The rule of thumb was on the flat part

Fitting across ten stiffnesses established that the tail exponent in a relaxed dimerised chain rises with stiffness where it can be measured at all — 0.41 at K = 1.1 to 0.66 at K = 2.8 — and that the four cases which appeared to settle it near two thirds have no asymptotic range and are measuring a noise floor.

It also closed by naming an unexamined choice in its own procedure. The exponent is a slope, fitted to the local decay rate against 1/b over a stretch of bonds. The far end of that stretch is set by how much tail there is. The near end is set by discarding the first few bonds, because the profile has not reached its asymptotic form there — and how many is a rule of thumb, three coherence lengths, rather than anything measured.

That worry is not idle here. The exponent turned out once already to be a threshold rather than a physical power, and the threshold in that case was the far end of the same window. A quantity that has been wrong once because of a window is a quantity whose other window deserves the same treatment.

The local rates are computed at every bond and the fit can start anywhere. So the honest version is the exponent plotted against where the fit is allowed to start. If there is a plateau, the rule of thumb does no harm. If there is not, the near-end cut is a second window with a second exponent in it, and the quantity has two arbitrary choices rather than one.

There is a plateau.

Half a per cent to nine #

At K = 1.1 the exponent is 0.4115 at six bonds and moves by 0.0022 over every start out to fifteen — half a per cent of itself. At K = 1.6 it moves by 0.026, five per cent. The worst case is K = 3.1 at 9.6 per cent, and that is the case already excluded as having no asymptotic range.

Against what? The exponent runs from 0.41 to 0.66 across the stiffnesses. That is a sixty per cent effect, and the near-end cut moves any one of them by under a tenth of it. So the finding survives the choice, and the choice is one choice rather than two.

That is a negative result about a worry, which is the least exciting kind of essay to write and the kind that has to be written anyway. An unexamined parameter in a published number is a liability until somebody sweeps it, and the sweep costs nothing here — every local rate was already computed and stored.

It is worth noticing which cases have the widest plateaus, because the pattern is the opposite of alarming. The softest chains, whose exponents are the most trusted, move least: 0.5 per cent at K = 1.1 and 1.5 at K = 1.25. The stiffest, whose exponents it had already excluded as unmeasurable, move most. So the sweep’s own worst numbers belong to the cases already set aside, and the cases the stiffness trend rests on are the flattest of all.

The rule of thumb is not idle #

The temptation on finding a plateau is to conclude the cut could have been anywhere. It could not.

Fitting from two bonds gives 0.403, 0.429, 0.452, 0.478, 0.499, 0.515, 0.526, 0.535, 0.536, 0.534 — every one lower than its value at six, and by up to 0.13. The whole sweep from two moves the exponent 7.5 times as much as the part from six outward does.

And it does something worse than lower the values. At two bonds the exponents span 0.403 to 0.536, a range of 0.133; at six they span 0.412 to 0.659, a range of 0.247. Starting too near the end would have halved the trend with stiffness. The near-end cut matters much more for the relation between stiffnesses than for any single number, and a study that had made the lazy choice would have found a weaker effect and might have concluded there was none.

So the rule of thumb is doing real work. It is avoiding a systematic error that is downward, that is largest where the finding is largest, and that would have flattened the result.

The mechanism is visible in the profiles and it is the one a closer look at the local decay rate showed: the local decay rate is not constant, it falls like the reciprocal of the distance from the end, and near the end it has not begun to. A fit that includes those bonds is fitting a different function over part of its range, and the part it gets wrong is the part with the largest 1/b — which is the leverage end of the regression. Two bad points at large 1/b drag a slope more than ten good ones at small 1/b hold it.

The plateau has a shape #

Flat to within a few per cent is not flat, and the residual structure is worth looking at because it is not the same structure everywhere.

