A particle in a box the alloy made

An equal binary alloy has a gap that depends on how the two components are arranged, even though its mean and its variance cannot. Three arrangements of one composition: the ordered one’s gap is exactly $2\delta$, the segregated one’s exactly $2\delta - 4$, and the random one’s has no closed form and falls as the chain grows — from 0.185 at 512 sites to 0.088 at 32,768 — because a longer chain holds a longer run of like sites, and a longer run is a narrower sub-band.

That was an argument about where a band edge is. It said nothing about what is at the edge, and the two questions have different answers: a defect level, which is a state living on one site and its neighbours, and a band edge is ordinarily a state living everywhere.

The states at this edge are the first kind, and what they are is unusually clean.

What a run is, taken on its own #

A run of $L$ like sites is a piece of pure material, and a piece of pure material with ends is a chain of $L$ sites. Its levels are the ones computed in the band limit: $2\cos(k\pi/(L+1))$, and its states are

$$\psi_k(j) \propto \sin\!\left(\frac{k \pi j}{L+1}\right),$$

which is the particle in a box, arriving here because a run really is one.

The quantity that says how much of a chain a state occupies is the participation ratio, $(\sum \psi^2)^2 / \sum \psi^4$ — one for a state on a single site, $L$ for a state spread evenly over $L$. For a sine profile the two sums are elementary:

$$\sum \sin^2 = \frac{L+1}{2}, \qquad \sum \sin^4 = \frac{3(L+1)}{8},$$

so the participation ratio is $\tfrac{2}{3}(L+1)$, with no parameter in it at all. A box state of a run of thirteen sites occupies 9.33 of them, and that is a prediction rather than a fit.

What the alloy’s gap states are #

Take a 400-site chain, half its sites raised by $\delta = 4$ and half lowered, arranged at random, and look at the eight levels nearest the middle of the gap. For each, find where its weight is centred and how long the run of like sites is at that place.

Six of the eight sit on the closed form to within three per cent, keep essentially all of their weight inside their own run, and are box states of a box the shuffle happened to make.

The measurement makes no reference to the disorder at all. Nothing here fits a localisation length, and nothing needs to know how the sites were arranged: the run is read off the chain, the closed form follows from its length, and the state matches.

Across five chains that is 38 of 40 gap states, with a worst departure of 3.1 per cent.

The two that are not #

The pair at −2.2128 and −2.2153 occupies 8.3 sites on a run of six, which is 1.78 times what a box of that size holds, and keeps only 65 per cent of its weight inside.

They arrive as a pair, split by 0.0025. That is the shape of the answer: the state is shared between two runs close enough to each other that a state on one reaches the other, and two coupled boxes give a symmetric and an antisymmetric combination split by twice the coupling. It is the two-impurity problem, arriving unbidden because a random arrangement eventually puts two runs near one another.

So the answer to how localised are these states is: as localised as their run, until there are enough runs that they start to find each other, at which point they stop being defect levels and start being a defect band. On a chain of 400 that has happened to one pair out of eight; on a chain of 32,768 with the same composition it will have happened to a great many more, and the sub-band edge computed from the longest run alone will have acquired structure the longest run does not describe.

Why a run and not a cluster #

There is a question hiding in the phrase the run the state sits on, and it is worth answering before the closed form is trusted anywhere.

A run of like sites in a one-dimensional chain is unambiguous: walk left until the site energy changes, walk right until it changes, and the stretch between is the run. That is a definition with no parameter in it. It is also a definition that only exists in one dimension — in two, the sites of one kind form a connected region of some shape, and the box it makes is no longer a length but an outline whose levels depend on its whole geometry.

So the closed form here is not a general fact about disordered alloys. It is a general fact about one-dimensional ones, and it holds because the only property a one-dimensional region has is its length.

