Sixteen sites pin the Fermi level

The essay that looked for a moment of the band to replace the gap ended on a level nobody had asked about. On the frustrated net’s sixteen-site torus, half filled, the two arrangements that trade the lead at a contrast of 3.790 both have their highest occupied level at exactly δ − 2, at every contrast from 1 to 8. The state carrying it has no amplitude at all on any lowered site, so the contrast only shifts it, and it is the same in both arrangements. The gap reads it and the binding sums up to it, and neither can see a difference in it.

The explanation offered was a local one. Such a state exists only if the raised sites’ own connections support a combination with eigenvalue −2 whose amplitudes cancel at every lowered neighbour — a property of how the raised sites cluster, and so, the essay suggested, a peculiarity of the triangular net. It left two questions: whether every half-filled arrangement has such a state at its Fermi level, and whether its presence is what made half filling so easy to search on that net.

Both have answers now, and neither is about the net.

Every arrangement, not two #

Two arrangements were inspected. There are 12,870 ways to raise eight of sixteen sites, and on a matrix of sixteen it is cheap to diagonalise all of them — the same exhaustive move that showed three arrangements were not enough to read a rule off.

Every one carries a state at δ − 2 with no weight on a lowered site. For 11,676 it is a single state; for the other 1,194 it is a plane of them. And at every contrast on a grid of thirty-one, from a quarter to eight, it is the highest occupied level of every arrangement. Not most; not the two contenders and their relatives; all of them.

So the level that looked like a clue to the winner was never information about any arrangement — in the same way that the second and third moments carry none at a fixed composition, but for a different reason. It is the same level, at the same energy, carried by the same kind of state, in the arrangement that binds best and the one that binds worst. That sharpens the moment essay’s reading of the gap considerably. The gap at the Fermi level — the quantity first offered as a measure of how well a structure binds — is the distance from the highest occupied level to the lowest empty one, and if the first is δ − 2 everywhere, the gap of every arrangement on this torus is a statement about its lowest empty level and nothing else — which is the level that contributes nothing to the binding. The lag between the gap and the winner was always going to be a lag.

Nine levels where the band says there should be none #

The reason is a dimension count, and the count is set before any site is raised.

A state with energy δ − 2 and no weight on the lowered sites is a vector $v$ on the raised sites with $(A + 2)v = 0$ on the whole net, where $A$ is the net’s adjacency: the rows for the raised sites say it is a −2 combination of their own graph, and the rows for the lowered sites say it cancels at each of them. So it is a vector in the kernel of $A + 2$ that happens to vanish on eight given sites. The kernel and the space of vectors living on the raised sites are two subspaces of a sixteen-dimensional space, the second of dimension eight. If the first has dimension nine, they must share a direction.

On the four-by-four torus it has dimension nine. The triangular net’s band is

$$\varepsilon(\mathbf{k}) = 2\left[\cos k_1 + \cos k_2 + \cos(k_1 + k_2)\right],$$

which runs from 6 at the centre down to a floor of −3 at the two corners where both components are a third of a turn. At −2 it has saddle points, at the midpoints of the zone’s edges, where the infinite net’s density of states has a logarithmic peak — a feature of the band that a finite torus can only sample, and how it samples it is the whole of what follows. A torus of $L_1 \times L_2$ sites samples the band at $k_i = 2\pi m / L_i$, and reaches the corners only if both sides are multiples of three. Four is not. The sixteen-site torus samples in quarter turns, its lowest level is −2, and nine of its sixteen sample points land exactly there — the three edge midpoints and six points between them. The band’s floor is missing, and the level just above it is crowded with more than half the net.

Two consequences follow directly from $A + 2$ having no negative levels. The first is the cancellation the earlier essay thought had to be arranged: if $v$ lives on the raised sites and is a −2 combination of their own graph, then $v^{\mathsf T}(A + 2)v = 0$, and for a matrix with no negative levels that forces $(A + 2)v = 0$ everywhere — the lowered rows cancel for free. Dropping them from the condition changes the pinned space for none of the 12,870 arrangements. The second is that $H - (\delta - 2) = (A + 2) - 2\delta P_L$, a matrix with no negative levels less one of rank eight, can have at most eight levels below δ − 2. The census adds the other half, which the inertia argument does not give: there are always exactly six or seven above it, so the pinned level is always the highest one filled.

