The distortion the ends decide

A chain cannot stay even computed the Peierls argument in the form it is usually given: a uniform half-filled chain gains more from alternating than the alternation costs, because the electronic gain goes as δ² times a logarithm and the elastic cost goes as δ². The gain wins for small δ however stiff the chain is, so the uniform chain is unstable.

Everything in that argument is about an infinite chain, and the picture it produces has two dimerised structures — short-long-short-long and long-short-long-short — related by a translation of one site and therefore degenerate. That degeneracy is the foundation of a great deal of subsequent physics: a boundary between the two is a soliton, and solitons are what a doped conjugated polymer is usually said to conduct by.

It is worth noticing how much of that structure rests on a symmetry the argument acquires only in the limit. The two dimerised patterns of an infinite chain are related by a translation, and a translation is a symmetry of an infinite chain and of nothing else. Every finite piece of one has boundaries, and boundaries are precisely what a translation does not preserve.

A finite chain does not have it. A chain of an even number of sites has an odd number of bonds, so the two patterns are not related by any symmetry: one has a short bond at each end and the other has a long one, and they are different molecules.

Holding the distortion fixed #

The difference between the two patterns is a property of the two ends, and measuring it cleanly means not letting anything else vary. So both are held at the same δ — the bulk optimum, 0.122 — and only the pattern is changed.

At sixteen sites the pattern with short bonds at the ends is more stable by 6.134 × 10⁻² per site. At thirty-two, by 3.3455 × 10⁻². At sixty-four, 1.6981 × 10⁻². Each doubling halves it, which is the signature of a quantity that lives somewhere fixed while the number of sites it is divided among grows.

Multiplied by the length it settles: 0.981, 1.071, 1.087, 1.08715, 1.08715, 1.08715. A fitted exponent over the last four sizes gives −0.99986 for the per-site difference, which is one.

An end is a bond that is not there, and it is worth a fixed amount of energy. Which phase gets it is decided by whether the bond that is missing was going to be a strong one or a weak one, and a chain that begins and ends with a strong bond is the one that wastes less.

The number is worth putting in context. One unit of the hopping is the whole binding a single bond contributes in this model at its own optimum, so 1.08715 for the pair of ends is very nearly one bond’s worth divided between them — which is exactly what the counting argument predicts and is a check on it rather than a coincidence.

The second phase does not always exist #

Holding δ fixed is the clean measurement and it hides something. Let both patterns find their own best distortion and the long-bond-first one turns out, below a certain length, not to distort at all.

At ninety-six sites the search returns δ = 3 × 10⁻⁹ for the long-bond-first pattern — the boundary of the search, meaning no distortion at all — against 0.13434 for the other. At a hundred and twenty-eight it returns 0.07909, and from there on both patterns exist.

The reason is arithmetic. A chain of n sites has n − 1 bonds; the short-first phase gets ⌈(n−1)/2⌉ short bonds and the long-first phase gets one fewer. That surplus is a fixed advantage, and the long-first phase has to make up a fixed deficit out of a gain that is proportional to the length. Below the length at which the gain covers the deficit, its optimum is δ = 0.

So a short chain has one dimerisation, not two. The degeneracy the soliton picture rests on is a large-system property, and the length at which it appears is computable: here, between ninety-six and a hundred and twenty-eight sites at an elastic constant of 1.6.

That is not a small number of atoms. A polyacetylene chain of a hundred and twenty-eight carbons is a real object, and the statement is that shorter ones have a single ground-state structure with no degenerate partner for a soliton to separate it from.

The ring, and the two families #

The ring is the control: no ends, no phase distinction, and the bulk answer directly. Its best distortion converges to 0.12215 from above, and the open chains approach it as their ends become a smaller share of them.

But the ring has a distinction of its own and it is a count rather than a boundary.

A half-filled ring of 4m has a degenerate pair of levels at the Fermi level, half filled. Any distortion that splits them lowers the energy at first order, so such a ring distorts strongly: a ring of eight goes to δ = 0.19948, of twelve to 0.16292, of sixteen to 0.14545, of twenty to 0.13588.

A half-filled ring of 4m + 2 has a closed shell and a gap. Splitting the levels costs at first order and gains only at second, so a small ring of this family does not distort at all: rings of ten and fourteen return δ = 0 exactly, a ring of eighteen manages 0.08304, and one of thirty 0.11750.

