A sign change is not always a zero

With a nearest-neighbour repulsion in the Hamiltonian, solved exactly on a ring of six, the give-back estimated variationally does not merely shrink. It reverses. At a neighbour repulsion of two the estimate and the solved answer agreed to a few hundredths; at a neighbour repulsion of four the estimate said forty-four per cent was given back and the solved answer said forty-six per cent was gained.

A quantity that is positive at one end of an interval and negative at the other has crossed zero somewhere in between, and that crossing has a meaning. A give-back of zero is a state whose correlation energy is genuinely a hundred per cent on-site even though the interaction is not — the one point on the axis where the headline claim about a Hubbard model is true of a model that is not a Hubbard model. Finding it should cost a handful of diagonalisations.

It costs about forty, and they do not find one crossing. They find two, a quarter of a repulsion unit apart, and only the first is the thing the sentence describes. The interval that paragraph named happens to contain the first and not the second, so its own arithmetic would have come out right — which is the least reassuring way for a method to work.

A ratio has two ways to change sign #

The give-back is not a quantity. It is a quotient of two quantities, and it is natural to write it as one because on the interval where it was first computed the denominator never comes near zero.

The numerator is the price of everything beyond contact: for each separation past the first, the change in the number of pairs sitting at that separation, multiplied by whatever the interaction charges there. On this model only one separation carries a price at all, because a nearest-neighbour repulsion is zero at two bonds and beyond. So the numerator is one term — the neighbour repulsion multiplied by the change in the pair count at one bond.

The denominator is the on-site price with its sign turned round: the on-site repulsion multiplied by the double occupancy the correlated state has given up relative to the free ring. That is the saving the fraction is a fraction of, and calling the result a percentage presumes it is a positive number.

That presumption is the same one the essay that chose a reference took apart in a different setting: a correlation energy is a difference from something, and quantities defined as differences from a chosen state carry that state’s arbitrariness into every number derived from them. A percentage carries it twice, once in each half of the quotient.

Neither presumption is safe once the neighbour term is strong. The state that minimises the whole interaction does not merely deplete its nearest-neighbour pairs; past a certain strength it starts putting double occupancy back, because two electrons on one site cost $U$ once while two electrons on adjacent sites cost $V$ at every one of the two bonds a site has on a ring. When the arithmetic of that trade turns over, the denominator turns over with it.

The double occupancy is the quantity the hole that is not repulsion established as the honest object here — a property of the wavefunction that can be counted without deciding first what the interaction is. What that essay did not have to consider, because it worked at one interaction, is that the same count can move in either direction once a second interaction competes with the first.

Where each part actually crosses #

Plotted separately the picture is unambiguous. The numerator falls through zero at $V = 3.8955$: at that neighbour repulsion the pair count at one bond is exactly what it was in the free ring, so the interaction beyond contact is multiplying nothing and the correlation energy really is a hundred per cent on-site. That is the hoped-for sentence, and it is true at one point on the axis.

The denominator falls through zero at $V = 4.1595$, a quarter of a unit further along. There the correlated state holds exactly as much double occupancy as the uncorrelated one — the on-site repulsion has pushed some out and the neighbour repulsion has pushed exactly that much back — so the saving the give-back is a fraction of has vanished. The numerator at that point is $-4.404$ against a scale of $19.18$: not small, not near zero, nowhere close to contributing nothing.

Both crossings are visible in the ratio, and in the ratio they look alike. A curve that goes positive-to-negative at the first and negative-to-positive at the second is what a root followed by a pole produces, and it is also what two roots produce. Nothing in the plotted give-back distinguishes them.

The three locations, side by side #

Set the two crossings next to the value the give-back was first priced at, and the arithmetic that separates them is one table wide.

The reading that matters is the third column against the fourth. At the root the price beyond contact is zero to the precision of the search and the on-site price is $4.203$ — a fraction with a vanishing top and a healthy bottom, which is what contributes exactly nothing means. At the pole the on-site price is zero and the price beyond contact is $-4.404$, which is nearly a quarter of the largest value it takes anywhere on the scan. Something is contributing a great deal there; it is simply being divided by nothing.

Between them sits $V = U/2$, where both columns are ordinary numbers and the give-back is an ordinary negative fraction. That is the value the exact solution was computed at, and nothing about the number there was wrong — the point here is only that the crossing inferred from a sign change is two crossings, and that the one described is not the one a search is most likely to land on.

