A net with no two-colouring

Composition was swept and found the half-filled arrangement among the easiest to search rather than the hardest, which was the opposite of the guess. It also offered a reason, and the reason was not a measurement:

on a bipartite net at half filling the two-colouring is the unique arrangement with every bond unlike, so the optimum is a single configuration with nothing competing with it.

That sentence makes a prediction about a net with no two-colouring. It is wrong.

The obstruction is a triangle #

A net is bipartite when its sites split into two classes with every bond running between them, and the obstruction to that is a cycle of odd length. A square net has none: every closed walk on it returns with an even number of steps. A triangular net is made of them.

Sixteen sites wrapped into a triangular net have forty-eight bonds and thirty-two triangles, two per site, each bond belonging to two of them. That gives a ceiling on how well any arrangement can do at making bonds unlike, and the ceiling is four lines of counting.

Three sites given two labels must repeat one of them, so at least one bond of every triangle joins two like sites and at most two are unlike. Summing over thirty-two triangles gives at most sixty-four unlike-bond-in-triangle incidences; each bond is in two triangles, so at most thirty-two of the forty-eight bonds can be unlike. Two thirds, exactly, with no energy and no search anywhere in it.

The square net’s ceiling is all thirty-two of its bonds, which is the same statement as its being bipartite.

So the two nets differ in precisely the way that explanation is about, and everything else about the census can be held fixed: same size, same contrasts, same number of starts, same ascent.

The prediction fails #

At the smaller contrast the triangular net’s half-filled composition has one basin, and all two hundred steepest ascents reach it.

That is as uniquely easy as anything in either census. The square net’s half filling at the same contrast is also one basin at a hundred per cent, and its hardest composition — five sites of sixteen raised — is two basins at twenty-one. So on the net with no two-colouring, half filling is the single easiest point of the sweep, by the same margin as on the net with one.

And it is not even the arrangement a count would pick. Its best makes thirty of the forty-eight bonds unlike, two short of the ceiling. If the ease came from an arrangement that maximises unlike bonds having nothing to compete with — which is the shape of that argument — then the arrangement that is uniquely reached ought at least to be that one. It is not.

So the explanation fails twice over: the structure it invokes is absent, and the property it attributes to the winning arrangement does not hold of the winner.

And the second failure is sharper than it sounds, because the ceiling is reachable. Two short of a bound would be a weak complaint if the bound were slack — a count over triangles is an upper bound and nothing promises an arrangement attains it. On this net one does. At the larger contrast the same composition finds two arrangements making thirty-two unlike bonds, which is the ceiling exactly; the section below has them. So an arrangement maximising unlike bonds exists on this net at this composition, the search does not land on it at the smaller contrast, and what it lands on instead is two bonds worse by that measure and uniquely reached by all two hundred ascents.

That is the shape a refutation wants. If the ease at half filling came from a count-maximising arrangement having nothing to compete with, the arrangement the ascent reaches would be a count-maximiser, and the count-maximisers are available for it to reach. It reaches something else, unanimously. Whatever makes half filling easy to search is not visible to a count of unlike bonds — which is the same conclusion the three-basin tie reaches from the other direction, at the other contrast, with the orderings disagreeing rather than the winner being wrong.

The finding it was an explanation for survives #

All four curves have the same shape. Every start reaches one arrangement at the sparse end, where there is too little to exchange for a landscape to exist. The hard region is between a quarter and a third full. And every curve recovers at half filling.

So the sweep’s result is not in question and only its account of it is. Half filling really is among the easiest compositions to search, on a bipartite net and on a frustrated one, at a small contrast and a large one — and nobody now has a reason why.

That is a better position than the one before, because the wrong reason was load-bearing. It was used to explain the recovery, and a recovery explained by something absent from half the cases is a recovery not yet explained at all. What can be said is narrower and is at least measured: the easy points of these sweeps are the ones where the arrangement is forced, and forcing can come from a two-colouring or from something else that these two nets share.

Where the frustration does show #

The triangles are not doing nothing. They do nothing at the smaller contrast.

