The number the tie got right

A ring of six ends with a number nobody should believe. The intensity contrast between the strongest line in a removal spectrum and the next one down, taken on a ring of six at half filling and extrapolated to infinite on-site repulsion, came out at exactly 1.000000 — not near one, not one to within the precision of the diagonalisation, but one to every digit the arithmetic carries, at five consecutive repulsions.

That is the floor of the quantity. A contrast is the larger of two weights over the smaller, so it cannot go below one, and a ratio that sits exactly on its own floor is almost always a ratio of something to itself. It was. The rank cut that separates fundamental from satellite had landed between two lines of the ion whose weights agreed to six parts in 10¹⁶, so the contrast was one line divided by its degenerate partner. The ring’s rotational symmetry collects the ion’s states into multiplets, and a multiplet is a supply of exactly equal weights for a cut to fall inside.

So the question actually being asked — does the contrast fall below two at half filling, meaning that satellites there stop being distinguishable by intensity alone — was left open. The measurement that answered it had turned out to be a measurement of the ring’s symmetry.

The obvious repair is to take the symmetry away.

A chain has no multiplets #

An open chain of six sites has the same number of orbitals as the ring, the same on-site repulsion, the same three fillings, and one symmetry element instead of twelve: reflection through its middle. That the ring’s twelve are what generate its multiplets is the ordinary reading of a degeneracy as a group-theoretic fact rather than a numerical coincidence, and it is the reading this substitution takes at its word. Its one-electron levels are non-degenerate at every point, and — much more to the point here — its ion’s many-electron levels are too.

That is the whole reason for the substitution and it is worth checking rather than assuming, because it is a claim about a spectrum of several hundred states and not about a symmetry group. Counting exactly degenerate neighbouring pairs in the removal spectrum at each of the eight repulsions, for each of the three fillings, on both geometries, gives the picture below.

The ring’s spectra are full of them: two at the lowest filling, up to twenty at four electrons, up to thirty at half filling. The chain’s have none. Not few — none, at any of the twenty-four combinations of geometry, filling and repulsion that the chain contributes. There is no pair for a cut to land inside, so whatever the chain’s contrast turns out to be, it is a ratio of two genuinely different numbers.

That is the condition the ring could not supply and it is arranged here by construction rather than by luck.

Where the contrast crosses two #

With the tie removed, the original question can be asked in the form it was meant to have. The distinguishability test wants a factor of two: two lines that differ by less than that are not going to be told apart by their heights in a real spectrum with a real background. So the question is at which filling the extrapolated contrast crosses two.

Two electrons on six sites give 2.905130, comfortably above. Four give 1.641697, below. Half filling gives 0.999999. So the crossing happens between two electrons and four, and half filling is not where the contrast stops satisfying the test — it is where the contrast reaches its floor, which is a different event at a different filling.

That distinction was invisible on the ring, where four electrons read exactly one at four of the eight repulsions and 1.28 at the others, flickering between a pinned value and a moving one as the cut wandered in and out of a twenty-fold degenerate block. On the chain both readings are clean and the ordering between them is stable across the whole range.

The three chain fillings also approach their limits from different sides, which is a small thing that turns out to be a useful check. Two electrons come down from 5.563 at the smallest repulsion to 2.921 at the largest, monotonically, with the fitted coefficient positive at 15.7. Half filling comes down too, at 9.64. But four electrons come up — 1.192, then a dip to 1.139, then a steady rise through 1.357, 1.490, 1.563, 1.602, 1.622 to 1.632 — and its coefficient is negative, −10.3. A limit approached from below cannot be an artefact of the same mechanism as one approached from above, and the three fillings between them use both signs. That is not a proof of anything, but it removes one whole family of explanations for why the numbers might all be wrong together.

The number the ring gave was right #

And now the finding this essay exists for. The chain’s half-filled contrast converges to 0.999999 — the same value, to six digits, that the ring produced by dividing a line by its own partner.

The ring’s argument was invalid. Its answer was correct.

This is worth stating carefully because it is easy to draw the wrong lesson from a refuted argument. The ring result did not show that the contrast at half filling is not one; it showed that the ring’s evidence could not establish that it is. Those are different, and the difference is the whole of what a refusal means. A guard that refuses an extrapolation is making a statement about the evidence in front of it. It has no access to the answer and is not entitled to an opinion about it.

