A decay that keeps slowing down

Relaxing a chain bond by bond instead of holding its alternation uniform gives something a uniform alternation cannot: a length. The distortion is largest at the end, decays inwards, and the distance it decays over runs from 1.51 bonds at a gap of 1.150 to 5.29 bonds at a gap of 0.143. Fitted against the gap, that is a power of −0.60, and the argument says −1.

The discrepancy could mean that the received relation is missing a term at the sizes real oligomers have. But an exponent fitted over a window has a different explanation for a fitted power that comes out shallow: the fit was averaging a local slope that was still moving, and the number belonged to the window rather than to the system.

That reading is checkable, and the check has an obvious form. Relax longer chains, fit over more of the profile, and watch the exponent walk up to −1.

It does not walk up to −1. Fitted the same way, over the first twelve bonds, the exponent on a chain of three hundred and twenty is −0.476 — further from −1 than the chain of ninety-six’s −0.550. Three times the chain, and the number moved the wrong way.

The profile is not an exponential #

A healing length is fitted by taking the excess alternation over the bulk value, putting it on a logarithmic axis and drawing a straight line through it. That procedure returns a number whatever the data does. It is a length only if the data is a straight line.

That is not pedantry here. An end binds a state where the middle does not, and the profile this fit is drawn through is the same end’s influence seen in the structure rather than in the spectrum. Drawing four relaxed chains of three hundred and twenty that way, the curves are visibly concave — every one of them, at every elastic constant, over every stretch of bonds.

Concave downwards on a logarithmic axis means a decay that slows down as it goes. Measuring the local decay rate — the derivative of the logarithm, taken as a centred difference at each bond rather than as a slope over a stretch — turns that into a number that can be watched:

Still rising at the fifty-eighth bond. A profile with one length would give the same number in every row.

So the question has no answer, because the object it asks about does not exist. There is no healing length on this chain. There is a decay whose rate depends on where it is measured, and the twelve-bond number is one of the many values it takes.

The local rate lies on a straight line against the reciprocal of the distance from the end. A rate of the form

$$\frac{1}{\lambda} + \frac{p}{b}$$

is the derivative of the logarithm of $b^{-p}\,e^{-b/\lambda}$, so the excess is a power law multiplying an exponential rather than an exponential. Two numbers come out of that fit, and both are useful.

$\lambda$ is the rate the decay is heading for, and it is a length with no window in it — the reciprocal of the intercept, which is the rate a bond infinitely far from the end would decay at. On the chain above it is 17.50 bonds, against 5.38 from a twelve-bond window and 12.93 from a ninety-six-bond one.

$p$ is the power in front, and it is not a nuisance parameter. It rises from 0.412 at the widest gap to 0.659 at the narrowest, and that rise is the mechanism behind the whole difficulty: a power law has no scale of its own, so the number of bonds needed before the exponential dominates grows with $\lambda$, and a window of fixed length sees a smaller and smaller fraction of the decay as the gap closes.

That is why the twelve-bond exponent is shallower than −1 rather than steeper, and why lengthening the chain does not help. The window is not too short for the chain. It is too short for the power law, and the power law gets worse as the thing being measured gets longer.

Nine answers from one profile #

The cleanest way to see that a fitted length is not a property of the chain is to fit one chain nine times.

One relaxed chain, one profile, one elastic constant. A factor of three, decided entirely by how much of the profile the fit was shown, and no window reaches the 17.50 the rate extrapolates to.

A quantity that moves by a factor of three under a choice the analyst makes is not a measurement, in the same way that a good residual is not evidence a model is the sample’s — and nothing in the fit says so — the residuals are excellent at every window, because a concave curve is locally straight everywhere.

