A cancellation between two separations

The give-back is a ratio of two quantities and each of them changes sign. The denominator is the on-site price, −U times the change in the pair count at contact; the numerator is what a neighbour interaction returns, V times the change in the pair count at one bond. On a six-site ring at an on-site repulsion of eight the numerator vanishes at V = 3.8955 and the denominator at 4.1595, and a bisection that does not know which it has found reports them in the same words.

The root has a clean reading, and it is the reading that essay gave it: it is the repulsion at which the pair count at one bond has come back to its free value. The numerator is V times that change, V is not zero, so the numerator vanishes exactly when the change does.

That reading holds because the numerator has one term in it. It has one term because the interaction reaches one bond. The closing question of the essay before it was what happens when it reaches two.

Two terms, and they point opposite ways #

Add a second-neighbour repulsion $V_2$ to the Hamiltonian and the give-back’s numerator becomes V·d(1) + $V_2$·d(2), with d® the change in the pair count at separation r. Two terms, and the second is not a small correction to the first.

Penalising pairs at two bonds depletes them. With a one-bond interaction the count at two bonds rises with V, and the reason is conservation of a kind: electrons pushed off their nearest neighbours have to go somewhere, and two bonds away is where they go. Put a price on two bonds and that redistribution stops paying, so the count falls — at a reach of a half it is −0.0241 at V = 2, where the one-bond interaction gives +0.0054 there.

That reversal is the mechanism behind everything below, and it is worth noticing that it is not a small effect with a sign. The correlation hole of this ring removes 1.27 pairs from contact and puts 1.11 of them one site away, so the redistribution is nearly complete and where the remainder goes is decided by what the Hamiltonian charges for. Two bonds is the only place left on a ring of six that is not the antipode.

So the two terms of the numerator have opposite signs over the range where the give-back is interesting, and the numerator’s root is where they cancel.

The root is no longer any separation’s free value #

At a reach of a half the numerator vanishes at V = 5.5151. There the change in the pair count at one bond is +0.185 and at two bonds −0.370: one separation still enhanced, the next still depleted, the two contributions 1.0215 and −1.0215.

That is the thing the earlier essay predicted and could not compute. The root of the give-back is not a statement about any separation’s pair count; it is a statement about a balance between two of them, and a measurement of the root tells nothing about either without the other.

The repulsion at which the one-bond count does return to its free value is still computable, and it is a different number: 5.8696 at a reach of a half against the root’s 5.5151. The two quantities coincided in the essay before it because there was nothing else in the numerator, and the coincidence was doing all the interpretive work.

The difference is a third of a unit of repulsion out of five and a half, which is six per cent — small enough to be dismissed and large enough to change what the number means. The same size of discrepancy separated the root from the pole in the earlier essay and that one mattered, because the two differ in kind and not in size. This one is the same: 5.5151 is a balance and 5.8696 is a pair count, and averaging them would be meaningless.

Which side it falls on reverses #

At a reach of a quarter the root is 5.1924 and the one-bond zero 5.0715, so the root is later by 0.121 and at the root the one-bond count is already depleted — −0.152 — with the two-bond count enhanced at +0.606. At a reach of a third the same arrangement holds, with a gap of 0.123.

At a reach of a half the root is 5.5151 and the zero 5.8696, so the root is earlier by 0.355 and the signs are the other way round.

The sign of the offset reverses between a third and a half, and the cause is in the previous figure: whether the two-bond count is enhanced or depleted at a given repulsion depends on how strongly two bonds are being penalised, and the numerator’s root has to sit wherever the two terms happen to cancel. There is no direction to the effect. A measurement of the root, in a model whose reach is not known exactly, cannot be turned into a statement about either separation even in its sign.

The window widens #

Both crossings move later as the interaction reaches further — root from 3.8955 to 5.5151, pole from 4.1595 to 6.2212 — which is the opposite of what adding a repulsive term suggests. The reason is competition: the second-neighbour term and the nearest-neighbour term want the electrons in different places, so a larger V is needed before the nearest-neighbour term wins by enough to change either sign.

