The sign a frustrated ring changes

A standard analysis took a magnetic susceptibility curve of one per cent precision, gave it four parameters — a coupling, a g factor, a monomer fraction and a temperature-independent term — and asked which combinations it fixes. Three directions are determined and one is not, and the one that is not is a product: $\rho \cdot \chi_{\text{TIP}}^{0.41}$ for a chain of eight spins.

The natural next system is the frustrated ring, and the question about it is worth sharpening. An odd ring is attractive because it has a low-temperature feature where an open chain has none, and the obvious thing to ask is whether a feature buys back the fourth parameter. The sharper question is whether it changes which combination is free — because a fit that is better conditioned along the same direction is a fit that reports the same wrong thing more precisely.

The sign is the parity #

The exponent is +0.412 for the open chain of eight. For rings of six, eight and ten it is +0.225, +0.278 and +0.285. For rings of five, seven and nine it is −0.857, −0.463 and −0.287.

Every odd ring is negative; nothing else is. The closest pair on either side of zero are +0.225 and −0.287, so there is no case in between and no sense in which one shades into the other.

That is a different combination, not a shifted one. On an even ring the curve leaves $\rho\,\chi_{\text{TIP}}^{+0.28}$ free — raise the monomer fraction and the fit compensates by raising the temperature-independent term. On an odd ring it leaves $\rho\,\chi_{\text{TIP}}^{-0.46}$ free — raise the monomer fraction and the fit compensates by lowering it. A paper quoting a correlation between two fitted parameters would report the opposite correlation on the two systems.

Two things in that paragraph need correcting, and the conclusion of its last sentence survives both. The combination whose exponent is quoted is the third singular direction, not the free one: the fourth is the least determined, it lies in the same plane, and its exponent is the negative reciprocal of the third’s. And because the correlation between the two parameters is dominated by that fourth direction, the compensation runs the other way from the one described. On the even rings the fitted monomer fraction and temperature-independent term are anticorrelated, at −0.88 to −0.89, and on the open chain at −0.90; on the odd rings they are correlated, at +0.77 to +0.92. The two systems do report opposite correlations. Which sign goes with which system turns out to depend on where the coupling sits in the fitted window, and a sweep across the coupling locates the poles where it reverses.

There is a way to see why a sign is the natural thing to change, and it is worth having because it makes the result less surprising without making it less useful.

The monomer term contributes a constant to $\chi T$ and the temperature-independent term contributes something linear in $T$. Neither is degenerate with the other on its own — a constant and a line are separable given enough range — so what makes them trade at all is the chain’s own contribution, which is neither constant nor linear and which the fit is using them to correct. The exponent is therefore a statement about the shape of the chain’s curve, not about the two nuisance parameters, and an odd ring’s curve has the opposite curvature to an even one’s over the window measured. The two nuisance terms are then pressed into service in opposite directions.

That reading predicts something the next figure checks: the effect should be largest where the odd ring’s curve differs most from an even one’s, which is at small rings.

It is the frustration, not the ring #

The rings of six, eight and ten are rings. They have the same boundary condition as the odd ones, the same number of couplings per spin, and the same absence of ends. They give positive exponents like the chain.

What the odd rings have that the even ones do not is frustration: with antiferromagnetic coupling all round, an odd ring cannot satisfy every bond at once, and its ground state and low-lying spectrum are shaped by that. So the deciding property is not the topology of the ring but whether the couplings can all be satisfied on it.

The size confirms it. Among the frustrated rings the exponent falls steadily — 0.857 at five spins, 0.463 at seven, 0.287 at nine — while the even rings sit near a quarter and do not move the same way.

That decay is what a frustration explanation predicts and a topological one does not. An odd ring’s frustration is a finite-size property: one unsatisfied bond among $n$, so its share of the physics falls as the ring grows, and by a ring of thirty the odd one would be indistinguishable from the even. The exponent follows it down.

It also means the exponent is not a number to quote. At five spins it is three times what it is at nine, and a value reported without the ring size is not a value.

What it buys as well as what it moves #

The weaker half of the question — does the feature buy information — has the expected answer. The open chain of eight leaves its worst parameter uncertain by 37.3 per cent; the frustrated rings leave theirs at 23.9, 22.1 and 21.9 per cent. Every one of them is better determined than the chain.

