The carriers a distortion was hiding

Asking what a thermometer can and cannot tell about a metal turns up, among other things, a suspicious result. A half-filled ring of 4m sites carries exactly one pair of thermal carriers at every temperature — a degenerate pair sits at the Fermi level, two electrons occupy two orbitals, and no amount of cooling empties them.

It then said what was wrong with that. “A degenerate pair at the Fermi level is not a stable situation. Such a ring distorts.” The carrier count was the carrier count of a structure that will not hold still.

Computing the two together — the distortion the ring settles at, and the carriers it has once it has settled — turns the flagged result into a different one. The pinned pair is not there. A cold ring of forty carries two hundredths of a millionth of a carrier, not one pair, and the pair only reappears at a temperature that has undone the distortion.

What is being minimised #

Everything else in this field is a ground-state energy, and a competition between a distortion and a temperature needs a free energy — so the electronic entropy is in it.

At each temperature the ring’s occupations come from a Fermi distribution over its own levels, the entropy is the usual $-[f\ln f + (1-f)\ln(1-f)]$ summed over states and doubled for spin, and the quantity maximised is the electronic free energy less the elastic cost $K\delta^2$. The alternation is found by a golden-section search over $\delta$ at each temperature, so every point is a genuine optimum rather than the end of a trajectory.

The one parameter is the stiffness $K$, and it is stated rather than fitted — the same 1.6 a chain cannot stay even uses.

Two things make this a different calculation from that one rather than the same calculation warmed up. The first is the entropy, which is what allows a distortion to be undone rather than merely reduced: an alternation that costs nothing at zero temperature can still be given up for the entropy of a partly occupied band. The second is that the carriers are read from the same occupations the free energy was computed with, so the two curves in the figure above are two readings of one calculation rather than two calculations that happen to be plotted together.

What the cold ring is #

At the lowest temperature tested the ring of forty alternates by 0.12317 and opens a gap of 0.49267.

Its carrier count is $2\times10^{-8}$, against 1.000000 for the same ring held flat. The distortion has removed the entire pinned pair, and it has removed it by the mechanism the pair’s existence was the condition for: a degenerate pair splits, one member goes down and is filled, the other goes up and is empty, and there is nothing at the Fermi level any more.

So the pinned-pair finding survives as a statement about the held-flat ring and does not survive as a statement about the ring. It is the same shape of correction as a half-filled band is not always a metal, except that the thing the one-electron picture missed here is not a repulsion but a nuclear coordinate.

And it is a third way of not being a metal, beside the two this field has already counted. The third way to be an insulator collected them; this adds one that is not a property of the electrons at all but of the structure they are allowed to choose. A calculation that fixes the geometry has ruled it out before it starts.

Why no spring is stiff enough #

The contrast with a ring of 4m + 2 is where the mechanism is, and it is a statement about orders rather than about magnitudes.

A ring of 4m has two orbitals at exactly the same energy holding two electrons. Alternating splits them linearly in $\delta$, so the electronic gain starts off proportional to $\delta$ while the elastic cost is proportional to $\delta^2$ — and a linear gain beats a quadratic cost for small enough $\delta$ whatever the coefficients are.

A ring of 4m + 2 has a filled shell with a finite-size gap above it, so alternating gains only at second order, and a stiff enough spring wins.

Measured: at elastic constants of 3, 6, 12, 24 and 48 the ring of forty still distorts — by 0.0301, 0.0108, 0.0047, 0.0021 and 0.0005 — and the ring of forty-two distorts at none of them.

That is a Jahn–Teller argument rather than a Peierls one, and the distinction matters. A chain cannot stay even is a statement about a logarithm in the thermodynamic limit and it holds for any half-filled chain. This is a statement about an exact degeneracy in a finite ring, it is first order rather than logarithmic, and it applies to one family of sizes.

The undoing, which is not a jump #

The alternation is undone at kT = 0.13579, found by bisection, and it goes continuously.

Fitted over the last decade before it goes, the alternation falls as the reduced temperature to the power 0.44 — near the one-half a mean-field transition of this kind gives, and quoted as a measurement over a finite range rather than as an exponent.

The undoing temperature is 0.2756 of the cold gap, which is the ratio this kind of transition is usually characterised by and is computed here from two quantities that share no arithmetic: one is a bisection on a free-energy optimum and the other is a diagonalisation.

And the carriers do not jump. By the time the distortion goes they are at 1.75 of the 1.81 the flat ring has at the same temperature — because the gap has been closing continuously and the carriers have been coming back with it. There is no temperature at which a measurement would see a sudden change in the carrier count, which is the practical version of the transition being second order.

What it says about the rigid-ring result #

The rigid-ring result’s central claim is untouched and is worth restating so that the correction is not read as a retraction.