And the pole moves further than the root, so the interval between them grows from 0.264 to 0.7061 — nearly threefold. That interval is where the two readings of the give-back disagree most sharply: the numerator has changed sign and the denominator has not, so the quantity is reported as a gain rather than a give-back while the structure beyond contact is in fact costing energy.

There is a reading of that growth worth stating. The essay below found U/2 sitting strictly between the root and the pole at every on-site repulsion, closing as 1/U — a pleasing result that made the two crossings look like two sides of one limit. With the interaction reaching two bonds the gap is three times wider at a fixed U, so whatever U/2 is the limit of, the approach to it is much slower than one reach suggested, and the naive balance point is correspondingly less useful.

The interval the essay before it found was the narrowest available, because it was measured at the shortest reach.

What a measurement of the root would be measuring #

It is worth asking what survives of the root as a quantity, because the answer is not nothing.

The root is still the repulsion at which the give-back changes sign, and the give-back is still what it was: the fraction of the on-site saving that the structure beyond contact hands back. That statement never mentioned a separation. What the earlier essay added was an interpretation — the root is where one bond is at its free value — and that is what has gone.

So the root remains a boundary between two régimes of a real quantity, and it is now a boundary whose position depends on the interaction’s reach as well as on its strength. At a fixed U of eight, changing the reach from nothing to a half moves the root by 1.62 units of V. Doubling U from 6 to 12 moved it by 3.12, from 2.845 to 5.962. So reaching one bond further is worth about half of doubling the on-site repulsion — and it is the lever nobody sweeps, because a nearest-neighbour model is the standard object and its reach is a fixed property of the model rather than a parameter.

That is the useful form of this essay’s finding for anyone reading a published extended-Hubbard number. A crossing located in a nearest-neighbour model is located in a model whose reach was chosen, and the choice is worth as much as a fifty per cent change in the repulsion.

What the second-neighbour term is #

It is a diagonal term, like the other two. A repulsion between electrons on two sites counts occupations rather than moving electrons, so it is diagonal in the configuration basis and costs one more pass over a list of pairs. The solver is exact in exactly the sense it was before.

The pairs are built rather than counted. On a ring of six the sites two bonds apart form six distinct unordered pairs; on a ring of four they form two, because {0,2} and {2,0} are the same pair there. A list built as “site i and site i+2 for every i” would double the interaction on the smallest ring, which is the kind of error that shows up as an answer of the wrong size rather than as a crash, so the list is constructed with a set and its length is checked against the geometry.

A reach of zero reproduces the earlier calculation exactly. Root 3.8955, pole 4.1595, and the root equal to the one-bond zero to six decimals — which is the check that this is an extension of that calculation and not a different one.

The reaches are chosen, not derived. A quarter, a third and a half are the values a 1/r tail would give if the second neighbour were four, three or two times as far as the first; on a hexagonal ring the geometric answer is nearer a half and on a straight chain exactly a half. Nothing here fits a reach to anything, and the essay’s findings are all about the dependence on it.

Where this stops #

A ring of six at half filling, U = 8. It is the smallest system with a separation of two bonds to price at all, and every number is that system’s. The ring’s own geometry decides which pairs are second neighbours, and on a ring of six the “third neighbour” is the antipode, which is not included — so the interaction here reaches two bonds and stops, rather than falling off as a tail. A genuine tail would add a third term and, on the evidence above, a third opportunity for a cancellation.

Half filling and one spin arrangement. The ground state is a singlet at every repulsion used here — which Lieb’s theorem fixes for a half-filled bipartite ring rather than a computation — so nothing above involves a change of spin state, and the filling is the one where the on-site repulsion has the most to do. Away from half filling the pair counts at every separation change for reasons that have nothing to do with the reach, which is why the comparison is made at one filling.

The state is the one the full Hamiltonian gives. V and $V_2$ are in the Hamiltonian, so the wavefunction knows about both, and the pair counts being differenced are that state’s against the uncorrelated one — which is the construction that makes a pair distribution a statement about the state rather than about a subtraction. That is the same construction the essay before it used and it is not the only one available: weighting the on-site state’s correlation hole with a longer-ranged interaction is a different quantity, and this argument has computed that too and found it worth forty per cent.

