The exponent was the window's

The carriers a distortion was hiding warmed a ring of forty until its Peierls distortion went away, and found that the alternation falls continuously rather than jumping: 0.0975 at $kT = 0.10$, 0.0163 at 0.135, and nothing at 0.14. It fitted a straight line through the logarithms over the last decade before the alternation vanished, reported a slope of 0.44, and said the number should be read as a measurement over a finite range rather than as an exponent.

It then asked the right question and predicted the wrong answer. Running the same measurement on rings of eighty and a hundred and sixty would say whether 0.44 belongs to the transition or to a forty-site ring — and, it said, an exponent that is stable across sizes would be the more surprising of the two answers, because a finite ring is exactly the system where a sharp transition ought not to survive.

The exponent is stable across sizes. It is stable across stiffnesses too. And it is not the exponent.

Where the number came from #

A slope fitted over a decade is an average of the local slope over that decade, and it is only an exponent if the local slope is constant there. Nobody checked, because a straight line through five points on a log-log plot looks like a straight line.

Taking the same ring of forty at the same stiffness and computing the alternation at reduced temperatures spanning two and a half decades, then taking the slope between each neighbouring pair:

It rises monotonically and it arrives at a half. There is no plateau at 0.44 anywhere, and the original window covered the part of the curve where the slope is still climbing steeply.

So the exponent is one half, which is what a free energy analytic in one order parameter is obliged to give. The distortion here is found by minimising a free energy over the alternation $\delta$; near the transition the free energy is a smooth even function of $\delta$, its quadratic coefficient changes sign, and the minimum moves as the square root of the distance from that point. There was never any room for anything else.

That is worth saying twice, because it means the answer was available without computing anything. A quantity found by minimising a smooth even function whose quadratic term changes sign goes as the square root, always — the same statement that makes the mean-field exponent a half for every order parameter of that shape, and the reason a half is called the mean-field value in the first place. What the computation adds is not the exponent but the distance over which the correction to it survives, and that turns out to be large: two and a half decades of reduced temperature, which is a great deal more range than anybody fitting a real measurement would have.

The correction is what a fit over a finite window is measuring, and it is not small until the alternation has fallen to about a twentieth of its cold value.

The number moves with the window, in order #

If 0.44 belongs to the window rather than to the system, then changing the window should change it in a way that has nothing to do with the ring. Fitting the same curve over five windows of one decade each:

The second row is the original window and reproduces its number. Every row is in order, and there is no window over which the answer stops depending on the window — which is the test for whether an exponent has been reached, and it is not met until the innermost decade.

An exponent quoted without the range it was fitted over is not a measurement. That sounds like a piece of standard advice; here it is a table with a spread of a quarter in it.

Why the stability across sizes was not what it looked like #

The natural expectation was a size-dependent exponent, on the reasonable ground that a finite ring has no sharp transition. The measurement says otherwise:

The fitted number moves by one and a half per cent across a factor of two in ring size and a factor of two in stiffness. Read as a measurement of the transition, that stability is impressive. Read correctly, it is the strongest possible evidence that the number is not a measurement of the transition at all.

Because here is what is actually stable: the whole curve. Dividing the alternation by its value at zero temperature and plotting against the reduced temperature puts all five cases on one another to within 3.41 per cent everywhere, and the two largest rings agree to 0.05 per cent.

A slope fitted over a stated window is a functional of the curve over that window. If the curve is the same for every case then the slope is the same for every case, whatever window is chosen and whatever the slope happens to mean. The stability was a consequence of the collapse, not of the exponent — and a stability that survives every change to the system is exactly what a quantity determined by the analyst rather than by the system looks like.

What the collapse itself says #

The collapse is worth more than the exponent it explains away.

It says the alternation has two scales and no more, which is the kind of statement a band’s second moment being a count of neighbours also makes: a temperature, which is where it goes, and a size, which is its cold value. Everything else about the curve is universal in this model — a ring of eighty at one stiffness and a ring of forty at another differ only in those two numbers.

The two scales are not independent either. The undoing temperature and the cold gap keep a fixed ratio — 0.2688, 0.2756, 0.2565, 0.2777 and 0.2690 for the five cases — which was reported for one of them and called the classic ratio. It is the same statement in a different currency: one scale, in two units.

So the honest summary of what the two calculations together have measured is:

One curve, one temperature scale, one amplitude, and an exponent of exactly one half. The 0.44 was the shape of the curve away from the transition, sampled by a fit that had no reason to know it was away from anything.

Three fitted powers, and which of them are safe #

A fitted power has been quoted three times in these arguments, and the arithmetic above says how to sort them.

The shape departure of a two-band system, reported by two bands and the shape of each as 2.2007 “over the small-coupling range”, against an argument that says exactly 2. That is a fit over a window, and the departure from 2 is in the same direction and of the same size as the departure here. It is the suspect one — and it has since been remeasured by taking the derivative at a point rather than the slope over a range, which gives 1.989 and −2.038 where the argument says 2 and −2. The window was the difference.

The healing length of a relaxed chain, reported by the chain distorts hardest where it stops as the gap to the power −0.60 against an argument that says −1, over five gaps spanning a factor of eight. That is a fit over a window, and its own essay says the discrepancy would mean the received relation is missing a term at the sizes real oligomers have. It might. It might equally be this.

The depolarisation ratio’s departure under a distortion, reported as an exponent of 1.9994 against an exact 2. That one is safe, and the reason is instructive: it was measured over a range in which the local slope does not move, and the check was that the answer agreed with a closed form to four decimal places rather than to one.

