The limit of one is a parity

A photoelectron spectrum is read, in the simplest account, as one band per orbital, at an energy Koopmans’ theorem relates to the orbital’s. A spectrum of a correlated molecule has more lines than orbitals, and telling the fundamentals from the satellites by height alone depends on a contrast: the weakest of the lines that belong to orbitals over the strongest of the rest. When there are N↑ electrons of one spin to remove, the fundamentals are taken to be the N↑ strongest lines, and the contrast is the N↑-th strongest over the next. Below a contrast of two, height stops being a reliable way to sort a spectrum.

At half filling that contrast was found to fall to exactly one on a ring of six, and the one turned out to be a tie: the rank cut fell between two exactly degenerate lines, so the ratio was a quantity divided by itself. An open chain of six has no degenerate lines at all, and its contrast went to one anyway — but not smoothly. It came within 0.0316 of one at U = 32, moved away to 0.0617 at U = 128, and only then began a steady approach. The degeneracy had invalidated the ring’s argument and not its number, and nothing yet said why the number was one.

Both questions have the same answer, and it is reached by looking at the spectrum one line at a time.

Every line is smooth #

A contrast is a ratio of two order statistics: whichever line is N↑-th strongest over whichever is next. Those two lines need not be the same lines at different repulsions. If lines cross in weight, the contrast is built from different pairs of lines on either side of the crossing, and its shape can turn wherever the ranks change even if every line changes smoothly.

So the lines were followed individually. At each of forty-one repulsions from 4 to 4096, spaced evenly in the logarithm, the removal spectrum of the half-filled chain of six was computed exactly, the strongest line in each cluster of energies was taken, and the six strongest were followed from the largest repulsion downward by continuity in their removal energy. Each is labelled by where it ends up: at ±0.45, ±1.25 and ±1.80.

From U = 19 upward, every one of the six is monotone in weight. The line tending to −0.45 falls from 0.388 to 0.353; the one tending to +0.45 rises from 0.290 to 0.353; the ±1.25 and ±1.80 pairs do the same, one member falling and the other rising. None of them turns. Whatever produced a minimum and a maximum in the contrast, it was not a turn in any line, and it was not a change in how quickly any line approaches its limit.

The minimum is a tie and the maximum a new satellite #

Where two followed lines exchange order, the exchange can be bisected to the repulsion at which their weights are equal. In the two windows where the contrast turned, there are four. Below U = 16 the contrast has a larger excursion of its own, rising to about 1.59 near U = 11; there the lines still cross in energy as well as in weight, following them by continuity is not reliable, and that part of the curve is shown but not taken apart.

At U = 17.91 the lines tending to −1.25 and −1.80 exchange ranks two and three. At U = 28.916 the line tending to +0.45 overtakes the one tending to −1.80 at ranks three and four — the cut itself — and their weights there are 0.313162 each, equal to eight figures. The contrast at that repulsion is 1.00000004. At U = 32.40 ranks two and three exchange again. And at U = 156.6 the lines tending to +1.25 and −1.80 exchange ranks four and five, so the strongest satellite — the denominator of the contrast — changes identity, and the contrast is 1.0632, its largest value in the window.

So the reading at U = 32 that looked anomalously close to one was the nearest sampled repulsion to an exact tie three units below it.

The lines at the cut on either side of the tie are worth naming, because they show how unlike the two sides are. Below U = 28.9 the third fundamental is the line heading for −1.80, which at infinite repulsion is the lowest level of the hole band, and the strongest satellite is the line heading for +0.45. Above it the two have swapped roles. A spectroscopist sorting lines by height at U = 25 and at U = 35 would call different lines the fundamental, with nothing in the spectrum’s appearance to say so, and the contrast would read close to one at both — not because the fundamental and the satellite are indistinguishable in general, but because at those repulsions two particular lines happen to have nearly the same weight. The rise to U = 128 was the contrast’s denominator falling while its numerator fell more slowly, until at 156.6 a different satellite took over as the strongest and the ratio turned. A chain with no degenerate lines anywhere has an exact tie at the cut, not because two levels coincide but because two weights cross — and eight sampled repulsions stepped over it.

