It was the count, not the frustration

A susceptibility curve of stated precision fixes three of a four-parameter model’s parameters and leaves one direction free. The product a curve measures found that the free direction is a product — the monomer fraction times a power of the temperature-independent term — and quoted the exponent. The sign a frustrated ring changes asked whether a low-temperature feature changes which combination is free, and found that it does, by changing a sign: every odd ring gave a negative exponent and every even ring and the open chain a positive one.

It attributed the sign to frustration. An odd ring of antiferromagnetically coupled spins cannot satisfy all its bonds at once; an even ring can, and so can a chain. And it wrote its own refusal into the check: an explanation appealing to the ring rather than to the frustration would give an even ring a negative exponent too, and it does not.

That refusal is a good one and it rules out the wrong alternative. It varies the frustration and holds the topology fixed, and it never varies the thing that turns out to matter — because the open chain it compared against had eight spins.

A chain of five #

An open chain of five spins is not frustrated. A chain is bipartite: colour the sites alternately and every antiferromagnetic bond joins one colour to the other, so the classical arrangement satisfying all of them exists and there is nothing left over. Frustration on a chain is not small, it is absent.

Its exponent is −0.792.

That is not a marginal number. The rings of five, seven and nine gave −0.857, −0.463 and −0.287, and the positive cases ran from 0.225 to 0.446. A chain of five sits squarely among the negatives, closer to the ring of five than the ring of five is to any positive case.

A chain of seven gives −0.440 and a chain of nine −0.272. Three unfrustrated systems, three negative exponents — and they sit alongside the rings of the same counts at −0.857, −0.463 and −0.287, within 0.065, 0.023 and 0.015 of them.

So the sign is not frustration’s. It follows the parity of the spin count, on both topologies, with no exception across the eleven systems computed: every even count positive, every odd count negative.

That is true of these eleven systems at this one coupling, and it is not a property of the count. On every chain and ring the parity and the ground state’s total spin agree, and they come apart on a bipartite cluster with unequal sublattices: a star of four spins is even, has a ground spin of one, and its exponent is −6.841. Swept across the coupling, every sign changes somewhere between 2 and 1000 K, and across sixteen clusters the parity rule predicts every sign at no coupling in that range.

The chains of four, six and eight give 0.287, 0.446 and 0.412; the rings of six and eight give 0.225 and 0.278. Eleven systems, two topologies, six counts, and one column predicts every sign.

What an odd count has that an even one does not #

The replacement explanation is worth stating before the measurements that support it, because it is short.

A collection of spin-½ centres coupled antiferromagnetically has a ground state whose total spin is zero when the count is even and one half when the count is odd. An even system has nothing left over; an odd one has, in effect, one unpaired spin, and that spin’s Curie contribution rises as the temperature falls.

The four parameters being fitted include a monomeric impurity fraction, whose entire signature is a rise in χT at low temperature, and a temperature-independent term, whose signature is a rise at the top. An odd system already has a low-temperature rise of its own. So the impurity’s signature is no longer distinguishable in the same way, the correlation between the two parameters changes character, and the direction the data leaves free changes with it.

That account makes no reference to bonds being satisfied, and it applies to a chain exactly as to a ring. The rest of this essay is the measurement that it, rather than frustration, is what the sign follows.

How much the topology does move it #

The frustration is not doing nothing, and it is worth measuring how much rather than dismissing it.

Closing the ring at an odd count changes the exponent by 0.065, 0.023 and 0.015 — a few per cent of its own size, and the effect shrinks as the ring grows, which is exactly what a finite-size property does. At the even counts, where there is no frustration at all, closing the ring changes it by 0.220 and 0.134, which is more. So the ring-against-chain difference is not even largest where the frustration is.

Against that, the two parities are separated by 0.4975 — the gap between the lowest positive exponent and the highest negative one. The parity is worth more than twice the largest thing the topology does anywhere, and it is worth thirty times what the topology does where the frustration is.

This is the comparison the frustration sweep could not make, and it is worth being clear about why. It had two topologies at odd counts and two at even counts, but never the same count on both topologies with the frustration varying and the parity held — which is exactly the row this table is built out of. The comparison costs one extra diagonalisation per count.

