The frustrated cluster with an even count

The parity account of the sign began from a confound, named precisely. The sign had first been attributed to frustration, and every case in that sweep had frustration and odd parity moving together, so any property of an odd count was an equally good explanation. Open chains of five, seven and nine — unfrustrated, odd — came out negative, and parity replaced frustration as the account.

Then it stopped one step short of its own lesson. The design it left behind has the same shape as the one it criticised. Every case in it is a chain or a ring, and among chains and rings frustration and odd parity are still aligned: an odd ring is frustrated, an even ring is not, and a chain is never frustrated at any length. The eleven systems contain no case where the two accounts disagree, and its closing section said what would be needed — a frustrated system with an even count — and named the tetrahedron.

A tetrahedron is not a chain and not a ring, and the exact solver could only build chains and rings.

Four systems the two accounts disagree about #

Frustration is a property of the coupling graph rather than of a count, and it is decidable. An antiferromagnet can satisfy every one of its bonds exactly when the spins can be split into two sets with every bond running between them — which is a two-colouring, so an antiferromagnet is unfrustrated precisely when its graph is bipartite. The two-colouring is searched for by traversal rather than assumed, and the search either finds one or returns the odd cycle that stops it.

That test reproduces what is already known where the two overlap: a ring of five comes back frustrated and a ring of six does not. It also builds the cases a scan of chains and rings cannot.

A tetrahedron of four, which is the complete graph on four vertices: every spin coupled to every other, six bonds, and no two-colouring, since any three of its spins form a triangle.

A square with one diagonal, which is the smallest frustrated even cluster there is — five bonds, and the diagonal is what breaks the colouring.

An octahedron of six, twelve bonds, and a triangular prism of six, nine bonds. Both contain triangles and neither can be coloured.

All four are even. All four are frustrated. The two accounts predict opposite signs for every one of them.

All four are positive #

The exponents are +0.2084 for the tetrahedron, +0.2136 for the square with a diagonal, +0.1924 for the octahedron and +0.2048 for the prism.

The parity rule survives. The frustration account, already replaced on weaker evidence, is now refuted on the arrangement it was never tested against — and the refutation is not marginal. The gap between the lowest positive exponent in the whole scan and the highest negative one is 1.0497, against a total range of about two.

The scan also reproduces the chain-and-ring numbers where the two overlap, which is the check that this extends that measurement rather than a neighbouring one: the ring of five comes back at −0.8573 against −0.857, the ring of six at 0.2254 against 0.225, and the chain of four at 0.2871 against 0.287.

The mechanism it was attributed to is wrong too #

This is where the parity account’s argument, rather than its conclusion, gives way.

It explained the parity rule by a ground state. An antiferromagnetically coupled cluster of an even number of spin-½ centres has a singlet ground state and nothing left over — the pairing superexchange produces; an odd one has, in effect, one unpaired spin, whose Curie contribution rises as the temperature falls and competes with the monomeric impurity term the fit is trying to determine — the same term a four-parameter curve cannot separate from a temperature-independent one. And it named the check: “showing that the exponent tracks the ground multiplicity rather than the parity as such would be the next thing to check”.

The tetrahedron performs that check without being asked to.

Its ground state is doubly degenerate — two states at one energy, both with total spin zero. That is the standard signature of a frustrated even cluster: with every pair coupled equally there is no unique way to pair the four spins into two singlets, and the two ways that exist are degenerate. Every other even cluster in this scan has a unique singlet ground state, and every odd one has a ground state of total spin one half with a degeneracy of four or six.

So the ground multiplicity takes the values 1, 1, 1, 2, 4, 6, 1, 1, 1 across the nine clusters, and the exponent’s sign takes the values +, +, +, +, −, −, +, +, +. The tetrahedron is the case where the two part company, and the exponent follows the parity rather than the degeneracy.

What it does follow is the ground state’s total spin, which is zero for every even cluster here — the tetrahedron included, despite its degeneracy — and one half for every odd one. That is the mechanism argued for alongside the parity rule, stated in the right variable: it is the moment the ground state carries that competes with the impurity term at low temperature, and a degenerate pair of singlets carries none.

