The chain that pays for nothing

A ring is not the bulk found the reason a Peierls chain’s middle is not its bulk. The relaxation that sets the bond lengths moves each bond by its bond order minus the chain’s mean bond order, over a stiffness K — and subtracting the mean holds the chain’s total length fixed. The ends carry extra alternation, as the chain distorts hardest where it stops found, and with the length fixed that extra alternation has to be paid for somewhere. It is paid for by every bond: a uniform compression c that goes as one over the chain’s length, −0.33/N at the softest stiffness and −0.081/N at the stiffest. Against the infinite chain at that compression, the soft middles agree to 10⁻¹¹, nothing turns back, and two tails describe every profile to its middle.

That repaired the reference. It left a question about the rule. A chain whose length is held fixed is a chain under tension, and the tension reaches every bond. Whether the tails measured against the compressed bulk — the exponents, the decay lengths — belong to how the ends heal or to the constraint that made the middle pay for them is not something the compressed bulk can answer, because the compressed bulk has the constraint built into it.

The essay named the experiment. Remove one line of the rule.

The same rule, with the length let go #

The fixed-length rule is

$$\delta_b = \frac{P_b - \bar P}{K},$$

with $P_b$ the bond order of bond $b$ and $\bar P$ the chain’s mean. The mean is what fixes the length: the δ sum to zero whatever the bond orders do.

The alternative replaces the chain’s own mean with a constant: the mean bond order of the infinite chain at its own, free, self-consistent alternation. Call it $P_0$. Then a bond in the deep middle of a long chain, with the infinite chain’s bond orders, moves by exactly ±$d_0$ — the free bulk’s alternation — and nothing anywhere is compressed. The ends still carry extra alternation, and now each end bond simply lengthens or shortens by its own excess bond order over $P_0$. Nothing is asked of the middle.

$P_0$ is a Bloch sum, and it carries the band’s own physics. At K = 1.1, where the free chain alternates strongly ($d_0$ = 0.285), it is 0.604. At K = 3.1, where it hardly alternates ($d_0$ = 0.011), it is 0.6364 — within three parts in ten thousand of 2/π, the bond order of a uniform chain, which is the value it must approach as the alternation vanishes. The alternation dies away with stiffness faster than any power — the electrons’ gain near zero alternation carries a logarithm, so the self-consistent alternation falls exponentially as the springs stiffen — and by the stiff end the free chain is within a hair of uniform and its mean bond order is the uniform chain’s. The rule’s own fixed point is checked: $P_1$ − $P_0$ over K returns $d_0$ to 10⁻¹⁰ at every stiffness.

The chain is the published one: 320 sites, Hückel, half filled, iterated to a change below 10⁻¹² a bond. It takes between 164 and 306 iterations, a few more than the fixed chain.

The middle stops paying #

The uniform part of a bond’s change — the mean of two neighbours, which cancels the alternation — tells where the length went. With the length fixed, it is −5.8 × 10⁻⁴ at K = 1.6 all along the chain: every bond is compressed by the same amount, as the decomposition into compression and tail showed. With the length free, it is 8 × 10⁻¹³ in the middle, which is zero, and it is non-zero only within a few decay lengths of each end, where the end bonds adjust locally to their own bond orders.

At every stiffness to K = 2.0 the free chain’s middle is the free bulk to better than 10⁻¹⁰: its uniform part is around 10⁻¹² and its alternation agrees with $d_0$ to a few times 10⁻¹¹. The fixed chain needed a bulk built at its own compression to reach that agreement. The free chain needs only the infinite chain as it is.

At the stiffest chains the middle is not bulk under either rule. At K = 3.1 the free chain’s middle sits 1.2 × 10⁻⁴ above $d_0$, the size of excess the fixed chain’s middle carries over its compressed bulk at the stiff end: two tails, each decaying more slowly than any window expected, still overlapping at bond 160. That is a property of the healing length against the chain’s length, and the rule does not touch it.

The healing is the same healing #

The excess over bulk is what the tails are fitted to, and the two rules give the same excess. At K = 1.6, 2.2 and 2.8 the free chain’s excess over $d_0$ and the fixed chain’s excess over its compressed bulk lie on top of one another on a logarithmic scale, down to the relaxation’s floor of 10⁻⁸.

The end bond itself hardly notices. Its alternation is 0.31757 free and 0.31775 fixed at K = 1.1, the largest difference of any stiffness; at K = 3.1 the two agree to five figures. The compression that the fixed chain spreads over 319 bonds is, at the end bond, a correction in the fourth decimal place.

