The give-back that turned into a saving

Half of it is given back at one bond settled a long-running question about correlation energy by refusing to decompose the energy and decomposing the hole instead. Where a Hubbard correlation energy is a hundred per cent on-site by construction — the interaction is zero everywhere else, so no structure anywhere else can contribute — the change in the number of electron pairs at each separation is a property of the wavefunction, and can be weighted afterwards by any interaction at all.

Weighted by one that reaches a neighbour, the answer was that the enhancement at one bond gives back forty-four per cent of the on-site saving. Not a correction: nearly half.

That number is a variational estimate. The wavefunction it was computed from minimises an interaction that lives on one site, and it was then handed a different interaction to be judged by. That essay says so and names the repair — solve the extended model outright, with the neighbour term in the Hamiltonian rather than in the weighting, and the difference between the two is the second-order term the bound is missing.

Six sites is small enough to do that exactly. The difference is not a second-order term.

One more diagonal term #

A nearest-neighbour repulsion counts electrons on sites rather than moving them, so it is diagonal in the same occupation basis the on-site term is diagonal in. Adding it to the exact solver is one more pass over the edges of the ring and nothing else: the whole configuration space is still written down and solved, nothing is truncated, and the levels that come out are the levels.

Two wavefunctions can then be compared rather than two calculations. The first is the variational one — the ground state of the on-site model, with its hole taken against the free ring. The second is the ground state of the model that has the neighbour term in it, with its hole taken against the same free ring. Both are priced with the same interaction. The only thing that differs is which Hamiltonian the state came from.

At small neighbour repulsions the two agree, and they agree in the way a variational argument says they should. At $V = 1$ the weighted give-back is 10.99 per cent and the solved one is 10.92; at $V = 2$ they are 21.98 and 21.50. The estimate is high by a few hundredths of what it estimates, which is exactly the behaviour of a bound whose error is second order in a small quantity.

At V = U/2 the sign changes #

At $V = 4$ — the value the nearest interaction uses, half the on-site repulsion — the weighted give-back is 43.97 per cent and the solved one is −45.57 per cent.

A negative give-back means the neighbour term is not taking anything back. The state that minimises the whole interaction has arranged its electrons so that the neighbour repulsion adds to the saving rather than subtracting from it: the pair count at one bond has gone from an enhancement to a depletion, so a positive interaction there is now multiplying a negative number.

The direct cause is visible in one number. The nearest-neighbour opposite-spin pair distribution is 1.52135 in the on-site-only state — a fifty-two per cent enhancement, which is the structure the whole give-back was computed from — and 1.07361 once the neighbour term is in the Hamiltonian. The structure being priced has very nearly been removed by the price.

The bound is not loosening quadratically #

The variational energy is available without any extra work, because the neighbour term is diagonal: the on-site-only state’s expectation of the full Hamiltonian is its own energy plus $V$ times its nearest-neighbour pair count.

The shortfall is 0.025, 0.117 and 0.962 at $V = 1$, 2 and 4. Doubling the neighbour term multiplies it by 4.6 and then by 8.2. A second-order error would give four both times.

So the expectation — that the difference between the two routes is the second-order term the bound is missing — is right about its origin and wrong about its size. There is a second-order term, it is visible at $V = 1$ and $V = 2$, and by $V = 4$ it is not what the discrepancy is made of.

What is at V = U/2 #

The reason the arithmetic stops being second order there is not about the arithmetic.

Turning the neighbour repulsion up drives the ring towards a charge-ordered state: electrons paired on alternate sites, avoiding their neighbours entirely, which costs the on-site repulsion and saves the neighbour one. On a ring of six there is no broken symmetry — the ground state is a superposition of the two staggered arrangements, so every site’s occupation is exactly one and no occupation difference reports anything. The correlation function does report it, and the alternating structure factor is what a finite ring can say instead.

That structure factor rises fastest at $V = 4.5$, which is 0.5625 times the on-site repulsion. At $V = U/2$ the double occupancy has already gone from 0.039 to 0.190 and the structure factor from 0.149 to 1.466, so the state at the interaction the variational estimate priced with is halfway through the crossover.

