A ring is not the bulk

The local decay rate on a stiff Peierls chain turned back: read to half the chain, it fell to a minimum and climbed again, and a model of two ends and a midpoint reference placed each minimum to within two bonds. The cause was the reference. Every excess in this argument is the alternation at a bond minus a bulk value, and the bulk had always been read from the chain’s middle — which on a stiff chain still carries both ends’ healing. Subtracting it forces the measured excess to zero at the midpoint, and the rate rises to meet the zero.

That essay proposed the obvious remedy. A periodic ring of the same length has no ends, so its alternation should be a bulk value that owes nothing to either; measure every excess against it and the turn should go, the stiff tails should be readable to the middle, and the soft chains — whose middles are bulk to every digit the relaxation carries — should not move at all. The last clause was the one that could go wrong.

It goes wrong, by thirty thousand times the relaxation’s floor. And the reason it fails is the more useful finding, because it says what the bulk of this chain actually is.

The chain’s length is held fixed #

At K = 1.1 the chain’s middle is bulk by any test: its profile meets the floor twenty-three bonds from the end, and the midpoint is 160. Its alternation there is 0.2852185. A free ring of the same length, relaxed by the same rule, has 0.2849345. They differ by 2.84 × 10⁻⁴, where the floor every rate in this argument is cut at is 10⁻⁸. Every stiffness shows the same thing, falling from that value to 1.1 × 10⁻⁵ at the stiffest; none comes within a thousand times the floor.

The difference is in the relaxation rule, and it was there from the first profile. Each bond’s hopping moves by its bond order minus the chain’s mean bond order, divided by the stiffness. Subtracting the mean is what holds the chain’s total length fixed — without it, every bond would contract together and the chain would simply shrink. But the mean is taken over every bond, and the bonds near the ends carry more alternation than the middle. Their extra has to come from somewhere, and it comes from everywhere: every bond in the chain is shifted by the same small amount, and the middle’s bonds are shifted with the rest.

The shift can be read off the relaxed chain directly. Neighbouring middle bonds alternate as +0.2834 and −0.2876 at K = 1.1 on a chain of 160 — not symmetric about zero. Their mean, −0.0021, is the shift. A bulk that has it — an infinite dimerised chain with hoppings $1 + c \pm d$, with $d$ found self-consistently by the same rule from a Bloch sum over 8,192 momenta — reproduces the soft chains’ middles to 10⁻¹¹: 5.2 × 10⁻¹² at K = 1.1, 2.1 × 10⁻¹¹ at K = 2.0. That is the bulk of this chain, and the ring is not it.

A shift that goes as one over the length #

If the shift is the ends’ share spread over the chain, it should fall as one over the length, and it does. At K = 1.1 it is −4.18 × 10⁻³ on a chain of 80, −2.07 × 10⁻³ on 160 and −1.03 × 10⁻³ on 320. Multiplied by the number of bonds it is −0.3302, −0.3299 and −0.3297 — the same number, settling as the chain grows. At K = 1.6 the product is −0.1852 to −0.1851, at K = 2.2 −0.1179 once the chain is long enough for its ends not to overlap. It is a fixed quantity per end, divided among however many bonds there are.

That makes it easy to underrate. On a chain of 320 at K = 3.1 the shift is 2.5 × 10⁻⁴ — a number that looks negligible beside an alternation of 0.011. But the quantity every essay in this argument measures is not the alternation. It is the alternation’s excess over bulk, followed down to 10⁻⁸, and against that the shift is enormous. A reference that misses it by 10⁻⁵ turns every profile’s last several decay lengths into a measurement of the reference.

The shift is a tension #

There is a cleaner way to say what the shift is, and it connects the relaxation rule to something mechanical. Holding a chain’s total length fixed while every bond is free to change is a constrained minimisation, and the mean bond order the rule subtracts is exactly its Lagrange multiplier — the price, in energy per unit of length, of the constraint. A Lagrange multiplier on a length is a force. The shift times the stiffness is the difference between the chain’s multiplier and a ring’s: the extra tension a chain of fixed length carries because its ends want more alternation than its middle.

Read that way, the numbers are unsurprising in hindsight. The tension is set by the ends and does not depend on how long the chain is; the shift it produces in each bond is that tension shared among all of them, so it falls as one over the length. A ring has no ends, carries no tension, and has no shift — which is why it cannot be the bulk of a chain that does. And a soft chain, whose ends heal within a few bonds, has the largest shift of the series, −0.33 over the number of bonds against −0.08 at the stiffest, because its ends’ extra alternation is concentrated and strong; the long, weak tails of the stiff chains ask for less.