That distinction cost nothing to find and could not have been found without a frustrated even cluster, because on chains and rings the degeneracy and the total spin agree in every case.

What the topology does do #

It would overstate the finding to say the exponent depends on nothing but the count, and the scan says so directly.

At four spins the exponents run from 0.2084 to 0.2871 — a spread of 38 per cent, with the open chain at one end and the tetrahedron at the other, in order of how many bonds each has. At six spins they run from 0.1924 to 0.2254, again with the most-connected cluster lowest. At five, a trigonal bipyramid gives −1.1809 against a ring of five’s −0.8573, which is 38 per cent again and in the same direction.

So more bonds means a smaller exponent at a fixed count, on all three counts, without exception. That is a real trend and nothing here explains it; what it does establish is that the trend never approaches the sign change, since the smallest positive exponent in the scan is five times further from zero than the spread within any count.

There is also a caveat here for the parity account’s second claim, which reads the decay of the odd exponents with size — 0.857 at five, 0.463 at seven, 0.287 at nine — as an unpaired spin’s moment becoming a smaller share of a growing system. The bipyramid of five sits at −1.1809, which is larger than any ring’s, so the decay is a property of the ring series rather than of the spin count, and a sweep over topologies at fixed count would move a point in that series by more than one step along it moves.

What survives of the standing claims #

Three calculations have now been spent on one exponent, and it is worth setting out what survives all three, because each corrected the one before it and the tetrahedron corrects two things at once.

The exponent’s sign is a property of the spin count, and that survives everything. Nine clusters on five topologies, two of which are neither chains nor rings, and the sign is positive for every even count and negative for every odd one with no exception and a gap of 1.05 between the two families.

And it did not survive a second coupling. Every reading here was taken at one coupling and one window. Swept across the coupling, every one of these clusters changes sign somewhere between 2 and 1000 K, through poles its ground spin places. A star of four spins — even and unfrustrated, with a ground spin of one because its sublattices are unequal — is negative at the coupling used here, at −6.841. Across sixteen clusters the parity of the count predicts every sign at no coupling in that range, and the ground state’s spin predicts every sign only between 20.1 and 47.0 K.

It is not frustration, and that is now established rather than inferred. Unfrustrated odd systems refuted the frustration account from one side; frustrated even ones refute it from the other, which is the half a design of chains and rings could not supply.

It is not the ground multiplicity either, which is new and is a correction to the parity account rather than to the frustration one. The mechanism proposed with it was right in substance and wrong in the quantity it named.

And the practical consequence widens again. The frustration account read the result as covering frustrated rings; the parity account widened it to any cluster with an odd count; the tetrahedron establishes that the topology is genuinely irrelevant to the sign, so an experimentalist fitting a four-parameter model to any antiferromagnetically coupled cluster of an odd number of spin-½ centres — chain, ring, cage or lattice fragment — inherits the same correlation structure. That is a much larger class than “frustrated rings”, and it is the class most published susceptibility analyses are drawn from.

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. The exponent is the ratio of the temperature-independent term’s component to the monomer fraction’s, in the third singular direction of the design. That is the second least determined of the four, not the free one: the free fourth direction lies in the same plane and carries the negative reciprocal of this exponent, so each sign here is the opposite sign for the free direction.

What is new is the coupling graph. A chain or a ring is fixed by a spin count and whether its ends join; a tetrahedron or a bipyramid needs an explicit list of bonds, and the Hamiltonian is built sector by sector over that list. Everything else — the sector construction, the diagonalisation, the Van Vleck susceptibility, the identifiability analysis — is unchanged. That the chain-and-ring numbers come back to four figures is the check on the bond lists, and it is a real check rather than a formality: a bond list assembled wrongly would produce a plausible spectrum and a plausible exponent, and the only thing that would notice is a case whose answer is already known.

The ground multiplicity is one more read of a spectrum that is already computed, and it is taken as the count of levels within 10⁻⁹ of the lowest.

Five results are checked. The design contains at least two cases the two accounts disagree about — a property of the case list, checked before any exponent is looked at, because a scan without such a case would confirm whatever it was already confirming. The bipartite test reproduces the frustration of an odd ring and its absence in an even one, so it measures what it is meant to. Exactly one of the two accounts survives, a statement that “both hold” and “neither holds” would each contradict. The degeneracy account fails and the total-spin account holds — both halves together, since either alone would be a story about one label. And the topology does move the magnitude at a fixed count, so the scan is not reporting a quantity that depends on the count alone.

