No moment sees the winner change

The gap follows the winner late followed two arrangements of a half-filled band on a triangular net of sixteen sites. Half the sites carry a site energy of +δ and half −δ; one orbital a site; a hopping integral of one between neighbours. The arrangement that binds best at small contrast makes thirty unlike bonds — raised sites next to lowered ones — out of the forty-eight the net has. At a contrast of 3.790 it is overtaken by one with thirty-two. The gap at the Fermi level, which picks the winner in eighteen of twenty cases elsewhere, keeps favouring the old winner until 4.849. For a whole unit of contrast the better binder has the narrower gap.

That essay named the natural next candidate. The gap reads two levels at the edge of the occupied band, and the binding is all eight occupied levels. If what decides the switch lies deeper in the band, a quantity that reads the whole band should see it on time. The quantities that read the whole band are its moments: the mean of the levels, of their squares, of their cubes and so on, each a single number summarising the density of states. They are what a structure’s neighbours fix and what a great deal of the theory of what holds a solid together is written in.

So the question is plain: does any moment of the density of states change its preference between the two arrangements at 3.790?

Three moments that are not about the arrangement #

Much of the answer comes from arithmetic before anything is diagonalised, because a moment is a trace. The k-th moment per site is the trace of the k-th power of the Hamiltonian divided by the number of sites, and the trace of a power counts closed walks: every way of leaving a site and returning in k steps, each step either a hop to a neighbour (weight one) or a stay (weight the site’s energy).

The second moment counts walks of two steps. Out and back along one of the six bonds each site has, or stay twice. So $\mu_2 = 6 + \delta^2$ for every arrangement whatever, since every site’s energy squared is $\delta^2$. That is the identity that fixes the second moment at the coordination, with the site energies added.

The third moment adds walks round a triangle, which are the same for every arrangement, and walks that carry site energies: out and back along a bond with one stay, or three stays at one site. Summed over the net, the first kind is a multiple of the sum of all the site energies and the second is the sum of their cubes — and at half filling, eight at +δ and eight at −δ, both are zero. So $\mu_3$ is the same for every arrangement at this composition too.

The fourth moment is the first that can differ. Its walks include going out along a bond and back with one stay at each end, which carries the product of the two site energies: $+\delta^2$ if the bond joins like sites and $-\delta^2$ if it joins unlike ones. Every other term in the fourth power is either the same for every arrangement or a multiple of the zero sum. Counting the orders in which the two stays can fall along such a walk, from either end of each bond, and turning one like bond into an unlike one, gives $\mu_4 = C(\delta) - 8\delta^2 U/n$, where U is the number of unlike bonds and C(δ) is common to every arrangement.

The identity holds to 10⁻⁹ for eighty seeded half-filled arrangements with between twenty and thirty unlike bonds, and for every basin the census finds at contrasts of 3, 3.5, 4, 5 and 6. The first moment of the band that can tell two arrangements apart is the count of unlike bonds, exactly — not correlated with it, not approximately it, but the same ranking with a factor in front.

And the count of unlike bonds is the predictor that an earlier census found failing: on this net at half filling the winner below 3.790 makes thirty unlike bonds where thirty-two were available. The fourth moment therefore prefers the thirty-two-bond arrangement at every contrast, and is wrong about the winner at every contrast below the switch. The deep-level reading’s first candidate has already been measured, under another name, and failed.

Twenty-one moments and not one sign change #

That leaves the higher moments, which count longer walks and carry products of three, four and more site energies. Nothing in the trace argument fixes their sign, and each is a polynomial in δ whose terms could trade places somewhere between 1 and 8. The obvious test is to compute each moment of each contender from its sixteen levels, across the range, and look for a change of sign in the difference.

There is none. At twenty-nine contrasts from 1 to 8, every moment from the fourth to the twenty-fourth is smaller for the thirty-two-bond arrangement than for the thirty-bond one, at every contrast. The winner changes once in that range, and no moment of the density of states changes with it.

This is a stronger failure than the gap’s. The gap changed its verdict late, a unit of contrast after the binding did. The moments never change theirs. Each one, taken alone, names the thirty-two-bond arrangement as different from the thirty-bond one in the same direction from a contrast of 1 to a contrast of 8, while the binding goes from preferring the thirty-bond arrangement by 0.011 per site at a contrast of 2 to preferring the other by 0.007 at 6.