On every other torus here at least one of the two conditions fails. Six by four reaches down to −2.73 and has nine levels at −2 against twelve raised sites. Eight by four reaches −2.83 with thirteen against sixteen; six by six reaches the floor itself, with fifteen at −2 against eighteen; eight by eight has twenty-one against thirty-two.

What the state looks like #

A dimension count says a state must exist without saying what it is, and drawing one makes the cancellation concrete.

On the winner at a contrast of 4 the pinned state alternates in sign around the raised sites, and every lowered site sits between raised neighbours whose amplitudes cancel. That is the familiar shape of a localised state on a frustrated lattice: on an infinite triangular net, an alternating ring of raised sites around a lowered one does exactly this. On this torus the picture is the same, but the reason it exists is not the clustering — the count of dimensions guarantees that some combination of this kind fits on any eight sites at all, whether or not they look clustered.

A plane of them, and four exact contrasts #

The 1,194 arrangements with a plane of pinned states are a second surprise inside the first.

Two pinned states and eight electron pairs leave a choice. Below a contrast of 2, every one of these arrangements has seven levels above δ − 2, so the eighth pair goes into one pinned state and the other is the lowest empty level: a gap of exactly zero, at every contrast in that range, in 1,194 arrangements that are each half filled with an even number of electrons. Then each loses it, at one of four contrasts — 750 at exactly 2, 288 at 16/7, 144 at 5/2 and the last 12 at 8/3 — where one of its own higher levels comes down through δ − 2 and both pinned states fill. From 8/3 on, none has a zero gap.

The contrasts are rational, which says they are crossings of levels whose energies are simple functions of δ on a torus this symmetric. That is a fact about sixteen sites as much as the pin is, and it is the kind of fact that makes a small census look more structured than the material it stands for.

Nowhere else is it universal #

A pinned state is not impossible on a larger torus; it is just no longer forced.

On six by four, 85 of 300 seeded half-fillings carry a state pinned at δ − 2, and in 56 it is the highest occupied level. On eight by four, 113 and 36. On six by six — the only torus here that reaches the band’s floor — 41 of 150 and 4. On eight by eight, a handful and none. Where the pin survives it is the localised ring the drawing shows, an accident of a particular arrangement; it is no longer a constraint on all of them.

And on a torus where $A + 2$ has negative levels the cancellation stops being free. Among seeded half-fillings of six by four there are raised sets whose own graph has a −2 combination that leaks onto a lowered neighbour — a state the sixteen-site torus could never have produced.

What the pin was doing to the search #

The second half of the question was whether the pin is what made half filling easy. On sixteen sites the frustrated net’s half-filled composition was a single basin, reached by all two hundred steepest ascents at the smaller contrast, as uniquely easy as the bipartite net’s two-colouring.

On a six-by-four torus it is not. Sixty ascents at a contrast of 1 end in four distinct optima, and at every contrast from 1 to 6 there are between four and nine. At a contrast of 1.5 the best of them is reached by four starts in sixty. Half filling on the frustrated net is not easy at this size.

The census over compositions had found half filling the easiest composition on both nets, and the frustrated net’s single basin was the strongest form of that. That does not prove the pin made it easy at sixteen. Two things change between the tori — the pin disappears and the net gets larger — and the square net’s own census went from two basins to twelve between sixteen and thirty-six sites, which is a growth no level on the frustrated net was needed for. What it does establish is that the single basin was not a property of the frustrated net at half filling; it is one of several things that were true of sixteen sites and are not true of twenty-four, and the pin is the one with a mechanism that reaches every arrangement at once.

The gap’s lag without the pin #

There is one finding of this argument that the pin does not explain away, and it is the one it started from.

On the six-by-four torus no level is pinned to make the gap a statement about the lowest empty level alone, and the best-binding arrangement found is still often not the one with the widest gap. Among the 21 distinct arrangements the searches found, the best binder changes near a contrast of 3.0 from one with 44 unlike bonds to one with 48 — the same kind of change the sixteen-site winner makes from 30 to 32 near 3.79 — and from there to a contrast of 6 it is never the arrangement with the widest gap, on sixty consecutive contrasts a twentieth apart. At a contrast of 4 its gap is 5.032, against 5.040 for an arrangement binding two parts in a hundred thousand less per site. Counting the stretch between 1.05 and 1.85, where two arrangements with 44 unlike bonds trade places, the best binder is not the widest gap on 77 of 101 contrasts. The disagreement between the gap and the winner, which the essay that first measured it found on sixteen sites, survives the loss of the pin. On sixteen sites it had a ready mechanism — the gap was reading a level that is the same everywhere and a level that contributes nothing. On twenty-four that mechanism is gone, and the lag is still there.