Opening the ring changes no sign #

The cleanest form of the test removes the frustration continuously rather than by comparing two systems.

Scaling the ring-closing bond from one to zero interpolates between a frustrated ring of five and an unfrustrated chain of five through a continuum of systems the exact solver handles. At zero the two are the same system, and they are the same to a part in 10⁹ in every singular value of the design — which is the check that the interpolation is between the two things it names rather than between two calculations.

Along the whole path the exponent moves from −0.857 to −0.792. It crosses nothing. If the sign were the frustration’s, this is the path on which it would change, and it does not change anywhere on it.

The path is also a check on something else. A weakened bond is a system with no independent result to compare against, so the interpolation could have been wrong at both ends and right nowhere; the two endpoint agreements — with the frustrated ring’s own result at one end and with the open chain at the other — are what make the interior believable.

The decay was the count too #

There is one more claim of the frustration account to revisit, and it is the one that made the frustration account persuasive.

The magnitude falls with size — 0.857 at five spins, 0.463 at seven, 0.287 at nine — and that was read as the frustration washing out, since an odd ring’s frustration is a finite-size property that vanishes in a long ring.

The chains do the same thing: 0.792, 0.440, 0.272. They have no frustration to wash out. What is washing out is an unpaired spin among a growing number of paired ones — an odd count of spin-½ has a doublet ground state and a net moment, an even count has a singlet, and the odd one’s moment is one spin’s worth however many spins there are. As a fraction of the whole it falls, and so does its effect on the low-temperature end of the curve where the monomer fraction and the temperature-independent term compete.

That is a better explanation than the one it replaces, and it explains the decay and the sign together rather than one each. It also predicts the ring-against-chain differences being larger at the even counts than at the odd ones, which is the otherwise puzzling row in the table above: at an even count both systems have a singlet ground state and the exponent is set by the excited structure, where closing a ring does more.

None of that is proved here. What is measured is that the sign and the decay both track the parity, on both topologies, and that neither tracks the frustration.

What was computed, and how #

The model is the same four-parameter one: an exchange coupling, a g factor, a monomeric impurity fraction and a temperature-independent term, fitted to a χT curve from 20 to 300 K at forty points with one per cent precision assumed on each. The spectrum is by exact diagonalisation in each total-spin-z sector, once per coupling and reused across the whole curve.

The exponent is read out of the third singular direction of the design matrix — the free one — as the ratio of its temperature-independent component to its monomer component.

The ring-closing bond takes a strength between zero and one, which is new here and scales that one bond’s coupling in both the diagonal and the off-diagonal part of the Hamiltonian. Every earlier caller gets one and is unchanged.

The check requires eight things: that every case in the frustration sweep comes back with the sign reported there; that the chain of eight, which was the control, is positive; that at least three unfrustrated systems are negative; that frustration therefore does not decide the sign; that parity does, on both topologies with no exception; that the parity gap exceeds anything the topology does; and the two refusals — that opening the ring crosses no sign anywhere on the path, and that a ring with a dead closing bond reproduces the open chain exactly.

The frustration sweep’s numbers are untouched #

It is worth setting down what does not change, because an argument that refutes an explanation can be read as refuting the result, and this one does not.

Every exponent in the frustration sweep comes back identically. The rings of five, seven and nine are −0.857, −0.463 and −0.287; the rings of six and eight are 0.225 and 0.278; the chain of eight is 0.412. Its central claim — that a low-temperature feature does not merely add information but changes which combination of parameters the data leaves free, and changes it by a sign — stands exactly as stated. So does its observation that the frustrated rings are better conditioned than the open chain.

What changes is the sentence explaining why, and the class of systems the result applies to. Read as a statement about frustration it covers odd rings; read as a statement about parity it covers every antiferromagnetically coupled cluster with an odd number of spin-½ centres, which is a much larger set and includes the open chains that are far commoner in the literature the model is fitted to.