At the softest stiffnesses the exponent falls slowly and monotonically as the start moves outward: 0.412, 0.411, 0.410, 0.409, 0.410. At the stiffest it rises to a maximum and then falls — K = 2.8 goes 0.659, 0.678, 0.689, 0.694, 0.697 at fifteen bonds, then 0.692, 0.681, 0.664 at thirty.

The maximum’s position moves with the stiffness, from nowhere at K = 1.1 to about twelve to fifteen bonds at K = 2.5 and above. That is a real feature of the profiles and it has a plausible reading — a fit that starts too near the end catches the non-asymptotic rise, and one that starts too far out is fitting a stretch where the excess is approaching the noise floor and the rates are corrupted from the other direction — the relaxation’s own noise floor, arriving from inside the window rather than at its edge. The best window is between the two, and where it is depends on how long the tail is.

That reading also explains why the softest chains have no maximum. Their tails are short — the chain distorts hardest where it stops and recovers quickest when the gap is wide — so the noise floor arrives before the non-asymptotic rise has finished being outrun, and the two errors never separate enough to leave a maximum between them.

Which means that six bonds is not the optimal start for the stiffer cases. Fitting from fifteen would give K = 2.8 an exponent of 0.697 rather than 0.659, and would widen the range across the series from 0.247 to about 0.29. The rule of thumb’s choice is conservative: it understates the effect it reports.

One consequence is worth spelling out for anybody who quotes one of these exponents. The number to quote is not the value at the flattest start, because “flattest” is itself a choice made after seeing the data. The number to quote is the value at a stated start with the plateau’s width beside it — 0.5168 from six bonds, ± 0.026 over every admissible start — which is a report rather than a selection, and is what a sweep makes possible.

How far out a fit is allowed to start #

Not every case admits every start. A fit needs at least six local rates to be worth calling a fit, and the soft chains run out early: K = 1.1 has no start beyond fifteen bonds, K = 1.25 none beyond twenty-five, and the rest reach thirty. That is the same noise floor that ends every tail — the excess falls below the relaxation’s own convergence and the points are dropped.

So the sweep is not the same length for every case, and a comparison across stiffnesses at a fixed large start would be comparing fits with different numbers of points behind them. Every number quoted above is at a start every case admits, and a start with fewer than six rates is dropped rather than fitted, because a two-point fit returns a number and the number is the two points.

What was computed, and how #

One relaxed chain of three hundred and twenty sites at each of ten stiffnesses, from the collection’s own stored relaxations — nothing here recomputes a profile. The local decay rate is a three-point logarithmic derivative of the excess at each bond; the exponent is the slope of that rate against 1/b, fitted by least squares over bonds lo to sixty.

Only the starting bond is varied. The far end, the noise floor and the profiles themselves are held fixed, which is what makes this a sweep of one choice rather than a different calculation. Twelve starts from two bonds to thirty, on ten stiffnesses, is a hundred and twenty fits, and none of them requires relaxing a chain again: every fit reads the same profile.

The check requires five things: that enough cases have a measurable exponent to sweep; that every one returns its reported value at the standard start of six bonds, so this is the same fit; that from six outward the movement is under fifteen per cent of the value, which is the plateau; that the whole sweep from two moves it more than half again as much, which is what makes the plateau a plateau rather than a flat function; and that the plateau holds case by case rather than on an average. What must be refused is a start past a case’s own reach, which must leave the fit undone rather than done on too few points.

What this does to the stiffness trend #

The conclusion about the tail exponent had three parts and this sweep touches them differently, so it is worth separating them.

That the exponent rises with stiffness where it is measurable. Strengthened. At every start from six outward the ordering is preserved, and at the starts nearer the flat part of each case’s own plateau the range is wider than the one reported.

That the four excluded cases are measuring a noise floor. Untouched, and consistent with what the ends of a chain do to a profile that has no room to become asymptotic. This sweep varies the near end and the floor is a far-end problem; the three cases with the largest plateau movement here are among those four, which is consistent but is not evidence either way.