Going from a contrast of one to a contrast of four, the bipartite net’s half filling goes from one basin at a hundred per cent to two at ninety-six — four points. The frustrated net’s goes from one basin at a hundred per cent to three at sixty-four — thirty-six points.

So the cost of having no two-colouring is real and it is a cost that only appears once the site energies dominate the band term. At a small contrast the energy is mostly the electronic one, the arrangement that wins is chosen by the whole spectrum rather than by a count of unlike bonds, and the frustrated net has a unique winner the same way the bipartite one does. At a large contrast the near-classical count takes over, and a count on a frustrated net has ties.

That is the finding about what a count cannot pick arriving from the other side. It found that a count of unlike bonds orders the arrangements at small contrast and fails at large — on a square net at four compositions, and on a triangular net at any contrast. Here the degeneracy the count produces is what shows up at large contrast, and the small-contrast case has no degeneracy because it is not being decided by a count.

What the sweep says about the hard region #

The recovery at half filling is one feature of these curves and the trough before it is the other, and the two nets put it in different places.

On the square net at a contrast of one the hardest composition is five sites of sixteen raised, reached from twenty-one per cent of starts. On the triangular net at the same contrast it is four, at forty-one per cent. At a contrast of four the square’s hardest is six at fifty-nine per cent and the triangular’s is six at forty-four.

So the frustrated net is not uniformly harder, which is the thing a reader would expect and is not what happens. It is harder at half filling and at the compositions near it; it is easier at five sites raised, by a factor of two in the hit rate, at the smaller contrast. The trough is in a different place and is a different depth, and no single statement of the form frustration makes the landscape rougher covers both.

That matters for how the result should be read. The sweep’s finding is about the shape of the curve — where the hard region is relative to half filling — and the shape survives. What does not survive is any reading in which the two-colouring sets the depth, because the net without one has a shallower trough than the net with one at three of the eight compositions.

Two at the ceiling, and one between them #

The three basins at that point are the sharpest instance of the count’s failure on record here.

Two of them attain the ceiling exactly — thirty-two unlike bonds of forty-eight, the maximum counting allows on a net with thirty-two triangles. They are not equally good: their bindings differ by 1.8 × 10⁻² per site, and one of them is reached by a hundred and twenty-eight starts and the other by two.

Between them, in energy, sits an arrangement making only thirty unlike bonds, reached by seventy.

So a ranking by unlike bonds would put the two ceiling arrangements first and second and the thirty-bond one third, and the energy puts the thirty-bond one second. The count does not merely fail to be the energy — it fails to order the arrangements, at the one composition and contrast where its ceiling is attained and the case for it looks strongest.

A bound reached is not an optimum found, and the ceiling is a bound.

The size of the disagreement is worth setting against what is being compared. The three bindings differ in the fifth decimal — 4.46092, 4.45978 and 4.44317 per site — so the count gets the ordering wrong by an amount that is small in absolute terms and is not small in the terms that matter: the whole spread of bindings across the entire composition sweep at this contrast is about three and a half, and a search that returned the wrong one of these would be returning an arrangement whose binding is wrong in the third significant figure. That is the accuracy at which a cohesive energy is ever quoted.

What was computed, and how #

The census is the sweep’s, unchanged: two hundred steepest ascents from random compositions, each swapping a raised site with a lowered one while the binding improves, grouped by the energy they reach. The net is built by the same routine with a different key, the triangles are enumerated from the edge list, and the two-colouring is attempted and reported rather than assumed.

Sixteen sites and not thirty-six, and the reason is worth recording rather than leaving to be inferred. The larger net has six neighbours a site against four, its ascent is correspondingly more expensive, and the sweep of it did not fit in the memory available. So this comparison is at one size on two nets where the sweep was at two sizes on one, which is the honest trade.

