exist unseen. The onsets are therefore upper bounds on where a second basin exists, and lower bounds on where two hundred starts will find one. Neither reading brings either onset to the count’s threshold.

So three contrasts that might have been one — where a count stops working, where a landscape acquires a second basin, and where the winner changes — are three different numbers on these nets. The count’s failure is a property of which arrangement wins; the onset is a property of which arrangements are local optima; the change of winner is a property of two particular arrangements’ binding. The frustrated net separates them cleanly.

How it was computed #

The nets are sixteen sites with periodic boundaries: the square net with four neighbours a site and thirty-two bonds, the triangular net with six neighbours and forty-eight bonds. An arrangement raises eight sites to +δ and lowers eight to −δ, with a hopping of one on every bond; its binding is the filled-level energy of a half-filled band per site. The census is two hundred steepest ascents from seeded random arrangements, each taking the best improving swap of one raised and one lowered site until none improves, with basins grouped by binding to nine decimal places. Its results are the ones the earlier census reported and are reused rather than recomputed.

The gap of a basin is the ninth-lowest level minus the eighth-lowest. The contenders are the census winners at 3.5 and 4, evaluated at any contrast directly. Both crossings are bisected fifty times on the fixed arrangements. The onset of a second basin is bisected ten times between the last grid contrast with one basin and the first with two, rerunning the census at each midpoint.

The checks, run wherever these figures are drawn. The winner has the widest gap in at least seventeen cases and not all twenty, and not at the triangular net’s half filling at 4. Up to 2.5 that composition has one basin of thirty unlike bonds, and from 4 its winner has thirty-two. The two contenders’ binding crosses between 3.5 and 4 and their gaps cross later; at every grid contrast the gap agrees with the binding about which contender wins except strictly between the two crossings, where it disagrees. The square net’s second basin appears between 1 and 1.25 and the triangular net’s between 2.5 and 3, neither within 0.25 of 1.5272. The refusal is the contenders’ identity: the thirty-bond arrangement must bind at 3.5 exactly as that census’s winner does, and the thirty-two-bond one at 4 exactly as that census’s winner does, or the arrangements being bisected are not the basins being counted.

Where this stops #

Sixteen sites. Both nets are four by four with periodic boundaries, and the triangular net’s frustration is that of a small torus. Whether the winner changes at the same contrast on a larger triangular net, or at all, is not measured, and the earlier census could not fit a larger triangular net in memory.

One electron count. The band is always half filled. The gap at the Fermi level is a half-filling quantity, and at another filling the relevant gap would be between different levels and could track the winner differently.

Found basins. The census reports the basins two hundred starts reach. The onsets depend on that, and a basin reached by fewer than one start in two hundred at some contrast would move them.

Two contenders. The comparison across the contrast follows the two arrangements that win at 3.5 and at 4. The census found a third basin at 4, with thirty-two unlike bonds, reached by two starts in two hundred, and it is never the winner at the contrasts sampled; the claim is about the change from one winner to the other, not about every arrangement’s gap.

And non-interacting electrons. Everything is a one-electron band. A repulsion between electrons on a site would change both the binding and the gap, and not necessarily together.

A predictor that is usually right is right for a reason that fails somewhere #

The gap was a good guess, and it is right most of the time, because across arrangements the gap and the binding are strongly correlated. A correlated predictor fails exactly where the two quantities it links respond to different parts of the problem — here the band edge and the whole band — and that happens where the winner changes. A test of a predictor that samples away from the changes in what it predicts will overrate it, and the places it fails are not scattered: they sit next to the transitions a model exists to describe.

The frustrated net makes the point cleanly because its winner changes within the range of contrasts studied, and it carries a second lesson in its three thresholds. Quantities that are named together — where a count fails, where a landscape roughens, where the ground state changes — invite being treated as one event, and on these nets they are three.

Still open: whether the whole band picks the winner, and a larger net #

The obvious open question is the replacement for the gap. If the binding at the transition is decided by the deep levels rather than the edge, a moment of the density of states — the second, which counts bonds and site energies, or the fourth, which sees the shape of the band — should pick the winner through the interval where the gap does not. The moments of every basin are available from the same diagonalisations, and whether one of them tracks the binding across 3.790 without lagging would say whether the lag is about the edge specifically.

The nearer question is the net size. A triangular net of thirty-six sites was too large for the census’s memory at its first attempt, and a smaller change — a six by four torus — would test whether the change of winner at 3.790 is a property of the triangular lattice or of the sixteen-site torus that cannot hold a larger ordered pattern.