The two families approach each other as the rings grow — the gap of a 4m + 2 ring falls as one over its size — and they never meet, because one family always has something at the Fermi level and the other never does. That is the same shell closure that decides aromaticity deciding a structural question, and it is the same one that pins a finite metal’s carrier count, as the metal a thermometer cannot find computes on the same rings.

What this does to the first-order picture #

The distortion the filling chooses found that the period of a distortion is set by the filling — period two at half, three at a third, four at a quarter — by margins of three to one. That result is about which pattern wins and it is unaffected by everything here, because it compares patterns in a ring.

A distortion needs two states found the condition under which a symmetric structure survives at all: the gap has to exceed twice the coupling squared over the stiffness. That is a second-order argument and it applies to the 4m + 2 rings above, which is why they distort late rather than not at all.

Two distortions in one coordinate found first- and second-order terms reinforcing rather than competing. The two ring families here are the pure cases of each: a 4m ring is a first-order distortion and a 4m + 2 ring is a second-order one, in the same coordinate, in structures four sites apart.

The smallest first-order case is cyclobutadiene, whose half-filled degenerate pair makes the square unstable against a rectangle. Every ring of 4m is that molecule’s situation repeated, and the chains in this essay are the open version of it with two ends added.

What a boundary condition is worth #

Put together, the three numbers say how much of a Peierls calculation is a bulk statement.

The end: 1.08715 in units of the hopping, fixed. That is roughly one bond’s worth of binding, which is what a missing bond ought to cost.

The distinction between a ring and a chain of the same length: 0.39873, also fixed, also an end effect and about a third the size — because both open phases pay it and the difference between them is what the larger number measures.

The onset of the second phase: between ninety-six and a hundred and twenty-eight sites.

None of those appears in a calculation done in the infinite limit, and all three are properties of objects a chemist would call molecules.

What lives at an end is a state localised there, which a ring does not have — and the end is the hardest place to bind for the same reason the end energy is positive. The two facts are the same fact about a boundary, one read as a state and one as an energy.

The gap is not what decides it #

There is one number in the two-phase comparison that does not do what a reader would expect, and it is worth pointing at.

The two patterns have the same gap. They are the same set of bond strengths in a different order, so their level distributions are very nearly identical and their band gaps agree to within the end effects. The difference between them is not a gap difference; it is a difference in how many strong bonds there are.

That separates this result from the other Peierls arguments cleanly. Every one of them runs through the gap — the gap is not the band width, the gap decides the stabilisation, the gap has to exceed twice the coupling squared over the stiffness. Here the gap is a bystander and the arithmetic is a count of bonds.

So an end effect is a chemical statement rather than an electronic one. A chain with a strong bond at each end has one more strong bond than a chain with a weak bond at each end, and one bond’s worth of binding is what the 1.08715 amounts to.

The spectra of the chains themselves are worth a sentence rather than a figure: every level of a uniform chain lies inside the same interval whatever its length, and the two dimerised patterns have the same gap as each other — they are the same set of bond strengths in a different order, so their level distributions are nearly identical and the difference between them is not a gap difference at all.

And which period a distortion chooses at a given filling is a separate comparison between patterns in a ring, unaffected by anything here: the end effect decides between two patterns of the same period and not between periods.

A fixed energy at a boundary is a soliton #

The finding — a fixed amount of energy living at the two ends, independent of the chain’s length — has a name in the material these chains model, and the name comes with measurements.

A long chain that has dimerised has chosen one of two patterns: long-short-long-short, or short-long-short-long. The two are equally good, so a real chain can contain regions of each, and where two such regions meet there is a boundary. That boundary is a soliton: a localised defect, costing a fixed amount of energy, whose cost does not depend on how long the chain on either side of it is.

Which is exactly the quantity measured here. The two dimerisations of a finite chain are the two patterns, the energy difference between them lives at the ends, and it is the same number whatever the length — because a boundary’s cost is a property of the boundary.

In polyacetylene those boundaries are real and they are the reason the material is interesting. A soliton in a polyacetylene chain costs a few tenths of an electronvolt to make, and it has two properties that no ordinary excitation has.

