The boundary belongs to the gap

A hundred lines and no way to sort them put two tests to a removal spectrum. A satellite is tellable from a fundamental if it is much weaker than any of them, or if it sits outside their range in energy — and both tests fail above some repulsion, on a ring of six at half filling, with the two failing at nearly the same place.

It said what that finding was and was not. It was enough to say a boundary exists; it was not enough to say what the boundary depends on, being one system at one filling. And it named the two candidates: the repulsion alone, or the repulsion measured against the band width.

Neither. It is the gap — which is the quantity Koopmans’ theorem is exact for nothing because of, arriving in a different question.

Four systems #

Rings and chains of four and six sites, all at half filling, all with the same on-site interaction. What differs is the size and whether the ends are joined — and, as a consequence, the one-electron spectrum each one has.

The boundary is not a property of the repulsion. Three systems, three answers, a factor of four apart, with the same interaction throughout.

Nor of the band width. A ring and a chain of six differ in band width by ten per cent and in boundary by a factor of four. A ring of four and a ring of six have the same band width and could not differ more.

It orders exactly with the gap. 0.890 → 2, 1.236 → 4, 2.000 → 8; and the ratio of boundary to gap runs 2.25, 3.24, 4.00, which is not constant but is not far off.

Why the gap and not the width #

The mechanism is the one the ring calculation implies and does not state.

A satellite is a final state of the ionised system in which one electron has been removed and another excited. Its energy is therefore a fundamental’s energy plus an excitation energy, and the cheapest excitation available costs the gap. So a satellite’s distance from the nearest fundamental is set by the gap, and its intensity — which comes from the overlap between the correlated ground state and that excited final state — is set by how much the repulsion has mixed the excitation into the ground state, which goes as the repulsion over the gap.

Both tests are therefore about U/Δ rather than U/W, and the band width enters only through where the fundamentals themselves sit. A wide band with a large gap keeps its satellites distinguishable to a large repulsion; a narrow band with a small one does not.

That the ratio is 2.25, 3.24 and 4.00 rather than one number is a real residue: U/Δ is the leading dependence and there is something else in it. Three systems cannot say what.

The case with no gap #

The ring of four is the case worth having and it is not a near miss.

Its half-filled one-electron ground state is degenerate — four sites give levels at 2, 0, 0 and −2, so the two electrons that would fill the middle have two orbitals to choose from — and the gap is zero to the last bit the eigensolver carries.

Its weakest fundamental is the same strength as its strongest satellite at every repulsion tried, including none at all: a contrast of 1.0000 at U = 0.25 through U = 8, and 1.0175 at U = 0.

A system with no gap does not have a large boundary. It has no boundary, because the two kinds of line were never distinguishable and there is nothing for a repulsion to spoil. Reading that as the boundary is at zero would be reading a limit into a case where the quantity does not exist — the same distinction an undetermined direction has from a badly determined one.

And it is a case a spectroscopist would meet: a metal has no gap — and an insulator band theory cannot see is the opposite corner of the same picture, where the one-electron gap is zero and the true one is not — and this is the arithmetic saying that satellite assignment in a metal is not a hard problem but an ill-posed one.

The two tests come apart #

There is a second correction to the one-ring result, and it is one its own check would have caught on a different system.

It found that its two tests fail at the same repulsion and required so — reasonably, since on its ring they do: the intensity contrast falls below two and the first satellite moves inside the fundamentals’ range at the same place.

On a chain of six they do not. A satellite sits inside the fundamentals’ energy range at every non-zero repulsion tried, down to U = 0.25, where the intensity contrast is still 25.78 and the two kinds of line are plainly distinguishable by strength.

So on that system the energy test is useless from the start and the intensity test works well, and a spectroscopist applying both would get contradictory answers. The reason is the same one: a chain’s levels are unevenly spaced and its fundamentals span a wide range with holes in it, so there is somewhere inside that range for a satellite to sit long before the repulsion has done anything.

The two tests are not two views of one boundary. They agree on a ring, which is the tidiest case, and the one-ring result generalised from it.