Three intervals, and one of them refuses #

The proposal was to find the crossing by bisection, which needs an interval whose endpoints disagree in sign. Two crossings on one interval is exactly the arrangement that destroys that condition: the function leaves on the side it arrived on.

The proposal named $[2, 4]$, and on that interval the proposal works exactly as written. The endpoints disagree in sign, the search converges on $3.8955$, and the answer is the root. Nothing needs correcting about it.

It works because $4$ happens to fall between the two crossings. That window is $0.264$ wide, and the interval’s upper end has to land inside it: at $4.1595$ the pole arrives, and past it the ratio returns to the sign it started with. Widen the search to $[2, 4.2]$, or to $[2, 4.5]$, or to $[2, 5]$ — which is what anybody unsure of the answer does — and the give-back is $+0.215$ at one end and positive at the other. There is no bracket at all. A bisection asked to start there stops before it begins, and the honest reading of that refusal is this interval contains an even number of crossings, which is not the reading anybody takes from a routine that declines to run.

Move the interval the other way instead, to $[3.9, 4.5]$, and the endpoints disagree again; the search converges on $4.1595$ and reports it in exactly the same form — a neighbour repulsion, to six figures, at which the solved give-back is zero. It is not zero there. It is undefined there, and the two answers sit a quarter of a unit apart in a quantity whose scale is measured in units.

So the proposal was right and was right by luck, and the luck is a quarter-unit window whose existence it had no way to know about. What makes that worth writing down rather than filing as a numerical inconvenience is that the wrong answer is the one a careful person is more likely to get. A bracket is usually narrowed by looking at a plot, and the plot’s most dramatic feature — the place the curve goes vertical — is the pole.

What is happening at the pole #

The second crossing is not an accident of the bookkeeping. It sits inside the ring’s charge-ordering crossover, which the exact solution had already located and drawn.

At the root the alternating structure factor is $1.143$ and the ring is barely ordered. At the pole it is $2.201$, one-and-nine-tenths times as much, on the steepest part of the rise. That is what a vanishing denominator means physically: the neighbour repulsion has driven the ring far enough towards a charge-ordered arrangement — electrons paired on alternate sites — that the double occupancy is back where the free ring had it.

So the pole is a location in the model rather than a defect of the ratio, and it is the more interesting of the two crossings as a statement about the ring. What it is not is a repulsion at which anything contributes nothing.

Both crossings close on the half #

One repulsion is one system, and the whole argument turned on $V = U/2$ being the interaction it had priced with. On a ring of six at $U = 8$, that value is $4$ — which is neither crossing, and sits between them.

That is not a coincidence of one repulsion. At $U = 6$, $8$, $10$ and $12$ the root sits at $0.4741$, $0.4869$, $0.4934$ and $0.4968$ of the on-site repulsion, rising towards a half from below; the pole sits at $0.5304$, $0.5200$, $0.5140$ and $0.5103$, falling towards a half from above. The naive location is the limit of both and is neither of them at any repulsion where somebody would compute.

The gap between them closes as the reciprocal of the repulsion. Multiplied by $U$ the separation is $2.03$, $2.11$, $2.06$ and $1.94$ — one number across a factor of two in the repulsion, which is what an effect proportional to $t^2/U$ looks like on a lattice whose hopping is the unit. So the ambiguity this essay is about is a finite-repulsion effect that never quite goes away, narrowing as $2t/U$ around a location that is exact only in a limit no calculation is done at.

What was computed, and how #

Everything here is one function evaluated at forty-two neighbour repulsions. Each evaluation is an exact diagonalisation of the full configuration space of six sites at half filling — four hundred configurations, no truncation, no self-consistency — followed by a pass over the ground state to count pairs at each separation.

The searches are grid-bracketed before they are bisected, which is what makes the count forty-two rather than two hundred. A bisection started on $[2, 5]$ spends its first several steps walking back to where the scan has already been, and on a system where each step is a second and a half that is the difference between arithmetic and an afternoon. The scan the figures draw does the bracketing; the bisection only refines.

Nothing here is variational and nothing is fitted, which is the standing condition on every number in this corner of the site and the reason the smallest many-electron calculation is done on six sites rather than sixty. The whole configuration space is written down, so a disagreement between two of these numbers is a disagreement about arithmetic rather than about which approximation was used.

