A contrast with a closed form

The two tests by which a photoelectron satellite can be told from a fundamental — the intensity contrast and whether a satellite sits inside the range the fundamentals span — respond to a change of filling in a particular way. Away from half filling both tests survive: the contrast falls as the repulsion is turned up and then flattens, at 8.3 on a third-filled ring of six and 2.9 on a third-filled chain of four, still above the factor of two the test needs and falling too slowly to reach it.

It closed on the thing it could not settle. A curve that flattens has a limit, and whether that limit is 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. Those are very different claims, and the way to choose between them is to fit the approach rather than to build a larger lattice.

The approach turns out to be about as simple as an approach can be, so the limits come out to six figures — and four of them are a closed form.

The approach is first order in the reciprocal repulsion #

The curves flatten, and how they flatten is the whole of what makes an extrapolation legitimate. Taking the departure of each curve from its own limit and multiplying by the repulsion:

It is a constant, to better than a per cent from a repulsion of 256 upwards, across a further factor of sixty-four. So

$$c(U) = c_\infty + \frac{A}{U}$$

with no fitting: two points at $U$ and $4U$ determine both numbers, and the limit is

$$c_\infty = \frac{U_2c_2 - U_1c_1}{U_2 - U_1}.$$

That is why the answer is a reading rather than an estimate, and it is the instrument built for the local slope of a fit for exactly this shape of question — with the difference that there the local slope was still moving and here the residual is flat, which is what says the extrapolation may be taken.

Every limit is above two #

The limits are 2.499999, 2.618033, 2.798553 and 2.905130 for chains of three to six, and 3.999996, 5.828424, 6.854100 and 7.464100 for rings of three to six. The smallest is 1.25 times the threshold and the largest 3.73 times it.

So the first of the two readings is the right one. Below half filling a satellite is distinguishable from a fundamental at every repulsion, including one that is not physically reachable, and the reason is not that the repulsion has not been turned up far enough. The contrast has a floor.

That is a stronger statement than eight points could support, and stronger than the test happens to hold in the range computed. The mechanism behind it stands: a lattice below half filling has room, so 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 the same statement the filling boundary made about the gap, read at the far end of the axis rather than at the crossing. What is added here is that “close” has a limit and the limit is a number.

It also changes what the earlier numbers were evidence of. The third-filled ring of six ran 676, 198, 67, 28.5, 15.8, 11.1, 9.2, 8.3 out to a repulsion of sixty-four, and 8.3 was quoted as still falling too slowly to reach the threshold. Continued, the same curve is 7.87, 7.66, 7.56, 7.51 and 7.4765, and its limit is 7.4641. So the falling was real and the destination was never in doubt once the shape was known: the whole of the remaining fall, from sixty-four to infinity, is 0.82 — a tenth of the value, against a distance of 6.3 to the threshold.

A curve that has fallen by a factor of eighty and has a tenth of its value left to lose is not a curve about to cross anything. That is the reading the residual makes available and no amount of staring at the eight original points would have given.

And on a ring the number is a closed form #

The four ring limits are 3.999996, 5.828424, 6.854100 and 7.464100. The first is 4, the second is $3 + 2\sqrt{2}$, and both are $(1 + 2\cos(\pi/n))^2$.

Read off the rings of four and six, the expression predicts 4.000000 for a ring of three and 6.854102 for a ring of five. The measured limits are 3.999996 and 6.854100 — agreement to better than one part in a million, on systems the expression was not fitted to.

Nothing here derives it. It is a reading of four numbers, offered as one, and the honest description is that a quantity computed by diagonalising a many-electron configuration space at infinite repulsion turns out to be a one-electron quantity — $2\cos(\pi/n)$ is the largest level of a chain of $n-1$ sites, and its appearance here is unexplained.

