The sign rule holds between two poles

A susceptibility curve fitted with four parameters — an exchange coupling, a g factor, a monomeric impurity fraction and a temperature-independent term — fixes three combinations of them and leaves one nearly free, and each combination is a product of powers. The power the temperature-independent term carries has been followed through three calculations. On rings it came out negative for odd counts and positive for even ones, and the sign was put down to frustration. Open chains of odd length broke that, and the parity of the count replaced it. A tetrahedron, frustrated and even, came out positive, and the account was restated: the sign is a property of the spin count, “and that survives everything”, with its mechanism named as the total spin of the ground state.

Two things were never done. The parity of the count and the spin of the ground state were never separated, because every even cluster tried had a singlet ground state and every odd one a doublet — on those clusters the two rules are the same rule. And no sign was ever read at a second coupling. The last of the three calculations said so in its own caveats: every cluster was evaluated at one point in parameter space, and “whether the rule survives a different coupling or a different monomer fraction is a separate sweep”.

This essay runs both.

An even cluster whose ground state carries a spin of one #

A cluster of spin-½ centres coupled antiferromagnetically on a bipartite graph — one whose spins split into two sets with every bond running between them — has a ground state whose total spin is fixed by a counting theorem. Lieb and Mattis proved it: the ground spin is half the difference between the sizes of the two sets. The theorem does not mention the parity of the total. A balanced bipartite cluster of six spins has a singlet ground state; an unbalanced one of six need not.

The cheapest unbalanced cluster is a star: one centre coupled to every leaf and nothing else. A star of four has sets of one and three, so its ground spin is one; a star of five has one and four, so three halves; a star of six has one and five, so two. The complete bipartite graph on two and four spins — every member of one pair coupled to every member of the other four — is six spins with a ground spin of one. On two and three it is five spins with a ground spin of one half, and on three and three it is six spins with a singlet, which is the balanced control. The same sublattice count decides how many levels a vacancy leaves at the middle of a band, where Lieb’s related theorem fixes the spin of a half-filled bipartite structure in the same way.

The exact solver reproduces the theorem on every bipartite cluster before any exponent is read: nine of the sixteen clusters are bipartite, and on all nine the computed ground spin is half the imbalance. A fan of six spins — a centre joined to a path of five — is added as a frustrated singlet, so that the new clusters do not all share a graph family.

At the usual coupling the spin rule wins, by one cluster #

Every reading so far was taken at one point: an antiferromagnetic coupling of 50 cm⁻¹, a g factor of 2, two per cent monomer, a temperature-independent term of 2×10⁻⁴ and a curve from 20 to 300 K at forty points and one per cent precision. At that point the nine clusters already published come back with their exponents to nine figures, which is the check that the new clusters are being read by the same instrument.

The star of four gives −6.841. The complete bipartite graph on two and four gives −3.812. Both are even, both unfrustrated, both carry a ground spin of one — and both are negative, where the parity rule predicts positive. The spin rule predicts negative for any ground state that is not a singlet, and it is right on both.

The other new clusters sit where both rules agree. The graph on two and three, odd with a ground spin of one half, gives −0.891, beside the ring of five’s −0.857. The balanced graph on three and three gives 0.213, beside the ring of six’s 0.225. The star of five, odd and with a ground spin of three halves, gives −18.838. The fan gives 0.291.

The star of six does not fit. It is even with a ground spin of two, and it gives +28.893. The parity rule is right about it and the spin rule is wrong.

So the tally at this point is that the parity rule misses two clusters and the spin rule one, and the obvious report is that the spin rule wins, fifteen to fourteen. What stops that report is the size of the numbers. Every singlet gives an exponent near 0.2 and every doublet one near −1, while the stars give 6.8, 18.8 and 28.9. An exponent is the ratio of two components of a direction, and a ratio that large means the monomer fraction’s component is close to zero. A sign read off a denominator that small is worth reading again somewhere else before anything rests on it.

Every cluster changes sign somewhere #

The somewhere else is the coupling. Each cluster was fitted at seventy-two couplings spaced evenly on a logarithmic scale from 2 to 1000 cm⁻¹, with every other setting of the working point held. The direction was followed from one coupling to the next, and each change in the sign of its monomer component was narrowed by halving the interval twenty-four times.