Twelve things are checked. That the square net is bipartite with no triangle and the triangular one is neither, so the comparison is between the presence and absence of a two-colouring rather than between two arbitrary lattices. That the triangle ceiling is two thirds of the bonds, computed from the triangle count and the bonds-per-triangle rather than quoted. That on all four sweeps the half-filled composition is no harder than that sweep’s hardest point. That the frustrated net’s half filling at the smaller contrast has one basin and every start, and that its arrangement is below the ceiling. That at the larger contrast the same composition has more basins and does attain the ceiling. And the three parts of the tie: that two arrangements reach the ceiling, that they differ in energy, and that one below the ceiling sits between them.

The sparse compositions are the control. With one, two or three sites raised there is little to exchange, every start must reach the same arrangement, and all four sweeps report one basin at a hundred per cent — which is what says the census is measuring a landscape rather than a search.

Where this stops #

Sixteen sites is small and the frustrated net’s landscape is the one most likely to be size-dependent. A triangular net’s degeneracy at half filling grows with the net in a way a bipartite net’s does not, so the three basins found here are three at this size and the count may well grow. The basin count was found going from two at sixteen sites to twelve at thirty-six on a square net; there is no reason to expect a frustrated net to be gentler.

The two contrasts are two points and the crossover between them is not located. What is established is that the frustrated net is undegenerate at one and degenerate at four, and where between them that changes is a bisection nobody has run. Whether the change is gradual or is a threshold would say whether the band term suppresses the degeneracy or merely outweighs it.

And a basin here is an energy. The census groups ascents by the binding they reach, to nine decimals, so two arrangements related by a symmetry of the net are counted once. That is the right thing for a question about how hard the search is, since a search that lands on either has succeeded, and it is the wrong thing for a question about how many arrangements the optimum has. The two questions are not distinguished anywhere in these essays and the frustrated net is where they come apart, because its optima are the ones most likely to be symmetry-related.

The generalisation #

The habit is to find the case an explanation forbids, and to run it even when the finding it explains is not in doubt.

The sweep had a result and a story, and the story was plausible enough that nobody would have checked it — it invokes a real structural fact, it predicts the right thing on the net it was written about, and the result it explains is robust. What makes it checkable is that it names a structure, and a structure can be removed. Removing it left the result standing, which is the outcome that marks the story as decoration.

The corollary is about what to do next in that situation. A finding with a refuted explanation is in a worse epistemic position than a finding with none, because the refuted one has already absorbed the attention that would have gone to finding a real cause. The useful move is to notice what the two cases share rather than what the failed explanation named, since the shared thing is the only candidate left standing, and here that is not the two-colouring but the forcing — at half filling both nets have an arrangement the ascent cannot improve from anywhere, and why is now open.

Who found it, and when #

Frustration on a triangular lattice is Wannier, 1950, and Toulouse’s formulation is 1977; the two-colouring obstruction is König’s theorem, 1916. The ascent, the census and the composition sweep come from the essays before this one. What is computed here is the triangle ceiling for this net, the two censuses side by side, and the three arrangements at half filling whose energies and unlike-bond counts disagree about their order.

The number worth carrying is not thirty-two. It is one: the number of basins a net with no two-colouring has at the composition a two-colouring was supposed to explain.

Still open: what the forcing is #

The obvious open question is what the two nets share at half filling that the two-colouring was standing in for. Both have a single basin reached by every start at the smaller contrast, and on the frustrated net the arrangement reached is not the one with most unlike bonds — so whatever forces it is a property of the spectrum rather than of the adjacency. The cheapest probe is the gap: the arrangement that wins at half filling on either net is the one that opens the largest gap at the Fermi level, and computing the gap for every arrangement the census finds would say whether the winner is the gapped one and whether the runner-up is close in gap as well as in energy. That is one diagonalisation per basin over numbers already in hand.

The nearer question is the crossover. Two contrasts are two points, and the frustrated net’s half filling goes from one basin to three between them while the bipartite one barely moves. Bisecting on the contrast until the second basin appears would give a number — and the number would be comparable with something already measured, since the count of unlike bonds was found failing above a contrast threshold on the square net. Whether the degeneracy appears at the same contrast as the count’s failure, or somewhere else, would say whether the two are one phenomenon.