What survives in higher dimensions is the argument rather than the formula: a connected region of one component is a piece of that material with a boundary, its levels are the levels of that region, and a level near the middle of the gap belongs to a large region. The gap edge is still set by the largest region and the states there still live on it. What is lost is the ability to write down where the level is without solving the region, which is exactly what makes the one-dimensional case worth having: it is the case in which the whole account is arithmetic.

There is one more thing the definition buys. Because a run is found from the site energies alone, the two halves of every row in the table above come from different objects — one from a list of site energies, the other from an eigenvector — and neither computation can see the other. A stabilisation measured from somewhere is this collection’s standing warning about quantities computed against references chosen after the fact; there is no such freedom here, because the reference is an integer read off the chain.

The contrast does nothing, which was not expected #

A gap state is a bound state, and a bound state in a deeper well ought to be more tightly bound. Turning the contrast up from $\delta = 2.5$ to $\delta = 8$ takes the gap from 1.11 to 12.10 — a factor of eleven — and the weight the state keeps inside its own run goes from 91.3 per cent to 91.0.

It does not move. The prediction was wrong and the reason is worth having.

The barrier confining the state is not the depth of the well. It is the ends of the run, where the next site is of the other kind, and a site of the other kind is off-resonance by $2\delta$ — which is already large compared with the hopping at the smallest contrast that opens a gap at all. Making it larger changes a barrier that was already effectively complete.

What the contrast does change, slightly, is the participation ratio: 9.04 sites at $\delta = 2.5$ and 9.25 at $\delta = 8$, creeping up towards the box value of 9.33 from below. A state in a finite well is slightly compressed relative to the infinite one, and a deeper well compresses it less. The whole effect is two per cent.

The control, which had to be the ordered arrangement #

A participation ratio of nine on a chain of four hundred is only evidence of localisation if the same measurement returns something large for a state that is not localised — and the obvious control does not work.

Every state of a disordered one-dimensional chain is localised. Taking a level from the middle of a sub-band of the same random alloy and measuring it returns a small number too, which proves nothing about the gap state in particular.

So the control is the ordered arrangement of the same composition: the same site energies, the same $\delta$, in alternating order rather than shuffled. Its band-edge state occupies 133.7 sites of 400 — a third of the chain, and growing with the chain, which is what an extended state does.

That contrast is the check, and it is the one that makes the participation ratio a measurement rather than a formula: the arrangement decides, and it decides by a factor of fifteen.

What this says about a large alloy #

The random alloy’s gap, computed from the longest run, leaves open what a very large alloy is best described as. The answer supported here is:

A split band with a defect band in the gap. The two sub-bands are the two components’ bands, narrowed — and narrowed rather than shifted, which is the distinction a band’s width being a count of neighbours makes exactly: a run of six has fewer neighbours available at its ends than a bulk site does. Between them are box states of runs — one level per run per box quantum number, at energies set by the run’s length through the closed form — and as the chain grows the runs become numerous enough that neighbouring ones couple and the levels smear into a band of their own.

That is not a description anybody draws, and the reason is that it is three descriptions at once: a band structure, a set of defect levels, and a coupling between them. Each of the three is something computed separately, and the alloy is the case where all three are present in one system with no parameter distinguishing them.

Why the gap closes as slowly as it does #

The box picture explains more than the states; it explains the rate at which the gap shrinks, and the rate is the distinctive part of the whole phenomenon.

A run of $L$ like sites carries levels at $2\cos\bigl(k\pi/(L+1)\bigr)$, and the one nearest the gap is the run’s own band edge, at $2\cos\bigl(\pi/(L+1)\bigr)$. Expanding that gives

$$2 - \frac{\pi^2}{(L+1)^2} + \dots,$$

so a run’s edge approaches the infinite-run limit as the inverse square of its length. The gap of the whole chain is set by the longest run it happens to contain, so the gap approaches its limiting value at that rate too.

Now ask how the longest run grows. In a random chain at equal composition, a specified run of $L$ like sites occurs with probability $2^{-L}$ per starting position, so the longest one in a chain of $N$ sites is near $\log_2 N$ — a length that grows with the logarithm of the chain rather than with any power of it.

