The error, against the repulsion it is an error about

One of the figures on when repulsion is in the model: Hubbard systems small enough to diagonalise exactly: the singlet a one-electron model cannot find, the coupling between two spins, and a gap where band theory says there is none.

Six essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

A mean field cannot get out of the way

Four electrons on four sites, solved exactly and in a spin-paired mean field, at seven repulsions. The exact energy saturates — the electrons stop meeting, so a stronger repulsion costs them almost nothing — and the mean field’s rises linearly for ever. The fourth column is the error divided by the energy it is an error in.

The prediction against the measurement, for three systems. The slopes are measured between the two largest repulsions computed; the closed form is n²/4N and has no diagonalisation in it. The largest disagreement is 0.0285, and it is in the direction the argument requires, because the exact state’s own double occupancy has not quite reached zero.

How often two opposite spins sit on the same site, for four electrons on four sites. At no repulsion the two descriptions agree exactly, because the exact state is a single determinant there and the mean field is looking for one. From then on the exact state’s double occupancy falls toward zero and the mean field’s does not move at all.

The reference decides the correlation

The energy each reference misses, against the repulsion, for four electrons on four sites. Below U = 2 the two curves are the same curve — the unrestricted search returns the restricted answer, and the two definitions of the correlation energy agree to the last bit a double holds. Above it they part, and one grows while the other falls.

The spin of the two references against the repulsion. The restricted one is a singlet at every point, exactly. The unrestricted one is a singlet until the instability and then is not, reaching ⟨S²⟩ = 1.99 — where a singlet is 0 and a triplet 2. The reference that reports almost no correlation is very nearly half a triplet.

What each description does as the repulsion rises. Both references are trying to get the double occupancy right: a restricted determinant cannot reduce it at all and a polarised one can, by putting the two spins in different places — and the correlation energy each is assigned is the difference between what it manages and what the exact state does.

The warning a cheap calculation gives

Thirty-two systems. Along the bottom is how far the mean field’s own symmetry breaking has moved between the reference and the target — a number available before any exact calculation. Up the side is how wrong the transferred correction turns out to be.

The polarisation against the modulation, at four repulsions. Every curve is flat, then bends, then reaches exactly zero — and the field it does so at rises with the repulsion.

For three systems: the rate at which the mean field’s error grows with the repulsion, measured between the two largest repulsions computed, beside the count n²/4N that predicts it. The prediction uses the electron number and the site number and nothing else — no energy, no diagonalisation, no wavefunction. So the size of the gap this essay is carrying from one system to another is cheap to estimate, which is exactly why the size is not the useful thing to know about it.

The half of the square a ring of four cannot show

The mean field’s spin polarisation against the on-site repulsion, at half filling and no site-energy modulation, for three systems. Two have a threshold. The third does not.

The one-electron levels of each system, with the two either side of half filling marked. Only one of the three has them at the same energy.

A correlation correction computed at a repulsion of six on a chain of four, carried to every other repulsion. The dashed line is where the broken solution appears.

The second number is the error, rearranged

The composite’s error against the change in the correlation energy, at every point on both axes. They lie on the diagonal, and they lie on it exactly.

The scatter under the spin-polarisation diagnostic, with the pair that breaks it circled. The two points agree in the diagnostic and differ by a factor of forty-six in the error.

Nine candidates on a logarithmic scale, with the floor the statistic itself imposes. Only one reaches it, and it is the one that costs the answer.

Five failures in five different places

Every system on the square, with each cheap diagnostic’s failing pair joined. No line shares an end with another.

How many failures each system is implicated in. Every one appears in exactly one.

Each failure with the two systems that produce it.