But the ranking is not by frustration. A ring of six is the best-determined system here, at 17.3 per cent, and it is not frustrated at all. So conditioning and the free direction are two separate things that a single number cannot report — which is the difference between a condition number and a direction, arriving on a system chosen for a different reason. How many parameters a curve is worth sets that distinction out.

The singular values say where the differences live. The top two are the same everywhere — the curve fixes the g factor and the coupling well on every system, which is why those two are the ones anybody quotes. The third is where the systems differ, by a factor of three between the best and the worst. The fourth is worth almost nothing anywhere: between 0.025 and 0.056, against a leading value of about twelve.

So no system here carries four parameters. What changes between them is only which three, and what the fourth is a combination of.

The combination loses a term as well as a sign #

There is a second difference and it is not in the exponent at all. On the unfrustrated systems the third direction is $\rho \cdot \chi_{\text{TIP}}^{+p} \cdot J^{0.11\text{–}0.15}$ — the coupling is in it. On the frustrated rings it is a pure $\rho$–$\chi_{\text{TIP}}$ product with no $J$ term above the tenth that gets printed.

So the frustration does not only flip the exponent; it removes the coupling from the combination. On an odd ring the third direction is entirely about the two nuisance parameters and says nothing about the physics, which is the tidier situation to be in: the quantity a paper cares about has been separated from the two it does not.

The table is the summary and the point of it is what fails to sort it. The condition number does not — the best and worst are both unfrustrated. The ring size does not. Being a ring rather than a chain does not. The exponent’s sign sorts it exactly, and it is the only column that does.

What this changes about reading a paper #

The result is about analysis rather than about magnetism, and it is worth saying what a reader should do differently.

A molecular magnetism paper reporting a four-parameter fit almost always quotes $J$ and $g$ with error bars and mentions $\rho$ and $\chi_{\text{TIP}}$ in passing as corrections. That practice is right — the first two are the well-determined directions on every system here, with singular values around twelve and two against 0.2 and 0.04 for the other two — and it is right for a reason the fit itself does not report.

What the practice hides is that the two corrections are not independently determined on any of these systems, and that the way they fail to be depends on the compound. On an even ring, an author who has over-estimated the monomer fraction has also over-estimated the temperature-independent term, and the two errors partly cancel in the fitted $J$. On an odd ring the same over-estimate goes with an under-estimate of the other, and the errors add. So the same fitting procedure applied to two compounds propagates its nuisance-parameter error into the physics with opposite signs.

That is checkable in any published fit without re-measuring anything: the correlation matrix a least-squares routine already computes carries the sign, and it is almost never printed. An orbital carries no angular momentum is another case where the quantity that decides an interpretation is one the standard output does not show.

What was computed, and how #

The model is the standard one: $\chi T$ from an exact diagonalisation of a Heisenberg chain or ring, scaled by $g^2$, mixed with a monomer fraction $\rho$ of free spins, plus a temperature-independent term linear in $T$. Forty points from twenty to three hundred kelvin, at one per cent precision each.

The parameter point is a real compound’s: a coupling of fifty kelvin, $g = 2$, two per cent monomer, and a temperature-independent term of $2\times10^{-4}$. Nothing about the conclusion depends on that choice except through the derivatives taken at it, which is stated because a null direction is a local object.

The identification is a singular value decomposition of the sensitivity matrix, the same decomposition that exposes an underdetermined force field. A singular value is how much the curve constrains one independent direction; the direction itself, written with its largest component set to one, is the product printed above.

The exponent is that direction’s $\chi_{\text{TIP}}$ component divided by its $\rho$ component, so it is a ratio of two entries of a unit vector and carries no scale of its own.

Four things must hold. Every frustrated ring’s exponent must be negative and every unfrustrated case’s positive — which is the finding, and would fail if the parity were not what mattered. The magnitude must fall monotonically with size among the frustrated rings. And every frustrated ring must be better determined than the open chain, which is the weaker half of the question stated so that it can fail separately.

The control is the even ring, and it is what makes the statement about parity a claim rather than a description. An even ring shares everything with an odd one except the frustration, so an explanation appealing to the boundary condition would give it a negative exponent too. It does not.

What a value quoted without its ring means #

The decay of the exponent with size deserves one more paragraph, because it is the part that generalises past this model.