How the census was taken #

The sixteen-site torus is the one every essay in this argument used: one orbital a site, nearest-neighbour hopping of one, site energies of +δ on the eight raised sites and −δ on the eight lowered ones as in the first of the mixtures, two electrons in each of the eight highest levels. Every one of the 12,870 arrangements is diagonalised directly at each contrast. The pinned space is the null space of $M^{\mathsf T} M$, where $M$ is $A + 2$ restricted to the raised columns, so it is computed from the adjacency alone and does not depend on δ; the version without the lowered rows is the null space of $(A_{RR} + 2)^2$.

The other tori are built by the same rule, and each one’s levels are checked against $\varepsilon(\mathbf{k})$ at its own wavevectors to $10^{-9}$. The draws on them are seeded random half-fillings. The six-by-four landscape is the same steepest ascent the sixteen-site censuses used — exchange a raised and a lowered site while the binding improves, take the best exchange each time — from sixty seeded starts at each of eleven contrasts, with optima grouped by binding to eight decimals. Arrangements related by a symmetry of the torus bind identically and are counted once.

What the numbers rest on: every arrangement on sixteen sites has a pinned space of dimension one or two; its highest occupied level is δ − 2 at every contrast checked; it has seven levels above when the space is a line and six or seven when it is a plane, with the gap exactly zero precisely where a plane straddles the Fermi level; the four crossing contrasts are 2, 16/7, 5/2 and 8/3 to six decimals; dropping the lowered rows changes nothing on sixteen sites; no other torus here forces the pin by counting; and on six by four half filling has more than one basin at every contrast. The refusal is the leaking state: on six by four a raised set must turn up whose own −2 combination does not cancel on a lowered neighbour, or the lowered rows would never have been tested by anything.

Where twenty-four sites still stop #

Twenty-four is not large. A six-by-four torus is not square and samples the band in sixths along one side and quarters along the other; its own grid has its own crowded levels, nine at −2 and others elsewhere. It removes the pin without removing finite size, and every number on it is a number about twenty-four sites.

The landscape there is a search, not a census. C(24, 12) is 2.7 million arrangements, and sixty ascents a contrast find the optima they find. A best arrangement no start reached would change the winner and the gap comparison, and nothing here rules one out.

One orbital, no repulsion. Every level here is a one-electron level. A zero gap in a half-filled one-electron band is an invitation for the repulsion between electrons to open one, and the 1,194 zero-gap arrangements are exactly where a model with repulsion would part company with this one.

A census of the instrument #

The habit worth carrying is to ask, of any regularity found on a small system, whether it is a property of the system or of the instrument’s resolution.

The δ − 2 level was found in two arrangements and read as a fact about them. The census took a few seconds and showed it was a fact about all of them, and the dimension count showed why: the sixteen-site torus is a sampling of the triangular band that happens to miss its floor and pile nine levels onto the next one. That is a property of the grid of wavevectors, which is to say of the choice of sixteen, and it reaches into every arrangement on the torus in the same way a coarse histogram reaches into every distribution drawn with it.

What survived the change of torus is what had a reason independent of the grid: the gap and the winner disagree on twenty-four sites as they did on sixteen. What did not survive — the universal pin, the zero gaps, the single basin — were things the grid was doing.

Still open: the lag’s mechanism on twenty-four sites, and a torus that reaches the floor #

The obvious open question is the lag itself. On sixteen sites it had a mechanism — the gap read a pinned level and an empty one — and on twenty-four that mechanism is gone while the lag remains. The level ledger the moment essay built, accumulating the binding difference level by level, can be run on the two leading arrangements at a contrast of 4 on six by four, and would say whether the deciding levels are still just below the Fermi level and the gap’s levels just above it, or whether without the pin the two readings share a level and disagree for some other reason.

The nearer question is a torus that reaches the band’s floor. Six by six is the smallest, at thirty-six sites, and it is the size the square net’s census was run at; a frustrated landscape there, with the band’s floor present and its own crowded levels at −2 short of half, would separate the two things that change between sixteen and twenty-four — losing the pin and gaining sites — which this comparison cannot.