A gap where band theory says there cannot be one

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.

Seven 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.

The hole that is not repulsion

Where the repulsion stops being a correction. Past a certain strength the system develops a charge gap that band theory says is not there, and the double occupancy has fallen far enough that moving an electron means creating a doubly occupied site. That regime is a different subject from the one this essay is about, and this site treats it elsewhere.

The insulator band theory cannot see

The charge gap of a half-filled four-site ring against the on-site repulsion, with the one-electron gap of the same ring beneath it. The one-electron answer is zero at every U — the repulsion is not in that matrix — and the exact gap reaches 5.99 in units of the hopping by U = 8, still rising.

The weak-repulsion end magnified. The one-electron gap is exactly zero at every point along this axis and the exact gap is not zero anywhere above U = 0 — so there is no threshold at which the insulator appears, and nothing about the band picture changes as it does.

And the strong-repulsion end, where the gap has become linear in U with a slope of one. Moving an electron onto an already-occupied site costs U, so the gap is U at large repulsion — a quantity the band picture has no term for at any coupling.

The third way to be an insulator

The second kind: the charge gap of a small Hubbard system against the repulsion, with the band-theory answer of zero marked at the left. Nothing in the structure has changed; the gap is the cost of putting two electrons on one site.

A hundred lines and no way to sort them

What the repulsion is doing to the system while it is doing this to the spectrum: opening a gap that band theory has no term for. The repulsion at which the spectrum stops being sortable is inside the range where that gap is opening.

Half of it is given back at one bond

The other quantity the same repulsion produces, and one that is not a contraction of anything: the gap. A gap is a difference of two exact energies and is not weighted by an interaction after the fact, which is why it transfers where a correlation energy does not.

The give-back that turned into a saving

A ring of six at an on-site repulsion of eight, as the nearest-neighbour repulsion is turned up. The dashed line is the interaction the variational hole was priced with.

The change in pair count at each separation on a ring of six, at a neighbour repulsion of two. The two wavefunctions agree closely: at this strength the neighbour term really is a perturbation of the on-site problem.

The same comparison at a neighbour repulsion of four, which is the interaction the variational estimate actually used. At one bond the two bars now point in opposite directions.

A sign change is not always a zero

The solved give-back against the neighbour repulsion. The curve is drawn broken at the second crossing because it is an asymptote rather than a passage through zero, and the points nearest it run off the band in both directions.

The two halves of the give-back plotted as themselves rather than as their ratio. Each crosses zero, and they do it in different places.

Both parts of the give-back at each of the three locations. At the root one column is zero and the other is not; at the pole they are the other way round; at V = U/2 neither is.