The rule that separates them is whether the quantity being fitted has a competing scale. A departure that is a pure power has a constant local slope and any window will do. One that is a power plus a correction has a local slope that moves, and then a fitted number is a report about the window. Both of the suspect cases have an obvious competing scale — the band width against the gap, and the lattice spacing against the healing length — and the safe one has none.

What a chemist should take from it #

A fitted exponent needs a window and a check that the window does not matter. The check is cheap — fit two windows and see whether they agree — and the failure it catches is silent, because the fit’s own residual says nothing about it. Every one of the five fits above has an excellent residual.

Stability across systems is not evidence of a physical exponent, in the same way that a good residual is not evidence that a model is the sample’s. It is evidence that whatever is being measured is the same for all of them, which is equally consistent with the measurement being a property of the procedure. Distinguishing the two needs the local behaviour rather than more systems.

And a collapse is worth looking for before an exponent is. The scaled curves lying on one another is a stronger and more useful statement than any single number taken off them, and it was available from the same computations.

What is quoted, and what is computed #

Nothing is quoted. There is no material here and no measurement: a ring of a stated size at half filling, a stated elastic constant, and a free energy that includes the electronic entropy. Every level, occupation, alternation, transition temperature, local slope and fitted exponent is computed.

The transition temperature is found by bisection on whether the free energy’s minimiser is at the boundary of its parameter. That is a property of a surface rather than of a thermodynamic limit, and it is what makes the exponent obliged to be a half.

The curves are cached between runs, and every restored curve is verified by recomputing its first point — a golden-section search over a free energy that diagonalises a ring at each step, run again and required to agree to a part in a billion. A cache of numbers computed by possibly other code is the one thing this collection refuses to accept without a check.

What this cannot say #

There is still no phase transition here in any strict sense, and the original caution stands unchanged. A ring of eighty is a finite system; what is computed is where a variational minimum leaves the boundary of its parameter. The exponent of one half is that surface’s exponent, and it would be one half for any smooth free energy of this shape whatever the physics.

A real Peierls transition is not mean-field. Fluctuations in one dimension destroy long-range order at any finite temperature, so a real quasi-one-dimensional material’s transition is driven by coupling between chains and has exponents this model cannot produce. What is established here is what the model says, and the reason that is worth establishing is that the model is what the number was taken from.

And the collapse is within one model. Five cases differing in two parameters is not a demonstration of universality; it is a demonstration that these two parameters are the only two, which is a much smaller claim and the only one the computation supports.

What was checked #

The local slope approaches one half on every ring and stiffness tried, and lies between 0.49 and 0.52 at the closest pair of points — checked for each case separately rather than for an average.

And it rises monotonically towards it, which is a different claim from being near it: a sequence scattering about a half would be consistent with noise, and one climbing towards it is not.

The scaled alternation is one curve to within four per cent across every case, and the two largest rings agree to a part in five hundred — the second checked separately, because it is the one that shows the size dependence has gone rather than merely being small.

A slope fitted far from the transition is well under a half and one fitted close to it is a half, with every window in between lying in order. The ordering is the check that makes the finding a finding: two windows disagreeing could be a numerical accident, and five in order cannot.

Why a log–log plot is such weak evidence #

The finding here is about one exponent on one curve, and the mechanism behind it is general enough to be worth stating as a caution about the whole practice of reading a power from a plot.

A power law is a straight line on logarithmic axes, and the temptation is to reverse that: a straight line on logarithmic axes is a power law. It is not, and the reason is that a decade is a short interval in log space. Any smooth positive function, plotted over a factor of ten in its argument, is close to straight on log axes — its curvature has only about 2.3 natural-log units to express itself in, and for most functions that is not enough to be visible against ordinary scatter.

So the points fall on a line is nearly always true and nearly always uninformative. What a fit over such a window returns is a chord of the local-slope curve — an average of the true exponent over the range chosen — and the number it reports is a property of the window as much as of the function.

The repair is the one performed here. Compute the local slope at a sequence of distances from the critical point and watch where it goes: a genuine power law has a slope that stops moving, and a curve that is not a power law has one that does not. The measured sequence here runs to 0.5020 as the transition is approached, and 0.44 is the chord over one particular decade — the same curve, read two ways, giving a number that converges and a number that does not.

Two practical consequences follow.

Report the local slope, not the fitted one, or report the fitted one with its window stated, since without the window it is not reproducible.

And be suspicious of an exponent close to a simple fraction. A chord can land near a half by arithmetic accident, and the fact that it did here — 0.44 against a true 0.50 — is what made the original number plausible enough to publish rather than obviously wrong.

Still open: the amplitude in closed form, and the healing-length exponent #

The obvious open question is the amplitude the collapse leaves behind. Scaling by the cold alternation is what makes the five curves one curve, so the cold alternation is the whole of what distinguishes them — and it is a function of the stiffness and the ring size that the same calculation already gives at every point. Writing it down in closed form, and checking it against the five measured values, would replace the collapse’s one amplitude with a formula and complete the description: a curve, a temperature and an expression.

The nearer question is one a neighbouring measurement raises. A chain with ends distorts hardest where it stops, over a healing length that goes as a power of the gap — and the exponent measured there was $-0.60$ against an argument that says $-1$, on chains of a few hundred and over five gaps. That is the same shape of measurement as this one: a slope fitted over a finite window, quoted as a power. Running the local-slope test on it would say whether $-0.60$ is the healing length’s exponent or the window’s, and this essay is the reason to expect the second.