And the root is found on a grid before it is bisected. Each evaluation diagonalises the whole configuration space, so a bisection started on the full range spends its first several steps walking back to where a scan has already been. The scan the figures draw is the bracket, which is why a search here is a dozen diagonalisations rather than a hundred.

The two régimes, named #

Putting the four reaches side by side, the give-back’s behaviour falls into two arrangements and the boundary between them is inside the range a real interaction occupies.

At a short reach the two-bond count is enhanced at the root. The second-neighbour penalty is too weak to reverse the redistribution, so by the time V is large enough for the numerator to vanish the one-bond count has gone past its free value into depletion and the two-bond count is still rising. The numerator vanishes because a depletion at one bond is being paid for by an enhancement at two.

At a long reach it is depleted. The penalty reverses the two-bond count from the start, so the numerator vanishes while the one-bond count is still enhanced, and the enhancement is paying for the depletion.

Between the two the offset passes through zero, which means there is a reach at which the root and the one-bond zero coincide again — where the earlier essay’s interpretation is accidentally restored. On the sweep it is between a third and a half. That is worth naming because it is the kind of coincidence a model fitted to data could land on: a reach chosen to reproduce a measurement, a root that then reads as a pair count, and no indication that the reading holds only at that reach.

A one-term numerator is a special case wearing a general name #

The habit: when a quantity’s zero has an interpretation, check whether the interpretation belongs to the quantity or to the number of terms it happens to have.

The root is where the structure beyond contact contributes nothing is a correct reading of a numerator with one term in it, and it is not a reading of the give-back. It survives no enrichment of the model at all: the moment the interaction reaches a second separation, the root becomes a balance and the reading becomes false in a way that keeps the same words. Nothing in the earlier essay flagged it, because a one-term expression gives no hint that its interpretation is about the count of terms.

The corollary is about which quantity to report. The repulsion at which one bond returns to its free value is still a well-defined number and it is still the interesting one — it is just not the root any more. Reporting both, and their difference, costs one more search and removes the ambiguity entirely, which is the same repair the essay before it made when it separated the root from the pole.

Who has computed what, and when #

The extended Hubbard model with a nearest-neighbour repulsion is Hubbard’s with one more term and dates from the 1970s; adding a second-neighbour term is standard in studies of charge order on chains and on two-leg systems. The pair distribution, the give-back and the separation of root from pole are this argument’s own, from the two essays before it.

What is computed here is the second-neighbour term in the exact solver, the numerator’s two contributions across four reaches, the root’s position relative to the repulsion at which one bond returns to its free value, and the widening of the interval between root and pole.

The numbers worth carrying are +0.185 and −0.370 — two pair counts at the repulsion where their contributions cancel, neither of them at its free value.

Still open: a third term, and the ring of eight #

The obvious open question is a genuine tail. This essay adds one term and finds that the root becomes a balance between two separations; a 1/r tail on a ring of six adds a third, at the antipode, and the numerator then has three terms whose signs need not be two against one. Whether a three-term numerator has one root or three is a question the same scan answers, and it decides whether the root of the give-back names a number at all in a model with a realistic interaction. The cost is one more pass over a list of pairs and the same dozen diagonalisations per reach.

The nearer question is the size, and it is blocked by the solver rather than by the argument. The scaling of the root–pole separation with the on-site repulsion was measured on one ring, and a second size would say whether the coefficient near two is a property of the model or of six sites — but the exact solver refuses above six sites by construction, because a dense diagonalisation of the configuration space grows as its cube and eight sites at half filling is a space of 4,900 states against 400. A ring of four is the size that is available, and it is a smaller change than the one the question wants: its second-neighbour pairs are the antipodal ones, so the term being added means something different there. That is the honest state of it — the question is well posed, the system it wants is one the solver will not build, and the system the solver will build answers a slightly different question.