At five spins the exponent is 0.857 and at nine it is 0.287 — a factor of three across four sites. Neither is a limit: the sequence is still falling at nine and there is no reason to expect it to stop before zero. So a paper reporting “the monomer fraction and the temperature-independent term trade as ρ·χ_TIP⁻⁰⋅⁵” would be reporting a property of a ring of about seven and nothing more general.

That is a familiar shape and it is worth naming as such. A fitted exponent that depends on the size of the system it was fitted on is not a law with a coefficient; it is a measurement of one system, and quoting it as the first is how a number outlives the case it came from. Band theory shows the same thing in a switch that turned out to be a crossover every lattice sits past, and ring aromaticity in an end correction that never settles as the molecule grows.

What survives the size dependence is the sign, which is the same on every odd ring measured and the opposite on every other system. A sign is the part of this result worth carrying, and the magnitude is the part that needs its ring quoted beside it.

Where the model stops #

Every spin here is a half, the coupling is isotropic and nearest-neighbour, and the ring is a ring of identical sites. Real frustrated magnets are frustrated by geometry — triangles and tetrahedra — rather than by an odd count on a line bent round, and a triangular lattice’s frustration does not wash out with size the way this one does. So the decay above is a statement about odd rings and not about frustration in general.

The diagonalisation is exact and the rings are small, which is the reason they are rings of five to ten. What happens at twenty is not computed, and the extrapolation that the exponent goes to zero is an extrapolation from three points.

The curve is noise-free and the analysis is a local sensitivity. A real fit has correlated errors, a background, and a temperature scale of its own, and all three would change the numbers without changing which direction is free — that last part being the only thing claimed here.

And the model has no anisotropy and no interchain coupling, either of which produces a low-temperature feature of its own that would be absorbed into $\rho$ by a fit that did not know about it. The monomer fraction in a real analysis is a rubbish bin, and its correlations with everything else are correspondingly less trustworthy than these. The moment a fit invents is an account of what that costs.

The generalisation #

A better-conditioned fit is not the same as a differently-conditioned one, and only the second changes what a measurement means. Adding a feature to a curve can shrink every error bar and leave the same combination free, in which case the extra precision is spent on a quantity that was already known and the unknown one is unchanged. Here the feature does both, and the two effects are visible separately: the condition number improves for reasons of parity-independent shape, and the direction rotates for reasons of frustration.

And a fitted correlation is a property of the system, not of the parameters. Two compounds analysed with the same four parameters and the same software return opposite correlations between the monomer fraction and the temperature-independent term, because one is a frustrated ring and the other is not. Anyone carrying a rule of thumb about which way those two trade — and such rules are carried — would be right on one and wrong on the other. The same warning applies to an underdetermined force field, where the undetermined directions are properties of the molecule and not of the coordinate set.

Who found it, and when #

The frustration of an odd antiferromagnetic ring is elementary and old, and the spin-half ring’s exact spectrum is Bethe’s for the infinite case and a small matrix for these. That a susceptibility fit with a monomer term and a temperature-independent term is underdetermined is folklore in molecular magnetism — every practitioner knows the monomer fraction is the parameter that absorbs whatever is left over — and the standard advice is to fix one of the four and fit three — which the model is what is fitted treats as a standing hazard rather than as a technique.

What is done here is to ask which three, and to find that the answer depends on the ring’s parity through a sign. The advice to fix a parameter is good advice and it is incomplete: which parameter it is safe to fix is a property of the system being fitted, and on these seven it changes.

Still open: frustration that does not wash out #

The obvious open question is a frustration that does not wash out. A triangular arrangement of three spins is frustrated for the same reason and stays frustrated at any size once it is tiled, so a small triangular cluster and a ring of the same count would separate frustration from odd. Exact diagonalisation handles any coupling graph, so it is a change of adjacency rather than of method.

The nearer question is where the sign changes. The exponent is a continuous function of the couplings, and a ring with one weakened bond interpolates between an odd ring and an open chain — so somewhere along that path the exponent passes through zero, and at that point the monomer fraction and the temperature-independent term are uncorrelated. A system sitting there would be one whose four parameters are as separable as this model allows, and finding it is a one-parameter scan of a matrix already being built.