That is not a paradox, and it says something precise about the binding. The binding at half filling is not a moment. It is the integral of the energy over the lower half of the density of states, and a cut-off at the Fermi level is a function no finite polynomial reproduces. Any expansion of the binding in moments has coefficients of both signs, so a sum of terms each of fixed sign can change sign where the terms change relative size. What the moments show is that no single term carries the switch. It lives in the balance between many of them.

How many moments the switch needs #

That suggests the obvious measurement: build an account of the band from its first so many moments, and ask where that account puts the switch. The standard way to do this is the Gauss rule for the density of states. For each m it gives m energies and m weights whose first 2m moments are exactly the band’s — the most any m-point account can match. Filling the rule to half, from the top node down, gives an m-point estimate of the binding. With as many points as the spectrum has distinct levels, the rule is the spectrum, and its binding is exact.

The two contenders have ten and eleven distinct levels: their sixteen levels fall into degenerate sets that the net’s symmetry leaves together. So the rules to compare run from three points, which match the first six moments, to eleven, which are exact for both.

The three-point account, which sees moments up to the fifth, never changes its winner: it names the thirty-two-bond arrangement throughout. The four-point account changes at 1.378 and changes back at 5.551. Five points give 2.097 and 5.189; six give 1.344 and 3.107; seven give 2.663 and 4.314. Below eight points no account of the band puts a change within 0.15 of the true switch. Eight points, matching sixteen moments, put it at 3.980, inside the interval where the gap lags. Nine points give 3.890, ten 3.687 — now on the early side — and eleven, which are the spectrum, give 3.790 exactly.

The approach is not steady. It comes from both sides, it picks up and drops spurious crossings far from the switch, and it reaches the right neighbourhood only when the account is within three points of being the whole spectrum. On a band of sixteen levels, that is an account nearly as detailed as the levels themselves.

The curves make a second point the crossings hide. The eight-point account puts the switch within 0.19 of the right place, but at a contrast of 2 it overstates the binding difference by a factor of three, and the six-point account has the wrong sign there. Getting the switch nearly right is not the same as getting the binding nearly right. The eight-point curve crosses near 3.8 by a coincidence of its own shape, not by resolving what the exact curve resolves.

Where in the band the difference is made #

The original proposal was that the switch is decided by the deep levels. The levels themselves can be asked directly: add up the binding difference one occupied level at a time, starting from the most strongly bonding and working up to the Fermi level, and watch the running total.

The deepest level favours the thirty-bond arrangement at every contrast, and so does the running total after two and after three levels. Then the fifth level swings it to +0.13 per site in favour of the thirty-two-bond arrangement, and the sixth and seventh bring it back almost to zero. What is left at the Fermi level is −0.0018 at 3.5, +0.0025 at 4.3 and +0.0049 at 5. The binding difference that decides the winner is a remainder, twenty-five to seventy times smaller than the swing the running total makes on the way.

So the deep levels do not decide it; they favour the loser at every contrast, including the ones where the other arrangement wins. Nor does the edge in the sense the gap reads. The eighth level, the highest occupied one, is exactly δ − 2 in both arrangements at every contrast from 1 to 8. It is carried by a state that has no amplitude at all on any lowered site, so the contrast only shifts it, by the same amount in both. It adds nothing to the difference. The gap is the distance from that level to the lowest empty one, so the gap difference between the two arrangements is entirely the difference in their lowest empty levels — a level that contributes nothing to the binding at all.

That is a sharper account of the lag than the one the gap essay could offer. The gap and the binding share no level that differs between the two arrangements. The binding’s verdict is settled by the sixth and seventh levels, just below the Fermi level, and the gap’s by the ninth, just above it. There was never a reason for them to change at the same contrast.

How it was computed #

The model is the sixteen-site triangular torus with one orbital a site, nearest-neighbour hopping of one, and site energies of +δ on the eight raised sites and −δ on the eight lowered ones, as in the census it continues, which found half filling the easy composition to search on both nets. The two contenders are the census’s winners on either side of the switch, identified by their unlike-bond counts and checked against the census’s own binding at 3.5 and at 4. Each arrangement’s sixteen levels come from a direct diagonalisation, with every eigenpair’s residual checked; the binding is two electrons in each of the upper eight levels, per site.

The moments are sums of the levels’ powers per site. The trace identities for the second, third and fourth moments are checked against those sums for every basin at five contrasts and for eighty arrangements drawn with a fixed seed. Sign changes are sought on a grid of twenty-nine contrasts from 1 to 8 in steps of a quarter.