That is the worst place on the axis to evaluate one state’s expectation of another state’s Hamiltonian, and it is where the variational estimate’s interaction was defined — not by design but by the ordinary convention that a nearest-neighbour term is about half an on-site one. The convention lands on the crossover.

Far past it the ring is unambiguously ordered: at $V = 8$ the double occupancy is 0.4922 against a maximum of a half, and the pair distribution at one bond is 0.0310. Every electron is paired, on alternate sites, and the on-site repulsion is being paid in full to avoid the neighbour one.

The ordered state is a fourth thing this ring can be #

The state past the crossover deserves naming, because it is not one of the three kinds of correlation usually separated.

A half-filled band is not always a metal because an on-site repulsion can hold one electron on each site and open a gap without any distortion. A third way to be an insulator is a band that is full because the sites are inequivalent. The state at large neighbour repulsion here is neither: every electron is paired, on half the sites, and the other half are empty. Its double occupancy is 0.4922 against the on-site-repelled state’s 0.039 — more than twelve times as many doubly occupied sites, in a model whose on-site repulsion is unchanged and is being paid.

That is worth stating plainly because it inverts the usual reading of a double occupancy. On the on-site axis a small double occupancy means a strongly correlated state; here a state that is more strongly interacting has a much larger one, because the interaction it is avoiding is somewhere else. A diagnostic read off one axis of a two-parameter model says nothing about the other axis, and the same caution applies to every correlation hole: the hole is a picture of what the electrons are avoiding, and which thing that is comes from the Hamiltonian rather than from the picture.

The consequence for the give-back is direct. Between the two states the pair count at one bond moves from an enhancement to a depletion, and there is no reason for a quantity that changes sign to have a well-defined perturbative estimate anywhere near where it does so.

The ring’s size is doing something here that is easy to mistake for physics. Six sites can hold three pairs on alternate sites and that is the only way to be fully ordered, so the ordered limit is a single configuration rather than a thermodynamic phase, and its double occupancy of 0.4922 is a hair under the exact one half that configuration would give. On a longer ring the same limit is approached the same way and the crossover sharpens; on an infinite one it becomes a transition with a location rather than a fastest-rising point. Nothing in the give-back argument depends on which of those it is — the sign change is in the pair distribution at one bond, and that is a local quantity — but the crossover language is a finite-ring statement and is used here deliberately in place of the word this state would earn on a lattice.

What was computed, and how #

Every energy is the lowest eigenvalue of a Hamiltonian built in the full occupation basis of three up and three down electrons on six sites — four hundred configurations, nothing truncated, nothing selected. The fermion signs are kept when an electron hops past another of the same spin, and the check that they are kept is that the matrix comes out symmetric, since the eigensolver refuses one that is not. The trace is separately required to count doubly occupied configurations and nothing else.

The pair distributions are expectation values in the ground state’s own coefficients, normalised by the site occupations, so an uncorrelated ring gives exactly one at every separation. That is the control and it is checked rather than assumed: every hole here is a difference against the free ring, and if the free ring’s distribution were not one, the differences would not be holes.

The variational energy uses the fact that the neighbour term is diagonal. That is not a convenience — it is what makes the comparison exact on both sides. A variational estimate computed by a second approximate route would leave the difference between the two ambiguous between the physics and the method.

The refusal is the direction of the bound: the variational energy must lie above the exact ground state at every neighbour repulsion. A variational energy below an exact one would mean the exact solver is not exact, and it is the one statement here that no amount of physical reasoning could rescue.

Both routes are checked against each other by rediagonalising one Hamiltonian and comparing its ground state against the other route’s, and that matters more here than usual: the two quantities being compared differ by a few per cent at small neighbour repulsion, which is well inside what a careless comparison could hide.