The chain supplies the evidence the ring could not. Its contrast approaches one from above and never reaches it, so the value is a limit rather than a coincidence of ranking.

The excess above one is 9.4 × 10⁻³ at the largest repulsion computed, well above anything the diagonalisation’s own precision could account for, and the product of the excess with the repulsion settles at 9.64 over the top three points — 9.773, 9.683, 9.638, 9.606, 9.606 across the last five. So the approach is a clean reciprocal in the repulsion, and the limit is one with a correction that is measurable rather than assumed. The ring’s excess, on the same axes, is exactly zero from a repulsion of sixty-four upward and cannot be plotted at all.

The approach is not monotone, and that matters #

There is a second thing in that picture and it refutes a habit rather than a claim.

The chain’s excess does not fall steadily. It is 0.2926 at U = 8, 0.2936 at U = 16, then drops abruptly to 0.0316 at U = 32 — and then rises again, to 0.0534, 0.0617, before turning over and falling as 1/U from U = 256 onward. The reading at U = 32 is closer to the limit than any reading taken between U = 64 and U = 256.

So a single measurement at a large repulsion is not an estimate of the limit, and a pair of measurements is not a bound on it. Anyone who had taken the contrast at U = 32, found 1.0316, and concluded that the limit is one to within three per cent would have been right by accident; anyone who had taken it at U = 128 and concluded that the limit is at least 1.06 would have been wrong. The turnover sits between them and nothing at either point announces it.

This is exactly why the extrapolation is done against a fitted residual rather than read off the end of the table. The fit uses only the repulsions from 256 upward, where the excess and the repulsion have become reciprocal, and it reports the spread of excess × U across that window as its own evidence.

What the guard is actually testing #

The residual test asks one question: over the tail of the range, is excess × U a single number? If it is, the departure from the limit is genuinely a first-order correction and the extrapolation is a subtraction rather than a guess. If it is not, the extrapolation is fitting a shape the data does not have.

All three chain fillings pass, with spreads of 0.9, 1.6 and 1.7 per cent. Two of the ring’s three fail, and the interesting part is that both failures are pinned cases and they fail for opposite reasons. At half filling the ring’s residuals are of order 10⁻¹⁰ — machine noise around a quantity that has stopped moving entirely, because the contrast is exactly one at five consecutive repulsions and the excess is therefore exactly zero. At four electrons the residuals are of order 10³ and swing through both signs, because the contrast is jumping on and off its pin as the ranking rearranges.

The guard reports both in the same words. It has to: the quantity it computes is a relative spread, and a spread is undefined about zero and enormous about noise. The ring calculation already found that and called it a misapplication; what the chain adds is a case where the guard permits — three of them — so the refusal can be read as informative rather than as the only thing the instrument ever says.

That is the point at which a guard becomes evidence. A test that refuses everything proves nothing, and until the chain the residual test had refused every case it had been given.

What was computed, and how #

Every number here comes from exact diagonalisation of the extended Hubbard Hamiltonian on six sites, once as a ring and once as an open chain, at eight on-site repulsions spanning a factor of 128, with the neighbour repulsion held at its usual value. For each geometry, filling and repulsion the ground state of the neutral system is found, the ion’s full spectrum is computed, and each ion state is given a weight from its overlap with the neutral ground state with one electron removed. The lines are then ranked by weight, the rank cut is taken at the position fixed earlier, and the contrast is the weight immediately above the cut over the weight immediately below it.

Degenerate pairs are counted as neighbouring lines whose weights agree to within the tolerance the ranking itself uses. Zero on the chain means zero at that tolerance; the smallest gap between neighbouring chain weights anywhere in the sweep is many orders above it.

That tolerance is the one place where a count of zero could be manufactured, so it is worth saying what would happen if it were wrong in either direction. A tolerance set too tight would report zero pairs on the ring as well, and it does not — it reports thirty at half filling, which is the number the twelve-element symmetry group predicts for that block. A tolerance set too loose would report pairs on the chain, and the way to see that it does not is that the ring’s counts change with the repulsion, from thirty down to ten, while the chain’s stay at zero throughout. An instrument that could not resolve the chain’s spacings would also fail to resolve the ring’s rearrangement, and the rearrangement is the thing the ring calculation spent its length describing.