Both crossings are then checked against the thing they are supposed to be. At the root the numerator is zero to within one part in a thousand of its own scale and the denominator is not; at the pole those hold the other way round. A search that returned a point where both were small would have been narrowed onto a coincidence, and the check that would have caught it is the one that never fails.

Where the model stops #

A ring of six is small, and two of the quantities here are known to be sensitive to that. The charge-ordering crossover is a crossover rather than a transition because a finite ring has no broken symmetry to speak of, so its location is a steepest point rather than a boundary, and the pole is pinned to that location rather than to anything sharp.

The scaling of the separation is read from four repulsions on one ring, which is enough to say like $1/U$ and not enough to say the coefficient is two rather than nine-fifths. What would settle it is the same four repulsions on a ring of eight, which is a Hilbert space of four thousand nine hundred rather than four hundred and is a different afternoon.

The pair counting itself is exact and is the calculation where the electrons are without subtracting built, so nothing in the decomposition is an estimate; what is small is the system it is evaluated on.

And the whole construction depends on a nearest-neighbour interaction, which is why the numerator has one term. An interaction reaching two bonds would have two, and could in principle vanish by cancellation rather than by the pair count at one bond coming back to its free value — a root that is not a statement about any single separation.

The generalisation #

The specific claim is about an extended Hubbard model on a small ring. The habit it argues for is not.

A great many quantities in electronic structure are reported as fractions, shares and percentages: how much of a binding energy is dispersion, how much of a barrier is correlation, how much of a gap is exchange. Each is a quotient, each has a denominator that somebody chose, and each is quoted as though the denominator were a constant of the problem rather than a computed quantity with a sign.

Where the denominator can change sign — and a correlation energy, a stabilisation, a saving relative to a reference can all change sign — the share does not merely become large. It becomes meaningless in a specific way: it changes sign, and it changes sign in the same direction and with the same appearance as a genuine cancellation in the numerator. A reader given the share alone has no way to tell which happened.

There is a second failure mode hiding in the same arithmetic, and it is worse because it produces a plausible number rather than an absurd one. Just off the pole the denominator is small but not zero, so the ratio is large but finite — and a share reported as three hundred per cent of the correlation energy comes from beyond contact reads as a striking result rather than as a division by a quantity that happens to be near zero. The give-back here passes through $-262$ per cent and $+632$ per cent at neighbour repulsions a tenth of a unit apart. Neither is a measurement of anything; both would print.

That is the same shape a mean field cannot get out of the way found in a different quotient, where a ratio’s denominator collapsed and the numerator was blamed. The diagnosis is always the same and it is always available: look at the two parts.

The cure is cheap and it is one that keeps coming back. Report both parts. A figure showing a numerator and a denominator that each cross zero, in different places, is a picture nobody misreads; a figure showing their quotient is a picture that hides which of the two events occurred.

Who found it, and when #

The extended Hubbard model with a nearest-neighbour term dates to the 1960s, and the charge-ordering crossover at $V \approx U/2$ in one dimension has been studied since Bari’s work in 1971 and refined by many people since; the fact that it is not exactly $U/2$ at finite coupling, and drifts, is standard in that literature.

What is not standard, because it belongs to a decomposition invented for this analysis rather than to the model, is that the give-back has a pole there. That follows from writing the correlation energy’s structure as a fraction of the on-site part, which is a choice about presentation and not about physics, and it is the kind of thing that gets found by trying to compute the number the previous paragraph promised.

The general point — that a ratio’s sign change is a root or a pole and that plotting the ratio destroys the distinction — is older than any of it and belongs to no field. It is worth restating because the arithmetic that hides it is written fresh every time.

Still open: the ring of eight #

The obvious open question is the ring of eight. The separation’s scaling with the on-site repulsion is measured on four points at one size, and one more size would say whether the coefficient near two is a property of the model or of six sites. The Hilbert space is twelve times larger, which puts a single point at something under a minute rather than under two seconds, so a four-point scan is an hour rather than a minute — expensive, and bounded.

The nearer question is the numerator with more than one term in it. An interaction reaching two bonds gives the numerator two contributions with opposite behaviour, and the root then need not correspond to any separation’s pair count returning to its free value: it can be a cancellation between one bond’s enhancement and the next bond’s depletion. Whether such a root exists on this ring, and whether it sits before or after the pole, is the same forty diagonalisations with one more term in the price — and it decides whether the correlation energy is entirely on-site here is a statement about a state or an artefact of an interaction with only one place to live.