That also qualifies what was established about the chain. “No exactly equal neighbouring pair at any of the eight repulsions” was true, and the statement it was taken to support — that the chain’s contrast of one could not come from a tie — was not quite right: the chain has ties, at particular repulsions, of a kind a finite sample of repulsions will miss.

Where the lines go #

The limit is a separate question from the approach, and the lines’ destinations answer it.

At U = 16384 the strongest line of each energy cluster, for the half-filled chains of two, four and six, sits at 2cos(kπ/(n + 1)) — the levels of a single particle hopping along a chain of n sites — to within a hundredth. On the chain of six those are ±0.445, ±1.247 and ±1.802; on the chain of four, ±0.618 and ±1.618; on the chain of two, ±1. The picture behind it is the standard one at strong repulsion: removing an electron from a half-filled chain leaves a hole moving through a background of spins, and the hole’s motion is that of one particle on the chain, dressed by the spins it disturbs.

That picture is worth dwelling on, because it is what makes the spectrum simple at the end of a sweep in which it was anything but. At small repulsion the removal spectrum is nearly the orbital one: N↑ strong lines at the orbital energies and weak satellites between them. At large repulsion every electron is localised on its site, the molecular-orbital description has dissociated, and the lines the spectrum shows are not orbitals at all but the n standing waves of a single hole on a chain of n sites. There are as many strong levels as sites rather than as occupied orbitals — six on the chain of six, where the orbital picture had three — and each is split into a cluster of spin states that share almost the same energy. The satellites of the weak-repulsion spectrum have become the other half of the hole’s band.

The levels of a single particle on a chain are symmetric about zero, so the lines come in pairs at ±E. And the weights within each pair converge. On the chain of six at U = 16384, the ±0.445 pair carries 0.3530 and 0.3529, the ±1.247 pair 0.2769 and 0.2768, the ±1.802 pair 0.2590 and 0.2588.

The difference within each pair falls as the reciprocal of the repulsion: on logarithmic axes the three pairs run parallel to a line of slope −1 from about U = 200. That is the same 1/U the contrast itself was found to approach its limit with, and it is not a coincidence.

One, when the cut falls inside a pair #

The contrast takes the N↑ strongest lines as fundamentals. At large repulsion the strongest lines are pairs of nearly equal weight, ordered pair by pair. Whether the cut after the N↑-th line falls between two pairs or inside one depends on whether N↑ is even or odd.

On the chain of six there are three up electrons. The two strongest lines are the ±0.445 pair; the third is one member of the ±1.247 pair and the fourth is the other. The cut falls inside a pair, the contrast is the ratio of two weights converging as 1/U, and its limit is one — 0.999999 by extrapolation. On the chain of two there is one up electron, the cut splits the only pair, and the limit is 1.000000.

On the chain of four there are two up electrons. The two strongest lines are the ±0.618 pair, at 0.4067 and 0.4066; the third and fourth are the ±1.618 pair at 0.3099 and 0.3097. The cut falls between pairs, and the contrast goes to the ratio of the two pairs’ weights: 1.3125.

So “the half-filled chain’s contrast converges to one” is a statement about the parity of the number of up electrons. It held on the chain of six because three is odd, and it holds on the chain of two. It fails on the chain of four, where the limit is well above one — and, like the ring of six’s exact one, it is a property of where a rank cut falls among lines the physics has organised into pairs, not a statement that fundamentals and satellites have become indistinguishable in any deeper sense.

How the lines were followed #

Each spectrum is the exact removal spectrum of the Hubbard chain at half filling: the ground state with N↑ up and N↓ down electrons, every eigenstate with one fewer up electron, and the weight of each as the squared amplitude of removing an up electron from any site into it, summed over sites. Lines closer than 0.12 in energy are grouped and the strongest of each group is kept, which merges the near-degenerate spin multiplets a hole leaves behind and keeps one line per hole level.

Following starts from U = 4096, where the six strongest lines are cleanly separated, and walks down the logarithmic grid, at each step taking for each followed line the group whose energy is nearest the one it had. Two followed lines exchange rank where the difference of their weights changes sign between grid points; the exchange is bisected sixteen times in the logarithm of the repulsion, following each line’s energy as it moves, to the repulsion where the weights are equal. The large-repulsion structure uses U = 4096 and 16384 and the first-order extrapolation the contrast has always been extrapolated with.