No cluster keeps its sign across the range. The nine singlets are positive up to their first change, which falls between 303 and 479 cm⁻¹, and negative for a stretch above it. Every cluster with a ground spin changes sign twice below 150 cm⁻¹ — negative between the two, positive outside — and at least once more above.

The star of six’s positive exponent is now explained without any rule at all. Its upper change of sign is at 47.03 cm⁻¹. The coupling at which it was read is 50 cm⁻¹. It was three wavenumbers past the point where its sign turns over, which is not a fact about stars, or about spin two, or about parity. Moved to 40 cm⁻¹, it is negative like every other cluster with a ground spin; moved to 65 cm⁻¹, the star of five is positive too.

The magnitudes near those couplings say what kind of change it is. The exponent does not shrink towards zero and cross it. It grows without bound on one side and returns from the opposite infinity on the other.

The sign changes through a pole, where two nuisance parameters separate #

A fit with four parameters has four singular directions, and the two least determined lie almost entirely in the plane of the monomer fraction and the temperature-independent term, with the coupling and the g factor nearly absent from both. Two directions in one plane that are orthogonal have exponents that are negative reciprocals, and across all sixteen clusters the product of the third and fourth exponents is within three per cent of −1.

That settles a naming error. The exponent these calculations followed is the third direction’s, and the third direction is fixed to between 1.0 and 16.1 per cent. The direction the data leaves nearly free is the fourth, fixed to between 11.5 and 121 per cent — the forty-per-cent product the first of these calculations found was the fourth. Because the two exponents are negative reciprocals, every sign reported for the third is the opposite sign for the free direction and the sign rules translate exactly; what they were rules about was misnamed.

It also says what a change of sign is. When the third direction’s monomer component passes through zero, the third direction is the temperature-independent term alone, the fourth is the monomer fraction alone, and the two parameters are uncorrelated.

The star of four’s upper pole is at 86.02 cm⁻¹. There the fourth direction’s temperature-independent component is 0.0009, and the correlation between the monomer fraction and the temperature-independent term is 0.006, against 0.558 at 70 cm⁻¹ and −0.566 at 105 cm⁻¹. Below 150 cm⁻¹ the temperature-independent component of the third direction never changes sign on any cluster, so every change of sign there is a pole of this kind and none is a zero, and at every one of them the fourth direction is the monomer fraction to within 0.014.

This is a known trap in a new place. A ratio can change sign through its numerator or its denominator, and a bisection reports both in identical words. Here the count was made rather than assumed.

It also answers a question the frustration calculation left. It proposed that somewhere between an odd ring and an open chain the exponent passes through zero, and that a system sitting there “would be one whose four parameters are as separable as this model allows”. That point exists. It is reached through a pole of the exponent rather than a zero, and every cluster with a ground spin has two of them below 150 cm⁻¹.

The ground spin places the poles #

With every pole located, the question of what the sign follows becomes the question of what places the poles.

The three clusters with a ground spin of one half — a ring of five, a trigonal bipyramid of five and the graph on two and three — put their upper poles at 125.7, 122.1 and 114.5 cm⁻¹. Two of them are frustrated and one is not, and the gaps to their first excited levels are 2.24, 2.00 and 3.00 in units of the coupling. The two clusters with a ground spin of one put theirs at 86.0 and 82.3 cm⁻¹, and they have different counts, different graphs and gaps that differ by a factor of two. The star of five, at three halves, puts its upper pole at 57.4 cm⁻¹ and the star of six, at two, at 47.0 cm⁻¹. The lower poles fall the same way: 17.2 to 20.1 cm⁻¹ for one half, 10.4 to 11.3 cm⁻¹ for one, 7.9 cm⁻¹ and 6.0 cm⁻¹.

So at a fixed window a cluster’s poles are placed by its ground spin to within ten per cent, and not by its count, its gap, its topology or its frustration. The mechanism the spin account proposed — a ground state carrying a moment whose Curie tail competes with the monomeric impurity’s — is the right mechanism, but it does not decide a sign. It decides the couplings at which the sign turns over.