The exponent, when the length has no window in it #

With $\lambda$ in hand the original question can be asked properly. Ten relaxed chains of three hundred and twenty spanning a factor of twenty in the gap, the extrapolated length for each, and the power that relates them:

Fitted against the gap, the three columns give −0.476, −0.899 and −1.029. The last of those is the answer the argument asks for, and the standard for an exponent requires more than a fitted number: the local slope between each neighbouring pair of points is −1.043, −1.028, −1.022, −1.025, −1.035, −1.046, −1.056 and −1.034, so the fit is not −1 by cancellation.

The healing length goes as the reciprocal of the gap. It always did. What was measured before was a window.

Which of the three numbers belongs to the chain #

Running all three readings on chains of three lengths separates them cleanly, and the separation is the point.

The first column barely moves and moves the wrong way: it is the fit’s number, and a longer chain cannot repair it. The second column moves a great deal, because a third of a chain is a different window on each chain — it is a mixture of the fit’s number and the chain’s. The third column climbs to −1 and stops, and it is short of −1 on the shorter chains for a reason the next section gives.

There is a second limit and it is that is genuinely the chain’s, and it is worth stating because it is the one a longer chain does fix.

A healing length is an excess over the bulk, and the bulk is read from the middle. When the length approaches the chain, the two ends’ profiles overlap in the middle and there is no bulk left to be an excess over. The excess is then understated at every distance, the fit comes out too steep, and the length comes out too short — silently, with a clean residual.

What that produces is a ceiling proportional to the chain. At an elastic constant of 4.0, whose true length is longer than anything here can hold:

A straight line through the origin, at about one twelfth of the chain, across a factor of seven in size. A quantity that scales with the apparatus is a measurement of the apparatus — and this is the reason the extrapolated exponent is −0.873 at ninety-six sites: the softest chains there are reporting the chain rather than the length.

Two instrument failures that look alike and are not #

The same symptom has now turned up twice, here and in critical exponents — a fitted power short of what the argument says — with different causes.

For the critical exponent the local slope was moving and arrived at the right answer. Fitting closer and closer to the transition gave 0.41, 0.47, 0.49, 0.50, 0.51, and the exponent was reached by shrinking the window. The fix was to look at a smaller window, and second-order exponents were repaired the same way — by taking a derivative at a point instead of a slope over a range.

Here the local slope is moving and shrinking the window makes it worse. Six bonds gives a shorter length than twelve, which gives a shorter one than twenty-four. There is no window at either end that returns $\lambda$, because $\lambda$ is not a slope anywhere on the curve — it is the limit of one.

The difference is which way the correction goes. A pure power with a correction that dies away is measured by getting close to where it dies. A power times a power law has a correction that dies only at infinity, and then the quantity has to be extrapolated rather than approached. Fitting $1/\lambda + p/b$ and taking the intercept is that extrapolation, and it is the same move as a local slope taken one step further: measure the derivative, then take its limit.

Where the −1 comes from #

It is worth saying why the answer had to be −1, because the computation is then a check rather than a discovery.

An end perturbs a chain that has a gap. The response of a gapped system to a local perturbation decays over the length an evanescent state at the gap edge has, and that length is set by how far into the forbidden region a state at the band edge can reach: the inverse decay rate goes as the group velocity at the band edge divided by the gap. In this model the velocity at the band edge is a constant of the hopping and the gap is what the alternation opens, so the length is proportional to the reciprocal of the gap and nothing else.

That is the same statement the gap is not the band width makes from the other side, and it is why the exponent is exactly −1 rather than approximately so. It is also why a power law in front is unsurprising: an evanescent tail in one dimension from a point perturbation carries an algebraic prefactor, and the fitted $p$ between 0.41 and 0.66 is that prefactor being measured rather than assumed. It is also a reminder of what a chain of two hundred is for: the band edge, the evanescent state and the gap are all read off a finite matrix, and no wavevector appears anywhere in the argument.

What is quoted, and what is computed #

Nothing is quoted. A chain of a stated length at half filling, a stated elastic constant on every bond, and no material anywhere.

Each profile is a fixed point rather than a search: the alternation of each bond is set to its own bond order minus the mean, divided by the elastic constant, and iterated under a mixing until nothing moves by more than a part in a million million. Every one converged, in a hundred and eighty-one iterations.

