A satellite that never loses its place

There is a boundary. Below some repulsion a satellite line in a removal spectrum is much weaker than any fundamental and sits outside their range, so the two kinds of line can be told apart; above it neither test works. Measured on four systems, the boundary orders exactly with the one-electron gap — and the objection to that result is plain:

Everything here is half filled, which is the case with the most correlation and the least room for the fundamentals. A quarter-filled system has fewer electrons, more empty levels and a different gap, and the same two tests applied there would say whether the boundary follows the gap when the gap is changed by the filling rather than by the geometry. That is the sharpest available test of the proposed mechanism.

It is, and the mechanism does not pass it.

What happens below half filling #

The exact solver is limited to six sites, so a quarter is not always an integer number of electrons. What is available is every filling from one electron of each spin up to half, on rings and chains of four and six — ten systems in all.

Four systems have no boundary at all. The intensity test holds at every repulsion tried, up to sixty-four times the hopping — a regime nothing physical reaches — and the contrasts there are 8.3, 3.2, 6.8 and 2.9, all above the factor of two the test needs and all falling too slowly to arrive.

Every one of the four is below half filling.

The word boundary is doing precise work there and is worth unpacking. It is the smallest repulsion in the list at which the weakest fundamental is less than twice the strongest satellite — so none up to 64 means the weakest line anybody would call a fundamental is still more than twice as strong as the strongest line anybody would call a satellite, at a repulsion sixty-four times the hopping. Koopmans’ theorem is exact for nothing even there; what survives is not the identification of a line with an orbital but the ability to sort the lines into two kinds.

Where the ordering fails #

The claim can be tested directly on a lattice measured twice. The six-site chain is the case:

A smaller gap with a larger boundary. If the boundary followed the gap, that pair would go the other way.

The ordering across four systems is not an artefact — run again here on a different set of systems, the half-filled cases still order with their gaps. What it is is a relation that holds along one axis of variation and not along the other, which means the gap was standing in for something that varies with the geometry and also with the filling.

What varies with the filling #

The share of the removal spectrum’s weight carried by satellites, at the largest repulsion tried:

There is less correlation in the spectrum away from half filling, and no repulsion repairs it. A half-filled lattice at large repulsion has one electron per site and every removal leaves a system that has to rearrange; a third-filled lattice has room, the electrons avoid each other without needing to correlate, and the removal spectrum stays close to the one-electron one however hard they push.

That is why the contrast flattens rather than crossing. The third-filled ring of six runs 676, 198, 67, 28.5, 15.8, 11.1, 9.2, 8.3 as the repulsion goes from 0.5 to 64 — falling, but by less and less, towards something above two.

What does survive #

One part of the gap account survives the change of filling, and it is the part that was a threshold rather than an ordering.

A system with no one-electron gap has no distinguishability at any repulsion. Two systems here have an exactly zero gap — a ring of four at half filling and a ring of six at two thirds, both with a degenerate level at the Fermi energy — and both have a contrast of 1.0 at every repulsion, including the smallest tried and including none at all.

So the honest summary of the two comparisons together is:

• a degenerate ground state destroys the distinction outright, at any repulsion, and that is about the gap;
• among systems with a gap, the boundary is set by how much correlation the filling admits, and the gap ordering at half filling was a proxy for that;
• and below half filling the distinction does not break at all in this model, at repulsions no material has.

What this does to the reading of a real spectrum #

The practical statement is worth extracting, with the caution about six sites attached to it.

A hundred lines and no way to sort them is the difficulty: a removal spectrum has more lines than the system has electrons, and there is no rule that says which are fundamentals. The gap account offered a partial answer — below some repulsion there is a rule, and the repulsion is set by the gap.

The filling narrows and widens it at once. Narrows, because the repulsion is not set by the gap and a spectrum’s own gap does not tell a reader whether the sorting can be trusted. Widens, because the filling does: a system with a shell far from half full has a removal spectrum whose satellites are faint and out of the way at every repulsion this model reaches.

That is a statement a spectroscopist could act on, and it is on the safe side of the usual caution. The systems where satellite structure is famously confusing — transition-metal oxides, the mid-shell cases — are the ones near half filling; the systems where a photoelectron spectrum reads like a list of orbitals are the closed-shell and nearly-empty-shell ones. The model reproduces that division and attributes it to the filling rather than to the gap, which is not where the gap account would have put it.

Why the proxy worked #

It is worth saying why four systems at one filling ordered so cleanly, because the answer is not coincidence.

At half filling, a system’s one-electron gap and its correlation strength are two readings of the same thing. A large gap means the electrons at the Fermi energy are far from the empty levels, so the repulsion has to be large before it can mix them; a small gap means it does not. So across half-filled systems, ordering by gap is ordering by how easily correlation sets in, and the four systems were four points on that line.

Change the filling and the two come apart, because the filling changes the correlation strength without changing the level scheme at all. The six-site chain’s levels are the same six numbers at both fillings; what changes is where the electrons sit in them and how much room they have.

There is a way to say the same thing that makes it a warning rather than an observation. Four points on a line are four points on a line, and the gap account rested on four systems and one filling — so the ordering it found was established over a set with one degree of freedom in it, and any quantity that varies monotonically with the geometry would have ordered just as well. The band width would have; the number of sites would have. What four systems at one filling could not do was tell them apart.

A relation established along one axis is a relation about whatever the two quantities have in common along that axis. a formula right for every molecule anybody would check it on is the same statement about a different sample.

The instrument this leaves behind #

Two systems of the same size and shape, measured at two fillings, is a controlled comparison — everything about the lattice is identical and one input changes — and it is the only kind of comparison in this field that can separate a property of the level scheme from a property of the electron count.