Every exponent where it was #

The tails were fitted exactly as before: A·[f(b) + f(N − 1 − b)] with f = b^(−p)·e^(−b/λ), one tail from each end, least squares on the logarithm, from bond 6, 10, 20 or 30 to the middle. Against the free bulk, the free chain’s exponents are the fixed chain’s.

From bond six they agree to under a thousandth at every stiffness — 0.4317 against 0.4314 at K = 1.1, 0.5088 against 0.5086 at 1.6, 0.6493 against 0.6484 at 2.5, 0.7262 against 0.7260 at 3.1. From any of the four starting bonds the largest difference is under three thousandths. The decay lengths agree to under a tenth of a bond.

The tension is a change of stiffness #

There is a reason the constraint could not have reached the tails, and it can be stated exactly. In the fixed chain every hopping is 1 + c plus its alternation. A uniform rescaling of all the hoppings in a Hückel chain rescales its energies and leaves its wavefunctions alone, so the bond orders — and through them the alternation — depend only on the alternation measured relative to the compressed hopping, a/(1 + c). Write the relaxation rule in that variable and the stiffness that appears is K(1 + c). The compression is not a new effect on the healing. It is the same chain at a stiffness lower by K·|c|: by 9 × 10⁻⁴ at K = 1.6, by 8 × 10⁻⁴ at K = 3.1.

That predicts a direction as well as a size. The exponents rise with stiffness, so a chain that is effectively a little softer should report a slightly smaller exponent. At all ten stiffnesses it does: the fixed chain’s exponent from bond six sits below the free chain’s every time, by between one and nine ten-thousandths. The slope of the exponent against stiffness near K = 1.6 is about 0.17, and 0.17 times 9 × 10⁻⁴ is 1.6 × 10⁻⁴ — the observed difference there is 2 × 10⁻⁴. On the stiffer chains the observed differences run a few times larger than the slope predicts, which is within the scatter that the fits’ sensitivity to their windows has already shown, and the sign never fails.

So the answer to the question the compressed bulk could not ask is that the constraint was never in the exponents. The tension that the fixed length puts on every bond moves the middle by a thousandth and the tails by nothing that a fit can see. The healing — how the ends’ extra alternation dies away into the bulk — is a property of the ends and the band, and it is the same whether or not the chain is allowed to change its length while it heals.

Nor was it in anything the earlier readings of the healing rest on. Every decay length, enhancement and fit window in the series was taken from fixed chains — including the check that discarding the first few bonds hides no second exponent — and a constraint equivalent to a change of stiffness under a thousandth shifts those by less than their last quoted figure. The instability itself — a chain cannot stay even at half filling — is a property of the bulk, and the free bulk and the compressed one differ only in where the zero of bond length is put.

That is worth having for its own sake, because the rising exponents the series has carried — from 0.43 at the softest stiffness to 0.73 at the stiffest — now stand on two independent constructions of the bulk rather than one. Neither the ring nor the compression nor the constraint is the cause of the rise.

The drift that was supposed to be a correction #

The exponents do have a habit that survived the repair of the reference. On the middling and stiff chains, starting the fit deeper lowers the exponent. At K = 2.2 it falls from 0.601 from bond 6 to 0.560 from bond 30; at K = 2.5 from 0.649 to 0.607. The window that was not a plateau met the same habit in a different reading of the same profiles.

The obvious explanation is that the tail form is the leading term of an asymptotic expansion and the drift is the next term. A pure b^(−p)·e^(−b/λ) fitted to a profile with a correction factor (1 + a/b) will report a p that depends on where it starts, because near the end the correction shifts the apparent power and far out it does not. Fit the correction, and p should stop moving.

It does not.

At K = 2.8 the plain fit’s exponent moves from 0.697 to 0.673 between bond 6 and bond 30. The corrected fit’s moves from 0.692 to 0.546 — six times as far. The fitted coefficient a grows from 0.08 to 4.9 over the same range, which is the correction doing what a free parameter does: taking up whatever the data in the window want, rather than describing something fixed about the profile.

The pattern holds wherever the drift was the question. From K = 2.2 to 3.1 the spread of p over the four fit starts grows when the correction is added: 0.041 to 0.053, 0.042 to 0.114, 0.025 to 0.146, and 0.008 to 0.118 at the stiffest. Only near K = 1.8 to 2.0, where the plain drift was smallest, does the correction leave it the same or trim it slightly. The free and fixed chains behave identically here too, to a few thousandths in every spread.

Why a correction cannot be told from a power #

The reason is in the shape of the fit’s cost. Hold p at a value, refit everything else, and record the error. Without a correction, the cost at K = 2.5 from bond 10 has a minimum at p = 0.644 and at p = 0.445 it is ten times that minimum. With the correction free, the minimum moves to 0.571 and the error at 0.445 is only 3.4 times it. Along the valley the correction’s coefficient slides from 6.3 at p = 0.40 to −0.9 at p = 0.70.