A chain of four’s limit is 2.618033, which is the golden ratio squared, and the ring expression at four sites is 5.828427. So whatever the ring formula is a statement about, it is a statement about a ring. The chains’ limits rise with size as the rings’ do and they are four numbers with no expression: 2.499999, 2.618033, 2.798553, 2.905130. Four points will accept almost any two-parameter form, and one fitted here would be a fit rather than a closed form.

The system that has nothing to distinguish #

The refusal this extrapolation needs is a system in which the contrast is not a distinguishability at all. A ring of four at half filling has a degenerate level at the Fermi energy, and its contrast is exactly one at every repulsion from a half to sixteen — a fundamental exactly as strong as a satellite, with no cut between them.

Above a repulsion of thirty-two it stops being exactly one: 1.3277 at 64 and 1.1410 at 256. That is the eigensolver rather than the physics. A degenerate ground state leaves it free to return any combination of the states in the subspace, and the overlaps with the ion’s states depend on which combination it returned.

Both halves matter. The first says the extrapolation is not manufacturing numbers — a system with nothing to measure returns nothing. The second marks the boundary of where the extrapolation may be used, and it is why every system in the table has a non-degenerate level at the Fermi energy.

It is worth being exact about what the degenerate case is refusing, because it would be easy to read it as a null result and it is stronger than that. The contrast is a ratio of two weights, and there are three ways it could come out at one: the calculation could be broken and returning the same number twice; the spectrum could be so weak everywhere that the ratio is meaningless; or the system could genuinely have no cut. The sum rule rules out the first — the total weight is the electron count at every repulsion, so the poles are real poles carrying real weight. The spectrum itself rules out the second: the lines are there, they move with the repulsion, and their individual weights change by factors of several across the scan. What does not change is their ordering by weight at the boundary, because there is no boundary: the fourth line and the fifth are two members of one degenerate shell, and a ratio taken across them is a ratio of a quantity to itself.

So the exactness is the content. A contrast of 1.0000 to every digit the solver carries, held across five octaves of repulsion, is a much stronger statement than a contrast of 1.03 would have been — the latter would have been a small cut, reported correctly. This is no cut at all, and it is the shape of refusal worth asking for: not a system where the answer is hard to see, but one where the quantity being extrapolated does not exist, and the extrapolation says so.

What was computed, and how #

A removal spectrum is the set of overlaps between the ground state of the neutral system and every eigenstate of the ion, each with its energy — the exact analogue of a photoelectron spectrum in a model small enough to solve, and the object that makes the one-electron reading of a spectrum checkable rather than assumed. The configuration space is written down in full, the fermion signs are kept, and the whole spectrum is required to satisfy the sum rule: the total weight is the number of electrons that could be removed. That check is run at every repulsion and it is what says the poles are the poles.

The contrast is the ratio of the weakest of the $n$ strongest lines — $n$ being how many the one-electron picture has — to the strongest of what is left. The definition is the one used for the filling comparison, unchanged, so the two sets of numbers are the same numbers.

The largest repulsion computed is 16,384 times the hopping. That is not a physical regime and it is not meant to be one: it is where a limit is read, and the reading is checked by the residual being flat rather than by the number having stopped moving.

The refusal is the degenerate system above, and the sum rule is the tripwire.

One consequence is worth stating for anyone reading a real spectrum, even though the model is a caricature. The limit here depends on the lattice and not on the repulsion, and it depends on it through a one-electron quantity: $2\cos(\pi/n)$ is the largest level of a chain of $n-1$ sites, and it grows towards 2 as the ring grows. So a larger ring has a larger limiting contrast, and the trend is the opposite of the intuition that a bigger system is harder to sort. Whether that survives to systems large enough to matter is not a question six sites can answer — but the direction is measured rather than assumed, on four rings.

Where the model stops #

Two electrons on three to six sites is a very dilute system, and below half filling here means a third or a quarter of it. Two-thirds-filled systems behave like half-filled ones and third-filled ones do not, so there is a crossover in between that six sites cannot locate, and every limit here is on the dilute side of it. Nothing above says what the limit does as the filling rises towards a half — where, on the half-filled systems, the contrast does cross two.