It carries spin without charge, or charge without spin. A neutral soliton has one unpaired electron and no charge; remove that electron and the defect carries a positive charge and no spin. That inversion of the usual pairing — charge and spin travelling together — is the signature that identified them, and it was seen in magnetic and optical measurements on doped polyacetylene before anybody had a picture of what was carrying it.

And it moves along the chain at almost no cost, because moving it means relabelling which bonds are long and which are short, and the two patterns are degenerate. That is what makes doped polyacetylene conduct: the carriers are these boundaries rather than ordinary electrons in a band.

So the number reported here — a fixed energy living at a boundary, unchanged by the length — is the formation energy of the object that carries the current in the first conducting polymer, and the length independence is what makes it a well-defined object at all.

It also explains the other finding. Below a hundred and twenty-eight sites the second dimerisation does not exist, which in this language says a chain shorter than that cannot contain a boundary: the two ends are too close for a defect to fit between them, and the defect has a width.

That width is a separate measurement, and putting the two together closes the description. A boundary between the two dimerisation patterns is not a sharp step from one to the other; the alternation reverses smoothly over some number of bonds, and that number is what stops a short chain from accommodating one. So the hundred-and-twenty-eight-site threshold and the healing length are the same statement — a defect with a width needs room, and a chain shorter than the width has nowhere to put it.

It also says why the per-site difference falls as one over the length with an exponent measured at $-0.99986$ rather than at anything more interesting. A fixed total divided among a growing number of sites is exactly a reciprocal, and the four nines are the statement that the total really is fixed — that nothing about the boundary depends on how much chain is on either side of it, which is the defining property of a localised defect and the reason it can be given an energy at all.

The contrast with the bulk quantities is the useful one to carry. An energy per site is a property of the material and an energy per boundary is a property of the boundary, and a finite chain contains both — which is why its total energy is not extensive until the boundary term has been separated out, and why separating it is the first thing that had to be done.

What this cannot say #

No relaxation of the ends. Both phases are held at a uniform δ throughout, so nothing is allowed to distort more near an end than in the middle. A real chain does exactly that, and letting it would reduce the end energy by an unknown amount without changing its existence.

One filling. Everything is at half filling, where the distortion of period two is the one that pays. The distortion the filling chooses is where the other fillings live, and the two-phase question would have to be asked again for each of them.

No repulsion. The received view is that the alternation of a long polyene is set by correlation as much as by the σ framework, and none of it is here.

No soliton. A boundary between the two patterns is a defect this model could hold, and computing one would need a chain long enough to have both and a way of joining them — which is the natural next calculation and not this one.

And the onset length depends on the elastic constant. At a stiffer chain the second phase appears later and at a softer one sooner. What does not depend on it is that there is an onset, since the deficit is fixed and the gain is proportional to the length.

What was checked #

The pattern with a short bond at each end is the more stable, at every length.

The difference falls as one over the length, with a fitted exponent between −0.98 and −1.02 over the four largest sizes.

The total settles, to better than half a per cent across those sizes — the statement that the whole difference is a fixed amount of energy at the ends.

Both open chains are less bound per site than the ring of the same length, because an end is a bond that is not there.

A chain of eight has only one dimerisation and one of a hundred and twenty-eight has two, with the second phase appearing at one length and staying.

The two ring families do not converge: a ring of 4m distorts further than one of 4m + 2 at every size compared, and the gap between them narrows without closing.

And an odd ring is refused rather than built, because a dimerisation of an odd ring would have its last bond disagreeing with its first.

Still open: the soliton, and a relaxed profile #

The obvious open question is the soliton. Both patterns exist above a hundred and twenty-eight sites, so a chain of two hundred and fifty-six can hold one of each with a boundary between them, and the boundary is a defect with a computable energy and a computable state. Defect states in the gap and the level between two of them are already computed, and the soliton is the same object made by a change of pattern rather than by a change of site energy. Whether its level sits exactly at mid-gap — which the received theory says it does, by a symmetry — is a check the same calculation could make.

The nearer question is the relaxation the model forbids. Holding δ uniform is what makes the end energy a clean constant, and a real chain’s distortion is larger near an end than in the middle because the end has less to lose. Allowing δ to vary site by site turns the calculation into a minimisation over as many parameters as there are bonds, which is not expensive at these sizes, and the profile it would produce — how far into the chain the end’s influence reaches — is a length not yet measured.