Over a window of 150 bonds, b^(−p) and b^(−p′)·(1 + a/b) differ by a slowly varying factor that the exponential’s own length and the amplitude can absorb almost entirely. A power and a 1/b correction are nearly the same function on the range a half chain of 160 bonds offers. Fitting both does not separate them; it lets the data trade one against the other, and where the trade lands depends on the window.

It is tempting to read something into where the corrected fits go when the window is deep. From bond 30, on the chains from K = 1.8 to 2.5, the corrected exponent is 0.495, 0.503, 0.500 and 0.497 — one half, near enough, on four chains. But at K = 2.8 it passes through one half and keeps falling — 0.514 from bond 40, 0.490 from 50, 0.448 from 80, 0.431 from 100 — and at K = 2.0 it climbs back above 0.54 once the fit starts beyond bond 50. A value the fit passes through on its way somewhere else is not a limit, and the ridge says why the fit is free to pass through it.

What a chain with free ends does not settle #

The free chain is free in one sense: its total length. Its ends are still the ends of a Hückel chain with the same hopping everywhere and no end-group chemistry; a real polyene’s terminal bonds differ from its interior ones for reasons this model does not contain, and those reasons would change the end’s excess, though nothing here suggests they would change how it decays.

The relaxation is still the linear one, a bond order mapped to a length through a single stiffness. A chain whose bonds stiffen as they shorten would heal differently near the end, where the bonds are most distorted, and the exponents are read starting six bonds from the end precisely to stay out of that region.

And the fit’s form is still a guess. Two forms were tested here — a power times an exponential, and the same with a 1/b factor — and the second was rejected as an explanation of the drift. That leaves the drift unexplained. It does not show that no correction describes it, only that the next term an expansion in 1/b would offer is not distinguishable from a change of power over the range available.

Who measured what #

The Peierls instability of a half-filled chain is Peierls’s, from the 1930s, and the Hückel chain with a linear bond-order relaxation is the model of Longuet-Higgins and Salem for polyenes, 1959, and of Su, Schrieffer and Heeger in its continuum and soliton form, 1979. The comparison between the fixed-length and zero-tension rules, and the test of the 1/b correction against the exponents’ drift, are new here.

How the chains were relaxed and read #

The free infinite chain was solved by iterating the relaxation rule over a Bloch sum of 8,192 points to obtain $d_0$ and $P_0$ at each of the ten stiffnesses. The open chain of 320 sites was then relaxed with $P_0$ in place of the chain’s mean, by the same linear mixing at three tenths and the same convergence criterion of 10⁻¹² a bond as the fixed chain.

The excess over bulk was the smoothed envelope of |δ| minus $d_0$, with points below the floor of 10⁻⁸ excluded. Both tail forms were fitted by Levenberg–Marquardt on the logarithm, the corrected form started from the plain fit’s solution with a = 0. The ridge held p on a grid of 21 values from 0.40 to 0.70 and refitted the amplitude, the length and the coefficient at each.

What had to hold, and could have failed: the free bulk’s $P_0$ had to return $d_0$ under the rule. Every free chain had to converge. The free middles had to be the free bulk to 10⁻¹⁰ wherever the stiffness was 2.0 or less. Free and fixed exponents had to agree to 0.005 from every starting bond, and the end alternation to 5 × 10⁻⁴. On every chain from K = 2.2 the correction had to widen the drift rather than narrow it. At K = 2.5 the cost of a wrong exponent had to fall by more than half once the correction was free, with the coefficient sliding by more than five along the valley.

Still open: a window long enough to separate the two #

The ridge is a statement about 160 bonds. A longer chain gives a longer window, and a power and a 1/b correction that are indistinguishable over 150 bonds may separate over 500. A ring is not the bulk estimated the length at which two tails at the middle fall to the floor — 711 sites at K = 2.8 and 1,102 at 3.1 — and a free chain of that length would be the natural place to fit both forms again and ask whether the corrected exponent stops moving.

The other question is what the drift is, now that the obvious answer has been taken away. The plain fit’s exponent falls with fit start on every chain from K = 1.6 upward and most steeply on the middling ones. Measured in decay lengths rather than in bonds, the fit windows on the stiff chains cover a few λ and those on the soft chains a dozen, so a drift that depends on how many decay lengths a window spans would look exactly like this — and the rule of thumb that was on the flat part is a reminder that where a window sits on a curve can matter more than how wide it is.