That last point runs against the intuition the rest of this argument has built. Everything difficult about the stiff chains has come from their long tails, and it would be natural to expect the reference’s error to be worst where the tails are longest. The tension is largest on the soft chains. What makes it matter most on the stiff ones is not its size but what it is compared with: the stiff chains’ excesses are the ones followed out to 10⁻⁸ over a hundred and fifty bonds, and a fixed error of 10⁻⁵ against a tail that small is the whole of the last several decay lengths.

A ring of 320 is not bulk either, at the stiff end #

The free ring fails for a second reason at the stiff end, and it is independent of the first. A ring of 320 sites has 160 allowed momenta, and the self-consistent alternation depends on how finely they sample the region near the gap, where the band’s energy changes fastest. At K = 1.1 the gap is large and 160 momenta are plenty; the ring and the infinite chain agree to the last digit. At K = 3.1 the gap is small, and a ring of 320 overstates the infinite chain’s alternation by three per cent — 0.011656 against 0.011302 — which is a further 3.4 × 10⁻⁴ of error on top of the missing shift.

So the proposed reference had two errors, one at each end of the series: the missing shift everywhere, and the ring’s own size at the stiff end. The infinite chain at the chain’s own shift has neither.

Against the chain’s own bulk, nothing turns #

Read against the shifted bulk, no profile turns back. At K = 2.2, where the rate against the middle reached its minimum at bond 99 and climbed, it now falls all the way to bond 137. At K = 2.5, 2.8 and 3.1, which turned at bonds 85, 75 and 69, the rate falls to the last bond before the middle.

Near the middle it does something the rate against the middle never did: it falls towards zero. That is the signature of two tails, not one. A profile that is the sum of a tail from each end is symmetric about the midpoint, so its slope there is zero and so is the local rate. The rate against the middle climbed because a wrong reference forced the excess to zero; the rate against the bulk falls because the true excess has a minimum there and stays positive.

The middle of a stiff chain is not bulk #

The shifted bulk also measures what the previous essay could only infer: how far each stiff chain’s middle is from bulk. K = 2.0 is bulk at its middle to 2 × 10⁻¹¹. K = 2.2 is 2.7 × 10⁻⁹ above — under the floor, which is why its profile was the one that turned latest and most gently. K = 2.5 is 6.3 × 10⁻⁷ above, K = 2.8 1.7 × 10⁻⁵ and K = 3.1 1.2 × 10⁻⁴: the stiffest chain’s middle is a hundredth of its bulk alternation away from it. The earlier essay computed that the reference was about half a per cent of the excess at K = 2.2’s turn, from a model. Here the size of the middle’s departure is measured, and the model’s number is confirmed rather than assumed.

Two tails, to the middle #

The form of the tail has been fixed since the decay was found to keep slowing: a power of the distance times an exponential, $b^{-p}e^{-b/\lambda}$. A chain has two ends, so the excess should be the sum of two such tails, one measured from each end. Fitted that way from bond six to the middle — three parameters, least squares on the logarithm — the sum lies on every profile to the middle, with an RMS in the logarithm under 0.013 at every stiffness and under 0.005 for all the stiff ones. The fitted tails at the midpoint reproduce the measured departures from bulk to within two per cent: 2.8 × 10⁻⁹ against 2.7 at K = 2.2, 1.22 × 10⁻⁴ against 1.22 at K = 3.1.

A single tail cannot do it on the stiff chains. From K = 2.5 up, fitting one tail from the near end alone leaves an RMS of 0.08 — seventeen to twenty-six times the two-ended fit’s — because a single decaying tail cannot turn flat at the middle. So the question the previous essay left, whether what lies past the turn is tail after all, has a definite answer: it is, and it is the other end’s tail.

The exponent keeps rising #

The published exponents, read from the local rate over bonds six to sixty with the middle as reference, rose from 0.41 at K = 1.1 to 0.659 at K = 2.8 and then fell, to 0.650 at K = 3.1. Every earlier essay treated the stiff end of that series as unresolved rather than as a fall, and the rule of thumb and the far end were each swept to show they were not responsible.

The two-ended fit against the shifted bulk gives 0.600, 0.648, 0.696 and 0.726 for K = 2.2, 2.5, 2.8 and 3.1. The series rises through the whole range of stiffness, and the stiff end continues the soft end’s trend instead of turning over.

Where the fit starts matters for the softer chains, as it always has on this tail: starting at bond 30 instead of 6 lowers K = 2.0’s value from 0.573 to 0.539, because the power-law-times-exponential form is the asymptotic form and the first few decay lengths still carry corrections to it. But at every start the series rises with stiffness from K = 1.8 on, and the stiffest value is the steadiest in the table — 0.726, 0.734, 0.734 and 0.726 from bonds 6, 10, 20 and 30. Its decay length rises too, from the published 37.6 bonds to 46.0.