That a least-squares extrapolation in 1/λ gives 0.608 at the limit, and should be reported as an extrapolation rather than a value. Weakened, slightly, and in the direction of caution. The seven measurable exponents each carry a few per cent of window dependence, and an extrapolation through seven points each uncertain by that much is uncertain by at least as much again. That was already the reason not to call it a value.

None of the three needs restating. What changes is that the first now has a stated robustness and the third has a stated source of error it did not have.

Where the model stops #

This sweeps one parameter and the quantity has others. The far end is still sixty bonds by choice; the noise floor is still 10⁻⁸, and raising it by three decades moves four of the ten cases by tens. Those are not swept here and the plateau found here says nothing about them.

The three-point derivative is also a choice — a wider stencil would smooth the local rates and change the fit — and the model is a one-dimensional tight-binding chain with a classical lattice, which is the standard caricature of a Peierls chain. What is measured is a property of a procedure applied to that model, not of a material. That limit has been explicit since the exponent turned out to be the floor, and nothing here moves it.

And “plateau” is being used loosely for a region flat to a few per cent. If the exponent were being quoted to three decimals the structure in it would matter; it is being quoted to two, and to two it is flat.

There is a third limit worth stating, because it bounds what the plateau can be used for. A plateau in a fitted exponent is a statement about the fit, not about the physics: it says that the stored profiles are described equally well by the same number over a range of starting distances, which is what one expects when the profile really is a single exponential over that range and the fit is simply being handed more or less of it. It would say the same thing if the profile were a sum of two exponentials whose fast component had already died by the sixth bond. Distinguishing those two would take a residual analysis the sweep does not perform, and the difference matters if anybody wants to read the exponent as a decay length rather than as a fitted parameter.

The generalisation #

Two things, and the second is the one that changes practice.

The first is the ordinary one: a parameter nobody has swept is a liability, and sweeping it is often free because the underlying quantities are already computed. Here the entire sweep is arithmetic over stored relaxation profiles — no new diagonalisation, no new minimisation — and it settles a question that could otherwise only be raised.

The second is that a plateau is not permission. Finding that a choice does not matter within a region is compatible with its mattering enormously outside it, and the way to report the result is to give the region and what leaving it costs. Here the region is six bonds and beyond, and leaving it costs half the effect being measured. A summary that said “the choice of start does not matter” would be true and would license exactly the mistake the sweep was run to check for.

The third-order lesson, which came for free, is that the conservative choice is sometimes identifiable. The standard start of six is not the flattest point of its own plateau, and moving to the flat part would strengthen its finding — so its number is a lower bound on the effect rather than a best estimate of it. That is worth knowing about any result that survives a robustness check: which direction the check would move it if it were tuned.

Who found it, and when #

The Peierls instability is from 1955, the coherence length in a dimerised chain is standard, and the rule of thumb of three coherence lengths is folklore. The relaxation, the local rates, the fit and the sweep are new arithmetic. The sweep is nearly free once the relaxed profiles exist, because it is a loop over one parameter of a fit and never relaxes a chain again — which is the general reason a window should always be swept before its result is quoted: the check costs almost nothing beside the calculation it tests.

Still open: the far end, and where the maximum sits #

The obvious open question is the far end, which is the other window and which nobody has swept. Sixty bonds was chosen for the same kind of reason six was, and the same sweep run on the far cut-off would say whether it too sits on a plateau — with the difference that the far end runs into the noise floor rather than into a non-asymptotic rise, so its failure mode is the noise-floor one described above and its plateau, if there is one, should end abruptly rather than gradually.

The nearer question is the maximum’s position. It moves from absent at K = 1.1 to about fifteen bonds at K = 2.8, and if it is the crossing point of a non-asymptotic error falling and a noise-floor error rising then it should sit at a fixed multiple of the coherence length rather than at a fixed number of bonds. The coherence lengths are computed for every case already; plotting the maximum’s position against them is one division, and if the ratio is constant then the rule of thumb has a defensible form — start at so many coherence lengths — with a measured constant in place of a remembered one.