Putting the two together: the gap’s excess over its limit falls as $1/(\log_2 N)^2$. That is an extraordinarily slow decay, and it is the reason the measured gap behaves the way it does. Between 512 sites and 32,768 the chain grew by a factor of 64 and the longest run only from about nine sites to about fifteen, so the predicted shrinkage of the gap’s excess is somewhere between 2.5 and 3.1 depending on how the run length is counted. The measured factor is 2.10.

Two points cannot separate that from a weak power law, and the agreement in magnitude is the most the argument can claim on this evidence. What the argument does supply, and no fit could, is a reason for the gap to be so reluctant to close: doubling the chain adds one site to its longest run, and one site to a run of fifteen is a small change to a quantity that already enters as an inverse square.

It also says the limit is approached and never reached. A chain of $10^{23}$ sites has a longest run near seventy-six, and its gap is still above the segregated arrangement’s by a term of order $\pi^2/76^2$. A random alloy in the thermodynamic limit has the segregated arrangement’s gap, and any real sample has a slightly larger one, by an amount that depends on how big the sample is — which is the sort of size dependence that is easy to mistake for a preparation effect.

What is quoted, and what is computed #

Nothing is quoted. There is no material here: a chain of 400 sites, half of them raised by a stated amount, arranged by a seeded shuffle. Every number is computed from that.

The closed form $\tfrac{2}{3}(L+1)$ is derived from two elementary sums and checked against the participation ratio of an actual eigenvector, which is the only sense in which it is verified — the two are computed by different routes and must agree.

The runs are found by walking the chain from the state’s centroid, which is a property of the site energies alone and knows nothing about any eigenvector. The state and the run are measured independently, which is what makes their agreement evidence.

What this cannot say #

It is one dimension. Localisation in one dimension is special: every state is localised at any disorder, which is why the control had to be the ordered arrangement rather than a state from the middle of a band. In three dimensions weak disorder leaves states extended and the whole question changes shape.

There are no electrons in it. These are one-electron levels, and which of them is occupied has not been asked. A gap state that is filled and a gap state that is empty are very different things for a material, and what a filled band is and is not is where that question lives.

And the runs are a caricature of an alloy. A real solid solution has short-range order — a tendency for like or unlike neighbours — so its runs are not the runs of a fair shuffle. That changes the distribution of run lengths and therefore the distribution of gap levels, and it does not change the closed form for any one of them.

What was checked #

Most of the gap states are box states of their run, stated as a majority of a census over five chains rather than as an observation about one.

And for those, the closed form needs no fitting — a worst departure of 3.1 per cent between a participation ratio computed from an eigenvector and one computed from an integer.

Some exceed it, which is checked as well: a census in which every state matched would mean the shuffle had been too kind to produce the shared pair, and the shared pair is half the finding.

The ordered arrangement’s band-edge state occupies more than eight times as many sites and more than a twentieth of the whole chain — the control, without which a small number would be arithmetic rather than evidence.

And the weight inside the run does not depend on the contrast, checked as a bound on the variation — two points across a factor of three in $\delta$ — because the prediction that it would rise was wrong and a check written to the prediction would have failed.

Still open: the gap’s density of states in closed form #

The obvious open question is the distribution rather than the individuals. A run of length $L$ appears in a random chain with a known probability, and each run contributes levels at energies the closed form gives — so the density of states inside the gap is computable in closed form from the run-length distribution, with no diagonalisation anywhere. Comparing that against the levels an actual diagonalisation puts in the gap would be a strong test of the whole picture, and it would say what fraction of the gap states the box description accounts for on a chain far too large to solve.

The nearer question is the coupling that made the shared pair. Two runs share a state when they are close enough, and close enough is a distance already measured for two impurities — the splitting against separation is drawn above. Working out at what chain length the typical gap between neighbouring long runs falls below that distance would give the size at which the defect levels become a defect band, as a number rather than as an expectation, and it is the size at which the longest-run account stops being the whole story.