Where the model stops #

Nine clusters is nine. They are small, they are highly symmetric, and every bond in them has the same coupling. A real frustrated compound has inequivalent exchange paths, and the sign rule is an observation across this set rather than a theorem — the same standing on which a moment between two integers was reported.

Spin-½ only. The parity rule and the total-spin account agree on spin-½ because an even number of half-integer spins can always couple to zero and an odd number cannot. On spin-1 they need not: a Haldane chain of even length has a singlet ground state for a quite different reason, and an odd-length one has an end-spin structure the sector construction here would have to be generalised to reach.

The exponent is local. It is read from one singular direction of one design matrix at one point in parameter space, and every cluster here is evaluated at the same point so the comparison is fair. Whether the rule survives a different coupling or a different monomer fraction is a separate sweep, and the caveat about that stands untouched.

And the four-parameter model is a caricature. A published susceptibility analysis carries a background, a diamagnetic correction and often a second exchange path. What transfers is the shape of the finding rather than the number, which is the standing caveat on fitting a model to a curve it is not.

The generalisation #

The transferable point is about how a design gets fixed after a confound is found in it, and the answer is that it usually is not.

The parity account did the hard half. It noticed that a proposed explanation co-varied with another property across every case available, added cases that broke the co-variation, and replaced the explanation. That is the right procedure and it worked.

What it then did was add three open chains — which broke the confound it had found and preserved the confound one level down, because a chain is never frustrated at any length. The new cases were chosen to vary the property under suspicion, and they varied it only in the direction that was already available. A design that has been repaired once is not thereby a good design, and the way to check is to run the same co-variation test on the repaired case list rather than on the original one. That test takes one line and it would have said, immediately, that frustration and odd parity were still perfectly correlated across all eleven cases.

The second half is about what a decisive case is, and it is worth stating because it is stronger than a case that merely varies something. Four of the nine clusters here are decisive in the strict sense: the two accounts make opposite predictions about them, so whatever the measurement returns, one account is refuted. That property can be checked before any computation is run — it is a property of the case list and the two hypotheses — and establishing it first is what stops a scan from being a larger version of the one it was meant to correct. The same requirement appears from the other direction where a control that could not be a mechanism outranked the mechanism and the control’s own uselessness was the finding.

There is a third point, smaller and more practical. The mechanism proposed with the parity rule was refuted by a case already named as its test, and the refutation cost one extra column in a table being computed anyway. A mechanism that has been argued for rather than measured should be measured in the same run as the rule it explains, because the marginal cost is usually one more quantity read from a calculation already done, and the two can come apart in exactly the case that makes the rule interesting.

Who found it, and when #

Frustration in triangular and tetrahedral antiferromagnets is a standard subject, the degeneracy of a tetrahedron’s singlet ground state is elementary, and the bipartite criterion for an unfrustrated antiferromagnet is folklore. The four-parameter model, the singular-direction analysis and every exponent above are new arithmetic.

The tetrahedron was the named test before it was run: what its exponent would have to be for the parity rule to survive was stated in advance, which is worth more than a scan with one more chain in it. A moment that counts electrons is where this line of argument started, and the distance between that and a fitted exponent’s sign is the distance four calculations cover.

Still open: why more bonds give a smaller exponent #

The obvious open question is the connectivity trend, which is now measured and unexplained. More bonds gives a smaller exponent at every count tried, monotonically and without exception, and nothing here says why. It is the kind of relationship that might be a property of the excitation spectrum’s density rather than of the ground state, and every level of every cluster is already computed — so plotting the exponent against the gap to the first excited state, or against the number of states below some energy, needs no new calculation.

The nearer question is the spin. The parity rule and the total-spin account are indistinguishable on spin-½ and would separate on spin-1, where an even chain’s ground state is a singlet for a reason that has nothing to do with pairing. That needs the sector construction generalised from two states per site to three, and it would say whether the sign follows a rule about counting or a rule about moments.