What this says to a spectroscopist #

Three things, and the third is the practical one.

The number to know about a system is its gap, not its band width. That is a different recommendation from the usual one, which was to compare the repulsion against the band width — the ratio every lattice-model paper quotes. A material with a large gap keeps its satellites clean to a large repulsion; a narrow-gap material does not, whatever its band width. Since the gap is measured routinely and more accurately than any correlation parameter, the diagnostic is available before the spectrum is.

The estimate is a factor rather than a number. The boundary sits between two and four times the gap on these three systems, so a material with a one-electron gap of 2 eV loses the distinction somewhere around 4 to 8 eV of on-site repulsion, which is the range real transition-metal oxides occupy — and that is why their satellite assignments are argued about.

And a gapless system is not a hard case but an ill-posed one, which is worth knowing before the assignment is attempted rather than after. In a metal there is nothing to separate, and the arithmetic says so at U = 0 rather than reporting a small boundary.

Why this changes the one-ring conclusion #

The one-ring conclusion was that the two tests fail together at a repulsion of about six on its ring, and that a spectroscopist could estimate which side of the boundary a material is on if the boundary could be tied to something measurable.

Both halves are now different.

The boundary is tied to something measurable, and it is the gap. That is better than could be hoped for: a band width is a derived quantity that needs a model to extract, and a gap is read off an absorption edge. So the estimate a spectroscopist wants is available, and it is twice to four times the gap.

And the two tests are not one boundary. A spectroscopist applying both to a chain-like material would find the energy test failing where the intensity test says the lines are cleanly separated, and would have to decide which to believe. The answer here is the intensity test, because the energy test is about where the outermost fundamentals happen to sit and that is a property of the band rather than of the correlation.

Neither correction could have been found on one system. That is the whole reason the one-ring result said what it had and had not established, and it is worth recording that the caution was the useful part of it: a finding stated with its own scope attached is a finding somebody can extend.

There is a fourth thing, and it is about how the two numbers are usually got. A gap is measured; a band width is fitted, usually to the same spectrum whose satellites are in question. So the recommendation is not only that the gap is the right quantity but that it is the independent one — the band width comes from the calculation the assignment is part of, and a diagnostic taken from inside the thing it is diagnosing is worth less than one taken from outside it.

What is quoted, and what is computed #

The gap that matters here is the one-electron gap, which is not the gap the repulsion opens — the insulator band theory cannot see is where the two are separated, and a system can have one and not the other.

Nothing is quoted. The systems are four small lattices, the interaction is on-site, and every spectrum is the exact removal spectrum of the exact ground state — the whole configuration space diagonalised and every pole’s weight computed.

The gaps are computed from the same lattices’ one-electron spectra, by a route with no interaction in it at all, so the two sides of the comparison come from different calculations.

The sweeps are cached between runs and every restored sweep is verified by its own shape before use, because a stored spectrum is a store of numbers computed somewhere else.

What this cannot say #

Nothing here is a material. Four lattices of four and six sites with one orbital each are a model of the question, and the two-to-four factor should be read as an order of magnitude rather than as a number to apply.

Four systems and three gaps. The ordering is exact and three points establish an ordering, not a law; the ratio of boundary to gap moves by a factor of nearly two across them, and what else is in it is not answered here.

The repulsion grid is coarse. The boundary is reported as the smallest repulsion tried at which the lines stop being distinguishable, and the grid runs 0.25, 0.5, 1, 2, 4, 8 — so each boundary is known to within a factor of two. That is fine for an ordering and would not be fine for the ratio.

One orbital per site, and no dispersion beyond nearest neighbours, so what a removal spectrum’s weights mean here is a model’s answer rather than a material’s. A real satellite structure has core levels, several bands and a screening response, none of which is here.

And the second test is about a range rather than a gap. Inside the fundamentals’ range depends on where the outermost fundamentals are, which is a property of the band and not of the gap — which is why it comes apart from the intensity test, and why the boundary reported here is the intensity test’s.