The m-point accounts are Gauss rules for the discrete density of states: the three-term recurrence of its orthogonal polynomials by the Stieltjes procedure, then the eigenvalues of the resulting tridiagonal matrix as nodes and the squares of their first components as weights. Each rule matches the first 2m moments and is filled from its top node down until half the weight is used. Changes of winner are located by bisection between grid points.

The table carries the other way to read the truncations: whether each names the right winner at the three contrasts that matter. Three points name the thirty-two-bond arrangement at 3.5, where it loses. Four points name the thirty-bond arrangement at 4.3 and 5.5, where it loses. Five and seven points are wrong inside the lag; six points are wrong below the switch. From eight points on, all three verdicts are right.

What must hold, and is checked: that $\mu_2 = 6 + \delta^2$ and $\mu_3$ is common to every basin and every sampled arrangement; that $\mu_4 + 8\delta^2 U/n$ is common to all of them, with more than one value of U present so that the identity is tested rather than assumed; that the contenders’ second and third moments are equal and every higher one to the twenty-fourth keeps its sign on the grid while the binding difference changes sign exactly once; that no account below eight points puts a change within 0.15 of the switch and the eight-point one puts it inside the lag; that the running total over levels reaches more than twenty times its final value at each of three contrasts; and that the highest occupied level of both contenders is δ − 2 with no weight on a lowered site at every contrast. The refusal is the eleven-point rule, which is the spectrum and must reproduce the exact binding to 10⁻¹⁰ and put its only change of winner at 3.790, and the three-point rule, which must not — if it did, the moments above the fifth would carry nothing.

What sixteen sites cannot say #

The net is small, and the count of distinct levels is what makes the truncations end. A sixteen-site band has at most sixteen levels, and these two have ten and eleven distinct ones, so a rule of eleven points is necessarily exact. On a larger net the density of states is closer to continuous, a Gauss rule of a given size is a coarser account of it, and the question of how many moments a switch needs would be asked of a band that never runs out. Nothing here says whether sixteen moments would still land within 0.2 on a net of a hundred sites.

The moment signs are tested on a grid. Twenty-nine contrasts from 1 to 8, a quarter apart. A moment difference that changed sign and changed back between two grid points would be missed, and so would one that changed sign below a contrast of 1, where the census that found the switch was never run.

Two arrangements, not every arrangement. The contenders are the census’s winners either side of 3.790. Whether some other pair of arrangements, on this net or another, has a moment that tracks its switch is not tested. What is established is that for this switch — the one the gap was shown to lag — no moment does.

One electron, one orbital, no repulsion. Every statement is about a tight-binding band. The binding of a real alloy includes the repulsion between electrons and the relaxation of the lattice, and neither changes a trace identity but both change which arrangement wins.

The winner is a remainder #

The gap was a predictor that read the wrong two levels. The moments are predictors that read the whole band and still cannot see the switch, because the switch is not in any one of them: the fourth is the count of unlike bonds, the ones below it cannot tell arrangements apart, and every one above it keeps its preference over the whole range. The binding changes sign because a sum of many terms of fixed sign changes the balance among them, and only an account that nearly resolves every level carries enough of that balance to place the change.

A quantity that summarises a band — its gap, its count, any one of its moments — can be a good predictor of which arrangement binds best and still be structurally unable to say where the answer changes, because near a change the binding difference is a remainder of cancelling contributions from nearly every occupied level. It is the same shape of result as a surface that is not a count of broken bonds, where the bonds that survive change what the missing ones cost, and it is the reason a mixture does not bind as the average of its ends and twelve basins appear where two were expected: the landscape is decided by small differences between large sums, which is exactly the regime a summary cannot reach.

And the surprise inside the calculation is that the level at the Fermi level itself belongs to neither argument. It is the same in both arrangements, a state confined to the raised sites that the contrast shifts rigidly. The gap, which reads that level, and the binding, which sums up to it, are both blind to it — and the two levels on either side that they do read are different levels.

Still open: the level that sits at δ − 2, and a band that does not run out #

The obvious open question is the highest occupied level. It is exactly δ − 2 in both contenders at every contrast, carried by a state with no amplitude on any lowered site. A state like that 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, which is the frustrated net’s peculiarity. Whether every half-filled arrangement on this net has such a state at its Fermi level, and whether its presence is what makes half filling easy to search here, is a count over the census’s basins that has not been made.

The nearer question is size. A six-by-four torus has twenty-four sites and a band with more distinct levels than any rule compared here. Running the same comparison there would say whether the eight-point account’s 3.980 was a property of the physics or of a sixteen-level band that a rule of eleven points exhausts.