The singlets’ poles are a different object and should not be read as the same effect at spin zero. At a singlet’s first pole the two nuisance parameters are still strongly correlated, between −0.61 and −0.69, and the fourth direction carries a temperature-independent component of 0.10 to 0.16. Those are changes of sign without separation.

Two of the four ground spins are represented by a single cluster each. The agreement within a ground spin is measured at one half and at one, and at three halves and two it is a prediction.

And the window places them too #

A pole is a coupling, and a coupling in a susceptibility fit is only large or small against the temperatures the curve covers. So the four clusters with a ground spin of one or more were refitted at three other windows.

Halving the whole window, from 20–300 K to 10–150 K, halves every upper pole: the ratios are 0.505, 0.505, 0.504 and 0.503. Lowering only the bottom, to 10–300 K, moves the upper pole by between 2 and 22 per cent, and moves the lower pole a long way: the star of four’s lower pole is at 6.00, 11.27 and 20.14 cm⁻¹ with the bottom of the window at 10, 20 and 40 K, a factor of 3.36 for a factor of four. The other three clusters give 3.41, 3.32 and 3.28.

So the upper pole belongs to the top of the window and the lower pole to its bottom. A sign is not a property of a cluster, nor of a cluster at a coupling, but of a coupling measured in units of the temperatures the curve was taken over. Two published analyses of one compound that fitted different windows can report opposite correlations between the same two parameters, and both would be right about their own curves.

Where each rule is right #

Every pole is now known, so for each rule and each coupling it is a matter of counting whether the rule predicts all sixteen signs.

The spin rule is right on all sixteen on one band, 20.1 to 47.0 cm⁻¹. Its lower edge is the ring of five’s lower pole and its upper edge is the star of six’s upper pole. Without the star of six the band runs to the star of five’s, at 57.4 cm⁻¹, and contains the coupling every earlier reading used.

The parity rule is right on all sixteen at no coupling between 2 and 1000 cm⁻¹. The stars and the graph on two and four need a coupling below about 10 cm⁻¹ or above about 86 cm⁻¹ to be positive, and the doublets need one between about 20 and 114 cm⁻¹ to be negative, and those two requirements never overlap.

So the history of this exponent reads differently from here. Frustration was never the explanation, and neither was the parity of the count. The spin rule is correct in a precise and limited sense: at a coupling between the lower pole of the clusters with a spin of one half and the upper pole of the highest-spin cluster present, every cluster’s sign is the one its ground spin predicts. The earlier calculations happened to sit inside that band for every cluster they tried, which is why each of them found a clean rule and why each rule looked like a property of the cluster.

The connectivity trend was the window’s #

The tetrahedron calculation ended on a measured trend it could not explain: at a fixed count, more bonds gave a smaller exponent, “monotonically and without exception”. Its proposal for explaining it was to plot the exponent against the gap to the first excited level.

At 20–300 K the four-spin exponents run 0.287, 0.221, 0.214 and 0.208 from three bonds to six, as reported. At 40–300 K the same four clusters give 2.31, 0.70, 3.07 and 1.83. The order is gone, and the square with one diagonal, in the middle of the bond count, has the largest exponent of the four.

The gap could not have explained it either: three of the four clusters have exactly the same gap, 2.00 in units of the coupling, and they spread from 0.70 to 3.07 at the warmer window. The trend was a property of one window. With the sign and the magnitude both depending on where the window sits, the only statement about a cluster that survives a change of window is where its poles are, measured in the window’s own units.

What survives of the standing claims #

Frustration does not decide the sign, and that stands. Nothing here restores it; the fan of six is frustrated and behaves as every singlet does.

The parity of the count does not decide it either. Two even, unfrustrated clusters with a ground spin of one are negative at the usual coupling, and no coupling in two and a half decades makes the parity rule right on every cluster.

The ground state’s spin is the right variable, attached to the wrong quantity. It places the poles, to within ten per cent at a fixed window across counts, topologies, gaps and frustration. The sign follows from the poles and the coupling together, and the spin rule is right on every cluster only between them.