The checks, run wherever these figures are drawn. On all three chains the large-repulsion lines lie within 0.01 of 2cos(kπ/(n + 1)), every −E level has a +E partner of equal weight to a tenth of a per cent, and the cut falls inside a pair exactly when N↑ is odd. The chains of two and six extrapolate to one within a thousandth and the chain of four to more than 1.3. On the chain of six, an exchange at ranks three and four between U = 16 and 48 has a contrast of one to a thousandth at its bisected crossing, an exchange at ranks four and five lies between U = 100 and 300, and every followed line is monotone from U = 19. The refusal is the sum rule: at every repulsion on the grid the removal weights must add to the three up electrons, or a line has been lost from the spectrum the ranks are taken over.

Where this stops #

Three chain lengths, and only the even ones at half filling. The parity rule predicts one for a chain of ten and not for a chain of eight. Both are beyond exact diagonalisation at the cost used here, and neither is computed; the rule is established on three chains and argued for the rest from the single-particle levels, which is an argument rather than a count.

Chains, not rings. The ring of six’s levels at large repulsion are not symmetric in the same way — a ring’s single-particle levels are 2cos(2πk/n), which for six includes degenerate pairs and the unpaired ±2 — and its contrast of one came through exact degeneracy. Whether rings of other sizes follow a parity rule of their own is not examined.

Grouping by energy. The strongest line of each cluster of energies stands for the cluster. At moderate repulsion the clusters are wider and a weaker multiplet member could in principle outrank a neighbouring cluster’s strongest line; the contrast used throughout is computed over all lines, and the followed lines agree with it at every grid point, but the grouping is a choice.

One removal channel. Only the removal of an up electron is computed. At half filling the down channel is identical by symmetry, and the addition spectrum is the removal spectrum reflected, so nothing is lost; away from half filling the channels differ, and the satellite whose place a filling fixes belongs to that regime.

And a height test. Everything here concerns sorting lines by weight. A spectrum sorted by whether satellites fall inside the fundamentals’ range is a different test with a different boundary.

A ratio of ranked things turns where the ranks change #

The general point is about ratios of order statistics. A quantity defined as “the k-th largest over the (k + 1)-th” inherits every crossing among the things being ranked, and its turns, minima and near-coincidences say nothing about the smooth behaviour of any one of them. Before reading a trend in such a ratio, the ranked objects have to be followed individually; otherwise a crossing reads as a feature and a tie reads as a limit. A sampled curve makes this worse rather than better, because a crossing between two grid points is invisible except as a kink, and the nearest sample to an exact tie reads as a near miss. Eight repulsions spaced by factors of two could not have found a tie at 28.916; forty-one spaced by factors of 1.19 bracketed it, and bisection placed it to eight figures. The resolution needed to see the structure of a ratio of ranked quantities is set by how close together its crossings are, which is not known until the ranked quantities are followed.

The second point is about limits. The chain of six’s contrast does go to one, and the reason is that at strong repulsion the spectrum organises itself into pairs of lines and the cut happens to fall inside one. That is a structural statement, and it is conditional on a parity. A contrast with a closed form below half filling was a statement about the lines’ weights; this one is a statement about where the cut falls among them, which is why changing the number of electrons by one pair changes the answer.

Still open: the pair ratio on the chain of four, and rings #

The obvious open question is the chain of four’s limit. At 1.3125 it is the ratio of the ±0.618 pair’s weight to the ±1.618 pair’s, and the levels 2cos(kπ/5) are the golden ratio and its reciprocal; whether the weight ratio has a closed form in the same numbers, and whether the chain of six’s pair weights share it, would turn the parity rule into a formula for the limit on every even chain.

The nearer question is the ring. A ring’s single-particle levels come in degenerate pairs of momentum ±k as well as in pairs of energy ±E, so its lines at large repulsion are organised twice over, and the ring of six’s exact one sits where both organisations meet. Following the ring’s lines individually, as here, would say which of the two structures its cut was falling into, and whether rings of four and eight obey a rule about parity, about momentum, or about both.