That is the practical consequence and it points the other way from a refutation. An experimentalist fitting a four-parameter model to a chain of an odd number of centres has the same correlation structure as one fitting a frustrated ring, and would not have known it from the frustration account — which matters because the model is what is fitted, and a correlation structure the fitter does not know about is a parameter the fitter will report with a confidence it does not have.

Where the model stops #

Eleven systems from four to nine spins is a small sweep, and the parity rule is an observation across it rather than a theorem. What would settle it is the ground-state degeneracy, which is what the parity argument above appeals to and which is not computed here — a doublet against a singlet is the mechanism, and showing that the exponent tracks the ground multiplicity rather than the parity as such would be the next thing to check. On spin-½ chains and rings the two coincide.

The exponent is also read from one singular direction of one design matrix at one point in parameter space, which is a local statement — the product a curve measures is a property of where the fit sits as well as of the system. Every case here is evaluated at the same point, so the comparison is fair; whether the parity rule survives a different coupling or a different monomer fraction is a separate sweep.

The four-parameter model is also a caricature of a real fit. A published susceptibility analysis has a background, a diamagnetic correction and often a second exchange path, and the free direction of a six-parameter fit is not the free direction of this one. What transfers is the shape of the finding rather than the number.

And “one per cent on each point” is an assumption about the data that has been carried since the four-parameter analysis introduced it. A different precision changes which parameters are determined and could change how many directions are free; it does not change a sign that comes from the ground multiplicity.

The generalisation #

The lesson is about controls, and it is specific enough to be actionable.

A control varies one thing. The frustration sweep’s control varied the frustration and held the topology fixed, which was the right instinct — and the property that actually mattered was neither, and was correlated with the frustration across every case it ran. Its sweep contained five odd rings, three even rings and one even chain, and in all nine the frustration and the odd parity agreed. There was no case in the design where they disagreed, and so no possibility of finding out.

This is the same shape of error as a correlation mistaken for an account, one level up: there a fitted relation was read as a mechanism, here a perfect classification was. A rule that gets every case right is not evidence for the reason given for it, if some other rule gets every case right too.

The cheap fix is to look at what a proposed explanation co-varies with in the cases already computed, before adding cases. Here it takes one line: the frustrated systems are exactly the odd-count ones, so any property of an odd count is an equally good explanation, and the way to separate them is an odd system that is not frustrated. A chain of five costs nothing to compute, and neither did the wrong conclusion.

The second half is worth stating because it is the happier half: the sweep’s finding is untouched. Every number it reported is right, its sweep is reproduced here exactly, and the free combination really does change which product it is. What was wrong was one sentence of explanation, and the parity rule replaces it with one that explains more.

Who found it, and when #

Frustration in triangular and odd-ring antiferromagnets is a standard idea, decades old. The four-parameter model, the singular-direction analysis and every number above are computed here, and the parity rule is a check on the frustration account.

The finding to carry is that a sweep with a confound in it is a sweep that cannot fail, and that the confound here was visible in the case list. It is also worth recording that the sweep’s number — the 0.4975 gap between the two families — is unchanged by any of this, and that its warning about what a curve is worth is the reason the exponent was being computed at all.

Still open: the ground multiplicity, and a frustrated even cluster #

The obvious open question is the ground multiplicity, which is the mechanism the parity rule points at and which has been argued for here rather than computed. Diagonalising each system and reading off the degeneracy of its lowest level, then plotting the exponent against that rather than against the parity, would say whether the rule is about the parity of the count or about the moment the ground state carries. On spin-½ the two agree; on spin-1 chains, where a Haldane gap makes an even-length chain’s ground state a singlet for a quite different reason, they need not — and the same diagonalisation handles any spin once the sector construction is generalised to it.

The nearer question is the triangle, which was proposed as the test and which is now a different question. It proposed a triangular cluster against a ring of the same count to separate frustration from oddness — but a triangle of three spins is a ring of three, and a larger tiled triangular cluster has a count that is not free to choose. What would work is a frustrated system with an even count: a tetrahedron of four spins is frustrated and even, and if its exponent is positive the parity rule survives its hardest test. That is one adjacency matrix away.