The practical consequence narrows. The tetrahedron calculation widened its conclusion to any antiferromagnetically coupled cluster of an odd number of spin-½ centres. What survives is a statement with the window in it. The correlation between the fitted monomer fraction and the fitted temperature-independent term always has the opposite sign to the exponent — at the usual coupling it is negative on every cluster whose exponent is positive and positive on every one whose exponent is negative — so inside a cluster’s own band a singlet gives an anticorrelation and a cluster with a ground spin a correlation; at a pole the two are uncorrelated; beyond it the correlation reverses. A fit invents a moment when its window is wrong for its model, and a fit’s correlation structure turns out to be conditional on the window in the same way.

What was computed, and how #

The model is the one every one of these fits has used: the exact susceptibility of the cluster, scaled by the square of the g factor and by one minus the monomer fraction, plus a free-spin Curie term for the monomer fraction and a term proportional to temperature. The spectrum comes from the exchange Hamiltonian built sector by sector over an explicit bond list and diagonalised exactly. The design is the logarithmic Jacobian of forty χT values with respect to the four parameters, and its singular values say how much the curve is worth.

Each pole is located on a grid of seventy-two couplings, with the third direction’s orientation carried from one coupling to the next so that a flip of the whole vector is not mistaken for a change of sign, and then narrowed by twenty-four halvings. A change of sign of the temperature-independent component is counted separately from one of the monomer component, so that the statement “every sign change below 150 cm⁻¹ is a pole” is a count.

Ten results are checked. Lieb and Mattis’s ground spins are reproduced on every bipartite cluster. The two even clusters with a ground spin of one are negative at the usual coupling, and the spin rule misses exactly one cluster there while the parity rule misses two. The nine published exponents are reproduced to nine figures. The third and fourth exponents are negative reciprocals on every cluster, and the fitted correlation between the two nuisance parameters has the opposite sign to the exponent on every cluster. Below 150 cm⁻¹ every change of sign is a pole, and at each the fourth direction is the monomer fraction alone. The singlets’ first poles are not points of separation. Clusters that share a ground spin share an upper pole to within twelve per cent, and the upper pole falls as the ground spin rises. Halving the window halves the upper pole and the lower pole follows the window’s bottom. The spin rule holds on a band and the parity rule on none.

The refusal is an open chain of four. It has no pole below 400 cm⁻¹, so the sweep does not manufacture changes of sign out of its own grid.

Where the model stops #

Sixteen clusters, and two ground spins with one cluster each. The claim that clusters sharing a ground spin share their poles is measured at spins of one half and one, and extended to three halves and two on single examples.

Spin-½ centres with equal couplings. A star with unequal arms, a cluster of spin-1 centres or a real compound with two exchange paths is outside what was computed, and each could move a pole.

One monomer fraction and one g factor. The poles were located with the monomer fraction at two per cent and g at 2; the windows were varied and those two were not. How the poles move with the impurity level is a separate sweep, and it matters because the impurity level is the thing a real sample does not choose.

And the model is a caricature, as a moment between two integers and every fit since have had to say. What transfers is the shape: a sign rule for a fitted correlation is a statement about where a coupling sits in a window.

Who found it, and when #

The ground-spin theorem for bipartite antiferromagnets is Lieb and Mattis’s, from 1962, and its extension to half-filled bipartite Hubbard structures is Lieb’s, from 1989. Poles of a ratio passing through infinity rather than zero are elementary. The four-parameter design, its singular directions, and every pole and band above are new arithmetic.

Still open: why the upper pole follows the ground spin #

The regularity in the upper poles is measured and unexplained. Multiplied by the ground spin plus one half, every upper pole below 150 cm⁻¹ lands between 114.5 and 129.0 cm⁻¹ — a spread of thirteen per cent across four ground spins, where the poles themselves spread by a factor of 2.7. At the upper pole a cluster’s excited multiplets have mostly left the curve and what remains looks increasingly like the Curie law of its ground multiplet, which is the same shape as the monomer term, so it is plausible that the pole marks a fixed ratio of the ground multiplet’s Curie constant to the temperatures in the window. Fitting a pure Curie paramagnet of each spin in place of the cluster would test that directly, and needs no diagonalisation at all.

The nearer question is the impurity level. Every pole here was read at two per cent monomer. If the poles move with the monomer fraction the band moves with it, and a compound whose purity is not known would not know which side of its own pole it was on.