What the crossing looks like in the spectrum itself is worth seeing before the practical consequence is drawn, because the change is not one of degree: the number of lines is different on the two sides of it, and a count is not something a resolution can improve.

And here is the consequence for anybody reading such a spectrum rather than computing it, which is where the boundary stops being an abstraction: the satellites have to be told apart from the fundamentals, and on one side of the crossing they are interleaved with them.

What was checked #

The census contains a system with no one-electron gap, which is what makes the third finding available at all.

That system’s satellites are never distinguishable, at any repulsion — checked across the whole sweep including zero, because a boundary at a very small repulsion and no boundary at all are different claims.

While every gapped system has repulsions at which they are, which is the other half.

The boundary sits at different repulsions on different systems, so it is not the repulsion’s alone.

It orders with the one-electron gap, checked as a monotone sequence over the three gapped systems.

And the two tests do not always fail at the same repulsion, which was found on one ring and taken as general — the correction made here.

The system with no gap and no boundary #

The fourth system is easy to read as a failed case and it is the most informative of the four, because it says what a satellite needs in order to be a satellite at all.

A boundary between satellites and fundamentals exists only if the two are distinguishable somewhere. In a system with no one-electron gap the fundamental lines are not separated from one another to begin with — the levels they come from are degenerate or nearly so — and there is no interval for a satellite to be outside of. The distinction has nothing to be a distinction between.

So the fourth system does not have a boundary at a very small repulsion, or at a very large one. It has none at any repulsion including zero, and that is the correct answer rather than a limit of the search.

Which sharpens what the other three measure. A boundary is not a property of the correlation; it is a property of the gap, and the repulsion decides where it falls rather than whether it exists. A gapless system has no boundary at any coupling, and a gapped one has a boundary at a coupling that scales with its gap — which is the essay’s finding, with its own tripwire attached.

Why the gap and not the repulsion is the surprise #

The received way to describe a correlated system is by the ratio of its repulsion to its band width, and that ratio is what every lattice-model paper quotes. It is the wrong ratio here, and it is worth being clear about why the right one is a surprise rather than an obvious correction.

A band width is a property of the whole spectrum. It is what sets the kinetic energy scale, it is what the repulsion competes against for the ground state, and it is what decides whether a system is a metal or an insulator in the first place. Every intuition about when correlation matters is built on it.

A gap is a property of two levels. It is much smaller, it is much more sensitive to the geometry, and two systems with the same band width can have gaps differing by any factor at all — which is exactly what a ring and a chain of six sites demonstrate.

So the finding is not that a familiar ratio needed refining. It is that the quantity governing this particular question is a different one, of a different kind, and the reason is mechanical rather than dimensional: a satellite’s separation and its intensity are both about one excitation, and an excitation costs a gap.

Whether other correlation diagnostics have the same structure is not answered here, and what a correlation energy is a measurement of is the nearest neighbour of the question — there too the quantity that governs the answer turned out to be one nobody was quoting.

Still open: the ratio to the gap, the filling, and a ring that distorts #

The obvious open question is the ratio the three points leave open. Boundary over gap runs 2.25, 3.24 and 4.00, and what varies with it is not the band width — so the candidates are the number of sites, the density of states at the Fermi level, and the number of excitations degenerate with the cheapest one. Adding a ring of eight would put a fourth point on the line and, more usefully, would give two systems with nearly the same gap and different sizes, which is the comparison that separates the first candidate from the other two.

The nearer question is the filling. 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 mechanism proposed here, and it needs nothing new — only a different electron count.

A third direction is the one the gapless case opens. A ring of four is degenerate at half filling and that degeneracy is a Jahn–Teller instability: the real system would distort, open a gap, and acquire a boundary. So the ill-posed case is ill-posed only for a structure held rigid, and computing the boundary of the distorted ring — reachable by alternating the bonds — would say whether a system that opens its own gap gets a boundary at the value that gap predicts. A distortion needs two states is where the instability itself is computed.