Where the model stops #

There is also nothing here about temperature. Every state above is a ground state, and a ring at a repulsion halfway through its crossover has excited states within a few tenths of the gap — which at room temperature would be populated and would average the pair distribution over configurations on both sides of the change of sign. The perturbative estimate was of a ground-state expectation and is compared with one, so the argument is closed on its own terms; what it does not license is carrying the number to a measurement made at a temperature.

Six sites is six sites. A crossover on a ring of six is a crossover and not a transition, and where it sits will move with the size — so the coincidence between $V = U/2$ and the steepest rise is a statement about this ring, not a universal boundary. What does not move with the size is the mechanism: a neighbour repulsion strong enough to reorganise the wavefunction cannot be priced on a wavefunction that has not been reorganised.

The neighbour term reaches exactly one bond and stops. A real Coulomb tail does not, and the variational hole was priced with a truncated tail as well, getting 46.91 per cent. Solving that model exactly would need terms at every separation, which is arithmetically no harder here and is a different model with a different crossover.

And nothing here is a molecule. This is a lattice with one orbital a site, hopping between neighbours, and a lattice hole is one bond wide by construction. The distinction between short-range correlation and the near-degenerate kind, which two kinds of correlation draws carefully, still cannot be drawn spatially here.

The generalisation #

The shape of the error is worth separating from the model it was found in.

A variational estimate of what a term costs is computed by taking a state optimised without that term and evaluating the term in it. That is a bound, it is always available, and it is nearly free. What it cannot see is the response — the state rearranging to reduce what it is being charged. When the term is weak the response is second order and the bound is good. When the term is strong enough to change what the state is, the response is the whole answer, and the bound is not a slightly high estimate of the cost. It is an estimate of the cost of not responding, which can have the wrong sign.

The same asymmetry appears from the reference side: what counts as correlation depends on what it is measured against, and a mean field cannot get out of the way of a term it was not told about. The version here is sharper only because the exact answer is available, so the size of what the bound cannot see is a number rather than an argument.

The practical form: a perturbative price on a correlated wavefunction is worth quoting only with the strength at which the state stops being that wavefunction. For this ring that strength is around half the on-site repulsion, and the variational headline number was computed at it.

Who found it, and when #

The extended Hubbard model, with a neighbour repulsion beside the on-site one, is older than most of the results quoted for the plain one, and the charge-ordered state at large neighbour repulsion is its standard feature. What is not standard is reading the crossover as a warning about where a perturbative estimate may be evaluated, which is what it is used for here.

The extended Hubbard model with an on-site and a nearest-neighbour term dates to the nineteen-sixties and its one-dimensional phase diagram — a spin-density-wave region and a charge-density-wave region separated by a line near $V = U/2$ at strong coupling — has been argued over since the seventies, with the order of the transition still not entirely settled and the boundary bending away from $U/2$ at weak coupling. The crossover found here at $0.5625\,U$ on six sites is that boundary seen through a very small window.

The observation that a correlation hole is a property of the wavefunction and the correlation energy a property of the interaction is older and is usually attributed to the electron-gas literature of the nineteen-fifties, where the same decomposition is done with a Coulomb tail throughout.

What does not seem to be standard is the specific arithmetic here: that the sign of a give-back computed variationally can be opposite to the sign of the same quantity computed self-consistently, at an interaction strength nobody would flag as extreme.

Still open: where the crossover moves, and a zero give-back #

The obvious open question is the boundary itself. The crossover’s location can be measured on rings of four and six directly, and the two points would say which way it moves with size — which decides whether $V = U/2$ is near the boundary for the reason found here or by an accident of six sites. Two points is not an extrapolation, but the direction is what the reading needs and two points give it.

The nearer question is the one the sign change makes available. If the solved give-back passes through zero somewhere between $V = 2$ and $V = 4$, there is a neighbour repulsion at which the structure beyond contact contributes exactly nothing to the energy — a state whose correlation energy is genuinely a hundred per cent on-site even though the interaction is not. Finding that value by bisection costs a handful of diagonalisations, and what makes it worth having is that it is the only point on the axis where the headline claim about a Hubbard model is true of a model that is not a Hubbard model.