The eigenvalues come from implicit QL on the tridiagonal form, and that is worth a sentence because it is what makes chains this long practical. A chain’s matrix is tridiagonal — site $b$ is joined to $b+1$ and to nothing else — so the reduction a general method spends its time on has already been done by the chemistry. It takes fifty-two milliseconds on a chain of a hundred and sixty where a general method takes twenty-one seconds, and the two agree: the same levels to a part in ten to the fourteenth, the same eigenvectors to a part in ten to the fifteenth, on the same chain. A general method would take four minutes for a single profile at three hundred and twenty sites.

The gaps are recomputed from the stored alternation through a second route whenever a cached chain is read back, which is a check the relaxation cannot pass by having gone wrong in a self-consistent way.

What this cannot say #

There is no electron repulsion here. Everything is one-electron, and a half-filled chain drawn this way is a metal before it distorts, which is exactly the case where that is least safe.

The extrapolation is a fit and inherits a window of its own. The intercept is taken from bonds six to sixty; taking bonds ten to eighty moves $\lambda$ by about a per cent on the stiff chains and by more on the softest, where the ceiling is already interfering. What the extrapolation removes is the systematic dependence on the window, not every dependence — and the evidence that it has been removed is that the exponent comes out at the value an independent argument predicts, not that the fit is stable.

And the two softest chains are near the ceiling even at three hundred and twenty sites. Their extrapolated lengths, 27.0 and 37.6 bonds, are a twelfth and a ninth of the chain; the last local slope, −1.034, is the one to be least confident about. The finding does not rest on it — the eight slopes before it are between −1.02 and −1.06.

What was checked #

The local decay length rises at every bond rather than scattering about a mean, and is more than four times larger at bond sixty than at bond four. That is the claim that there is no single length, and it is checked as a monotone sequence rather than as a ratio, because two values far apart could be noise.

The twelve-bond window gives the same exponent on a chain of three hundred and twenty as on one of ninety-six, to within 0.08. This is the check that the shortfall is not a finite-size effect, and it is the one that would have failed if the obvious explanation had been right.

The extrapolated exponent is between −0.95 and −1.10, and every local slope but the last is too — checked separately, because an average of slopes running from −0.8 to −1.2 would also be −1.

The measured length at an elastic constant beyond every chain here is a fixed fraction of the chain, to within twelve per cent across a factor of seven in size.

And the third-of-a-chain window moves with the chain by more than 0.15. That is the control: if every window gave the same answer there would be nothing here, and this is the check that fails if the ceiling is doing nothing.

Still open: the width of a domain wall #

The obvious open question is the soliton, which the decay profile now gives a shape to. A long chain can hold both dimerisation patterns with a boundary between them, and the width of that boundary is a length of the same kind — so the question is whether it is $\lambda$, the twelve-bond number, or neither. The two dimerisation patterns an open chain chooses between are what a wall would separate, and the energy of one is already computed. The relaxation reaches it by seeding the chain with a domain wall instead of a uniform alternation, and the comparison is between two independent measurements of one quantity rather than between a measurement and a fit.

The nearer question is the prefactor. $p$ runs from 0.41 to 0.66 and appears to be settling near two thirds rather than at a half or at one, and a power in front of an evanescent tail is the kind of quantity that has a closed form. Computing the response of a gapped chain to a site perturbation analytically — which is a sum over band-edge states already diagonalised here — would say what $p$ is obliged to be, and would turn the extrapolation from a fitted correction into a known one. That matters more than it sounds: a known $p$ means $\lambda$ can be recovered from a short profile, which is the only kind a real oligomer has.

And there is a third thing to do with the instrument rather than the question. Every defect state with a fitted length was fitted the same way, over a window chosen by eye, and the same concavity would be invisible in every one of them. Running the local-rate test on those profiles costs nothing and would say which of those lengths are lengths.