Three lines, then a hundred

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.

Eight 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.

More bands than there are orbitals

The test that works until it does not. The contrast between a fundamental and a satellite is large while the repulsion is small and shrinks as it grows, so the assignment is unambiguous exactly where nobody needs it to be — and the boundary is a property of the gap rather than of the repulsion.

The spectrum at no repulsion and at eight, drawn as sticks at the energy each final state costs to reach and with the intensity the matrix element gives it. Three lines become a hundred, on a molecule with six orbitals, and the strongest line is no longer where an orbital was.

The number of final states carrying any intensity at all, against the repulsion, with the orbital count marked. It crosses the orbital count somewhere below U = 1 and does not come back.

A hundred lines and no way to sort them

How many times stronger the weakest fundamental is than the strongest satellite, against the repulsion. It starts at nearly twenty-four and falls to a little over one, and the marked point is where the other test fails too.

The spectra themselves at two repulsions, with the fundamentals drawn tall and the range they span shaded. At the smaller repulsion the satellites are all outside the shaded band; at the larger one they are inside it, and the ones inside carry more than half the satellite intensity.

The quantity underneath both tests: how the removal weight is distributed over the final states, and how the distribution changes as the repulsion grows. A spectrum is sortable while this is concentrated and unsortable when it is not.

The boundary belongs to the gap

The repulsion at which the intensity test fails, against each system’s own one-electron gap. Three points in a line, and one that is not on it because it has no gap.

The original picture: how many times stronger the weakest fundamental is than the strongest satellite, against the repulsion, on the ring it was measured on. Everything above is that curve computed on four systems and compared.

Where the intensity test’s numbers come from: the weight each pole carries, which is how much of one electron it is. A fundamental carries nearly all of one and a satellite a fraction, until the fractions stop being small.

A contrast with a closed form

The contrast against the repulsion for eight systems with two electrons each, out to a repulsion sixteen thousand times the hopping. The dashed line is the factor of two the test needs.

The departure from the limit, multiplied by the repulsion, for every system. A first-order approach makes this a constant.

For each system: the contrast at the largest repulsion computed, the extrapolated limit, and how far that limit sits above the threshold.

A satellite that never loses its place

The intensity contrast against the repulsion, for a six-site ring and chain at each filling. Below the dashed line the intensity test no longer separates them; the half-filled systems cross it and the third-filled ones do not.

Every system and filling, with its gap, its boundary and the share of the removal weight its satellites carry.

The boundary against the gap, with a line joining the two fillings of each lattice. The heavy line is the pair that goes the wrong way.

A ratio of exactly one is a tie

The contrast on a ring of six at three fillings. Two electrons settle well above the factor of two; half filling falls through it and lands on exactly one from a repulsion of sixty-four upward. The hollow marks are the repulsions where the number is a tie.

The eight strongest removal lines at half filling and a repulsion of a thousand and twenty-four. Lines three and four are the two halves of one level.

The number of lines carrying weight at half filling, and how many neighbouring pairs in the weight ranking are exactly degenerate. The spectrum is contracting into multiplets.

The number the tie got right

The contrast against the on-site repulsion at three fillings, on a ring of six and on an open chain of six.

Exactly degenerate neighbouring pairs in each removal spectrum: up to thirty on the ring, none at all on the chain.

The chain’s extrapolated contrast against electron count, with the factor of two the distinguishability test needs.

The limit of one is a parity

The six strongest removal lines of the half-filled chain of six, each followed across the repulsion and labelled by the energy it tends to.

The chain of six’s contrast less one against the repulsion, with the bisected exchanges of rank.

The removal energies of the strongest lines at very large repulsion, against the levels of one particle hopping on a chain of the same length.