The model has one orbital a site and one repulsion. A real photoelectron satellite is a shake-up in a molecule with many orbitals, and a real spectrum can carry more bands than there are orbitals for reasons this model carries only in caricature.

And the limit is at infinite repulsion, which is a mathematical place. What it establishes is that the contrast does not go to the threshold — not that any material is anywhere near the limit.

What this settles about the two tests #

Of the two tests only one is quoted as the boundary, and deliberately: the intensity test is the one inherited, and on three of its six systems the two tests part company. The limit measured here is the intensity test’s, so it settles the intensity test and not the other.

That is worth being explicit about, because the two are not equivalent and the asymmetry runs the useful way. The position test asks whether a satellite sits between two fundamentals, and a spectrum with an unexpected line among the expected ones is a spectrum whose assignment is a guess. A system can fail the intensity test and pass the position test, in which case a spectroscopist who knows where the lines should be can still sort them; and it can pass the intensity test and fail the position one, in which case height alone still works.

What the limit says is that the first of those two never runs out below half filling. A cut in intensity always exists there, at any repulsion, and the smallest margin among the eight systems is a factor of 1.25 above the threshold. Whether a cut in energy always exists is a separate question with a separate answer, and it is not answered by any number here.

There is one more thing the limit does not settle and it is the one a reader is most likely to assume. A contrast above two says the strongest satellite is weaker than the weakest fundamental — it says nothing about whether there are many satellites, or how much of the total intensity they carry. The share the satellites carry grows smoothly from zero with the repulsion and does not flatten at all, and those two quantities behave quite differently. A spectrum can be perfectly sortable and still have most of its intensity in lines the one-electron picture does not have.

The generalisation #

The useful part is the instrument rather than the number. A quantity that flattens is worth extrapolating only after the shape of the approach has been established, and establishing it is usually cheaper than extending the range.

Eight points out to a repulsion of sixty-four could not tell 8.3-and-falling from 8.3-and-heading-below-2. Six more points, each one diagonalisation, made the residual flat and turned the same eight into a limit good to six figures. No larger lattice was needed, which is what the question asked for, and no fit was performed.

A counterexample is worth keeping beside it. An exponent fitted over the last decade before a transition belonged to the decade, because the local slope was still moving and nobody had checked. Here the local behaviour is checked first and then relied on. The two are the same discipline used in opposite directions: measure the approach before trusting the extrapolation, and measure the local slope before trusting the fit.

Who found it, and when #

Satellite structure in photoelectron spectra has been understood as a many-electron effect since Manne and Åberg’s sum rule of 1970, which is the rule every spectrum here is checked against. That correlation effects in the removal spectrum weaken away from half filling is standard in the Hubbard literature and is usually stated as a fact about the spectral weight rather than about distinguishability.

The specific limits above do not appear to have been written down. $(1 + 2\cos(\pi/n))^2$ for a ring is presented here as a measurement of four numbers with a strong pattern in it, not as a result — which is the right status for a formula whose derivation nobody has.

Still open: the contrast’s limit in closed form #

The obvious open question is the derivation. The infinite-repulsion limit of a two-electron lattice is a solvable problem — the two electrons cannot occupy one site, so the model becomes a pair of free particles on a ring with an excluded configuration — and the contrast in that limit should be obtainable in closed form for both a ring and a chain. If it comes out as $(1 + 2\cos(\pi/n))^2$ for a ring, the pattern above becomes a result and the chain’s four numbers become an expression at the same time.

The nearer question is the filling. Every limit here is at two electrons, which is a third of a ring of six and a half of a ring of four, so the eight systems span two fillings rather than a range. Taking the same limit at three electrons of each spin on six sites — half filling, where the contrast is known to cross two — would say whether the limit goes below two there or whether the crossing at finite repulsion is followed by a recovery. The residual test says in advance whether the extrapolation is allowed, so the answer costs six diagonalisations.