That is worth naming because such comparisons are rare. Most comparisons here are across molecules or across lattices, where a dozen things change at once and a rank correlation is the honest summary. A pair at two fillings is a comparison where the summary is a single arrow: this quantity went up and that one went down.

The six-site chain is the cleanest case of it. Between two-thirds filling and half filling its one-electron gap rises and the repulsion at which its satellites become unsortable falls — the opposite pairing to the one the half-filled systems show among themselves. One lattice cannot disagree with itself about its geometry, so whatever moved the boundary moved with the filling and with nothing else.

One such pair refuted a relation four uncorrelated points had established. Not because four points are too few, but because they varied along an axis that confounds two candidates and one pair varied along an axis that does not. A control outranking its mechanism is the same lesson learned from a set that could not be varied at all; here the set could be, and varying it was cheaper than adding to it.

What is quoted, and what is computed #

What a photoelectron spectrum measures is worth restating here, because everything above is about lines in a computed spectrum rather than about a measurement.

More bands than there are orbitals is where counting lines begins, and it is worth remembering that the object being sorted here is a computed list rather than a measured one.

Nothing is quoted. There is no spectrum and no material: a lattice of a stated size, one hopping, one on-site repulsion, and a stated number of electrons of each spin.

Every removal spectrum is an exact diagonalisation of the configuration space — a hundred and forty-four states for a half-filled six-site lattice, ninety for a third-filled one — with the lines and their weights read off the overlaps between the ground state of the neutral system and every eigenstate of the ion. The contrast is a ratio of two of those weights; the boundary is the smallest repulsion in the list at which the ratio falls below two.

The one-electron gaps come from a separate diagonalisation of the adjacency matrix, which shares nothing with the interacting calculation but the lattice.

The spectra are cached between builds and each restored set is checked by re-reading its own repulsion list, which is the cheap invariant that catches a set stored under the wrong key.

Two routes to one variable, and why both are needed #

The ordering across four systems and its failure under a change of filling is a methodological result as much as a physical one, and the method is worth stating because the same trap appears from another direction.

A correlation is evidence that two quantities move together. It is not evidence that one causes the other, and the standard way of finding out is to change the candidate variable by a route that changes nothing else.

Here there are two such routes and they disagree.

Change the system. Four different structures, four different one-electron gaps, and the boundary orders with the gap. That is a real correlation over four points.

Change the filling. One structure, two fillings, two different gaps — and the ordering reverses.

The second is the sharper test for exactly the reason a control is sharper than a correlation: changing the system changes everything about it, so a quantity that ordered with the gap across four systems may have been ordering with something else that varies alongside the gap. Changing the filling holds the structure, the couplings and the geometry fixed and moves the gap alone.

So the gap account survives as a description of four systems and does not survive as a mechanism. The gap is not what sets the boundary — or is not the only thing — and the evidence that it was came from a comparison in which too much varied at once.

The general rule this leaves is short and applies to every correlation in this collection. Vary the candidate two ways. A mechanism survives both; a confound survives the route it travelled on and fails the other, and the failure is the only thing that distinguishes them.

What this cannot say #

Six sites is not a solid. The exact solver is limited to it, and a satellite in a real photoelectron spectrum belongs to a system of 10²³ electrons where the level spacing is nothing and the distinction between a satellite and a fundamental is made by a lineshape rather than by counting lines. What this establishes is a property of a model that can be solved.

A boundary at U = 64 is not a boundary. The four systems reported as having none might have one at a repulsion of two hundred, and the honest statement is that they have none at any repulsion within a factor of sixty-four of the hopping — which is already far outside anything a material offers, since a real U/t is between two and ten.

The two tests do not always fail together, and only one is reported as the boundary. On the original ring the intensity test and the position test fail at the same repulsion; here they part company on three of the six systems that have a boundary at all — the six-site chain at two thirds loses the intensity test at U = 4 and never fails the position test at all. The boundary quoted throughout is the intensity test’s, and that choice is inherited rather than justified here.

And below half filling here means a third or a half of half filling. The two-thirds-filled cases behave like the half-filled ones and the third-filled ones do not, so the crossover in behaviour is somewhere between, and four systems cannot locate it. What the census establishes is that the two ends differ, not where they stop differing.

What was checked #

The census reaches more than one system below half filling, which is the check that the extension happened at all — a routine that silently fell back to half filling everywhere would produce a tidy table and no test.

At half filling the boundary orders with the one-electron gap, re-run here on a different set of systems. That had to hold: if it did not, the disagreement across fillings would be a disagreement with a result that no longer existed.

And there are lattices measured at both fillings, checked as a count, because the whole comparison is between two readings of one lattice and an empty pair list would leave nothing being compared.

Still open: the crossover in filling #

The obvious open question is the crossover the census cannot locate. Third-filled systems keep their satellites and two-thirds-filled ones lose them, so there is a filling in between where the boundary appears, and finding it needs a lattice with more fillings than six sites allow. Eight sites is a configuration space of four thousand nine hundred at half filling, which needs a different solver rather than a longer run — and it is the one extension of exact diagonalisation still to be made.

The nearer question is what the contrast is heading for. The third-filled curves flatten rather than crossing, and a curve that flattens has a limit: 8.3 and falling by tenths at the ring of six, 2.9 at the chain of four. Whether those limits are above two or below it is the difference between satellites are always distinguishable away from half filling and they are distinguishable up to a repulsion nobody has reached, and the two are very different claims. Fitting the approach — which is the instrument built for exactly this shape of question — would settle it without a larger lattice.