The extrapolation fits contrast = limit + c/U over the repulsions from 256 to 1024 and reports the residual spread as its own licence. The check on this computation requires three things and refuses if any fails: that the chain has no exactly degenerate pair at any point, that its half-filled limit lies below the ring’s four-electron limit, and that the guard permits every chain filling while refusing at least one ring filling. The last of those is the one that would break first if the guard were simply a test that always says no.

Where the model stops #

Six sites is small, and the tie on the ring was a property of a particular ring of a particular size. Nothing here shows that a longer chain converges to the same one; it shows that a chain of six does, and that the mechanism by which the ring reached one is absent. A chain of eight would be the check and it is roughly sixteen times the arithmetic.

The Hubbard model is also a single-band model with one orbital a site, so the “satellite” here is a many-body line in one band rather than the shake-up structure a real spectrum shows. The contrast it computes is the right shape of quantity — a ratio of two spectral weights across a rank cut — and the claim that it reaches one at half filling is a claim about this model. It is not a prediction about an experiment, and that distinction has mattered since the weights were introduced.

And the limit is a limit in the repulsion, which no material supplies. The useful reading is the one the picture of the approach makes: at eight times the hopping the contrast is thirty per cent above its floor, and at a hundred and twenty-eight times it is six per cent above. Those are the numbers a finite repulsion gives.

The generalisation #

What travels beyond this model is the relation between a refusal and an answer.

A great deal of careful computation consists of installing a test that can reject the thing being computed, and the value of such a test is measured by what it rejects — the same standard applied to a check that has never refused anything. But a rejection is a statement about a particular route to a number, and it is tempting to read it as a statement about the number. The ring’s result was refuted and the value survived, because the refutation was of the argument.

The repair was not a better guard. It was a second system with the same physics and none of the accidental structure, which is the standard move whenever a result might be an artefact of the thing that made it computable. Symmetry makes a calculation possible and then quietly supplies coincidences that look like findings, and the way to tell the two apart is to remove the symmetry and see what survives. Here everything survived except the argument.

There is a smaller lesson beside it. The non-monotone approach means the contrast has a feature — a minimum in its excess near U = 32 — that neither geometry’s story explains and that is not pursued here. It is not noise: it is 0.0316 against 0.0617 at four times the repulsion, a factor of two, far outside anything the diagonalisation could be uncertain by. Something in the ranking rearranges there, and the same rearrangement of weights that produced the ring’s flickering is the obvious suspect on a system where it cannot produce a tie.

Who found it, and when #

The Hubbard model dates from 1963 and the description of photoemission satellites as many-body structure in the ion rather than as one-electron levels is older than that in its qualitative form. What is computed here is not a result from the literature; it is new arithmetic on a six-site system, done to settle a question the ring calculation raised. The contrast and its rank cut are defined here rather than quoted, and the extrapolation and its guard were built earlier for exactly this purpose.

The finding worth carrying is the one about the guard, and it is a finding about method rather than about the model.

Still open: a chain of eight, and the minimum in the excess #

The obvious open question is the chain of eight. Everything here is a statement about a chain of six, and the claim that the half-filled contrast converges to one deserves a second size before it is treated as general. Eight sites at half filling is a much larger ion spectrum and the ranking already handles arbitrary size, so this is a matter of arithmetic rather than of new method — and the interesting outcome is not that the limit is confirmed but that the coefficient in the reciprocal approach, 9.64 here, either scales with the system or does not.

The nearer question is the minimum in the excess. The chain’s departure from one falls to 0.0316 at U = 32, rises to 0.0617 at U = 128, and only then begins its reciprocal decay — so there is a repulsion at which the contrast is anomalously close to its limit for a reason that has nothing to do with the limit. The two lines at the cut are computed at every repulsion already, so plotting their weights separately rather than their ratio would say whether one of them turns over or whether they cross, and the second of those would make the minimum a rank exchange on a system that has no degeneracies to exchange through.