How long a chain would have to be #

The previous essay’s last question was how long a chain has to be before its middle is bulk. With the tails fitted, that is one line: the length at which twice the tail’s value at the midpoint falls to the floor of 10⁻⁸.

It is 292 sites for K = 2.2 — so a chain of 320 is just long enough, which is why its middle sits under the floor — 456 for K = 2.5, 711 for K = 2.8 and 1,102 for K = 3.1. The estimate of eight hundred sites for the stiffest case, made when the excess was still read against the middle, was short by three hundred. And that length answers a different question from the one it was estimated for. With the shifted bulk as reference, a chain need not be long enough for its middle to be bulk; the two-ended fit reads the stiff profiles on 320 sites already. What the length measures now is how far this model’s chain is from the one earlier essays assumed they were measuring.

What the picture cannot show #

The bulk is this relaxation’s bulk. The shift exists because the rule holds the chain’s length fixed, and a model that let the length relax — or a real polymer, whose ends are free — would have a different bulk and no shift. That is not a defect of the reference; the reference is right for the profiles, which were computed with the rule. But the exponents measured here are properties of a chain at fixed length.

The tail form is assumed. Two tails of the form $b^{-p}e^{-b/\lambda}$ fit every profile to under half a per cent in the logarithm, and the softer chains’ exponents still move by a few hundredths with the starting bond. A tail with a further correction term would move them again. The stiff exponents are the steadiest numbers in the table, but they are the parameters of a form, not of a proof.

The ends interact beyond adding. On the stiffest chain the two tails overlap everywhere, and the fit treats their sum as exact. A relaxation that coupled the two ends’ healing nonlinearly would leave a residual concentrated at the middle; the fit’s RMS, a third of a per cent, bounds it without identifying it.

How it was computed #

The stored relaxations of a 320-site chain at ten stiffnesses — the ones the Peierls instability and its two coupled states set up — were recomputed to recover the signed bond changes that the stored envelope smooths away; the recomputed envelopes agree with the stored ones exactly. The shift is the mean of the two middle bonds’ changes. The bulk is the self-consistent alternation of an infinite chain with hoppings $1 + c \pm d$, from a Bloch sum over 8,192 momenta, converged to 10⁻¹² against 4,096. The free ring uses the 160 momenta a ring of 320 allows. The two-ended fits are Levenberg–Marquardt least squares on the logarithm of the excess, from a stated first bond to the middle, started from the quarter-window values.

The checks, made wherever these figures are drawn: the Bloch sum converged; the shift times the number of bonds within two parts in a thousand between 160 and 320 sites at three stiffnesses, and within one per cent from 80; the free bulk more than 10⁻⁵ from the shifted one at every stiffness; every soft middle within 10⁻¹⁰ of the shifted bulk; no stiff profile turning against it while every stiff profile turns against its own middle; every stiff middle above bulk by more at each stiffness, reaching 10⁻⁴; two tails leaving under 0.013 in the logarithm everywhere and a single tail ten times more from K = 2.5; the exponent rising with stiffness from K = 1.8 at every start; and the stiffest between 0.72 and 0.74 at every start. The refusal is the soft chains’ published exponents, which must not move by a thousandth when the shifted bulk replaces their middles — a reference that did more than it claims would move them.

Who found what #

Peierls argued in 1955 that a half-filled chain cannot stay uniform. The self-consistent treatment of the dimerised chain with a bond-order rule and an elastic constant is the Su–Schrieffer–Heeger model’s static limit, and the finite chain’s end effects and their healing are the subject of this whole argument.

What is computed here is the shift the fixed-length rule puts on every bond and its scaling with length, the bulk that includes it, the ring’s two errors, the ten profiles re-read against the shifted bulk, the two-ended fits and their dependence on where they start, and the chain length at which each middle would be bulk.

Still open: a chain with free ends, and the correction to the tail #

The obvious open question is the rule. A chain whose total length is free to relax has no shift, and its bulk is the free infinite chain’s. Its ends would heal differently — the extra alternation they carry would no longer be paid for by the rest — and whether the tail exponents of a free-length chain are the ones measured here, or whether the constraint has been in the exponents all along, is a relaxation with one line of the rule removed.

The nearer question is the tail’s form. The softer chains’ exponents fall by a few hundredths as the fit starts deeper, which says the asymptotic form has a correction the fit absorbs. A third term — a further power of one over the distance in the rate — fitted to the whole half chain against the shifted bulk would say whether the drift is that correction and whether the stiff exponents, already steady, move at all when it is included.