Its claim was about a thermometer, not about a ring: that a carrier count at a stated temperature cannot tell a metal from a gapped structure, because a small enough gap is thermally erased. That is a statement about two computed spectra and it stands.

What changes is one of its examples. The 4m ring was offered as a case where the carrier count is pinned at one pair however cold it gets, and the pin turns out to be an artefact of holding the ring rigid. The pinned pair is real for a ring that cannot move — a ring held in a lattice, or one whose distortion is frustrated by its surroundings — and is not a property of an isolated ring of that size.

And it makes an existing warning sharper. The metal a thermometer cannot find is about a measurement being unable to distinguish two structures. This is about a calculation being unable to, because it was not allowed to relax — which is a failure mode with no thermometer in it at all.

What a calculation should take from it #

A carrier count computed at a fixed geometry is a carrier count of that geometry. Which sounds obvious and is exactly the assumption the rigid-ring result was built on. The pinned pair is real for a rigid ring and absent for a free one, and nothing about the calculation announces which case it is in.

The signature to look for is a degeneracy at the Fermi level. Wherever there is one, a distortion gains at first order and the structure will take it — so a computed metal whose Fermi level sits on a degenerate pair is a computed metal that has not been allowed to relax.

And the temperature that undoes it is a fraction of the gap, not a multiple. 0.28 here, which means the distortion survives to temperatures well below the gap it opened — so a material with a gap of a tenth of an electronvolt is distorted at room temperature and one with a gap of a hundredth is not.

What this cannot say #

One electron and no repulsion. A half-filled ring is exactly where an on-site repulsion matters most, and the repulsion competes with the distortion for the same degeneracy: the insulator band theory cannot see opens a gap without moving a nucleus. Which of the two wins in a real material is not something this model can answer, and the answer is known to depend on the material.

A rigid lattice around nothing. The ring is free to distort at no cost beyond its own springs. A ring embedded in a solid pays more, and the pinned-pair result may well be the right answer for that case — which is not tested here.

The alternation is one number per bond pattern. The relaxation here is over a single alternation parameter rather than over every bond separately, which is what a chain relaxed bond by bond does and would change the numbers near the ends of an open chain. A ring has no ends, which is why one parameter is enough here.

And the temperatures are in units of the resonance integral. kT = 0.136 is not a temperature until β is given a value, and β is not determined by any spectrum to better than a factor of two and a half. So the ratio of the undoing temperature to the gap is the transferable number and the temperature itself is not.

Nor is there a phase transition here in any strict sense. A ring of forty is a finite system, and a finite system has no sharp transition — what is computed is the temperature at which a variational minimum moves to the boundary of its parameter, which is a property of the free-energy surface rather than of a thermodynamic limit. The exponent quoted above should be read the same way.

What is quoted, and what is computed #

Nothing is quoted. There is no material named in this essay and no measurement. The ring, its filling, the stiffness and the temperatures are the model’s parameters; every level, occupation, entropy, free energy, alternation, gap, carrier count, transition temperature and exponent is computed.

Two of the numbers are computed twice by different means, which is what makes the ratio between them worth quoting. The undoing temperature comes from a bisection on where a free-energy optimum reaches the boundary of its parameter; the cold gap comes from diagonalising the ring at the alternation that optimum settled at. Neither is the other read back to itself, and their ratio — 0.2756 — is the only dimensionless number in the essay.

What was checked #

A cold ring of 4m distorts rather than sitting with a degenerate pair at the Fermi level.

And the distortion removes the pinned pair entirely — both halves, because a distortion that reduced the carriers by a little would satisfy the first check alone.

The alternation does not grow with temperature, at every step of the scan, and is gone at the highest temperature tested — the shape of the curve, checked rather than eyeballed.

The distortion is undone at a definite temperature, found by bisection.

And it goes continuously rather than as a jump: the alternation just below the transition is under a tenth of its cold value, so the carrier count is already most of the way up before the distortion goes.

The undoing temperature is a sizeable fraction of the cold gap and smaller than it. Measured: 0.2756.

A ring of 4m distorts at every stiffness tested and a ring of 4m + 2 at none of them, which is the first-order-against-second-order argument turned into a pair of checks that could each fail on its own. It is also the tripwire: a minimiser that always returned a distortion would pass the first and fail the second, and one that had lost the electronic gain would do the reverse.

A ring of 4m has a pair of levels at exactly the same energy holding two electrons and a ring of 4m + 2 does not, which is a count of electrons rather than a property of the size. That exact degeneracy is what makes one family of ring sizes a special case here, and it is the same degeneracy that decides whether a ring is aromatic.

What the distortion looks like to a diffractometer #

The distortion computed here is a structural change, and structural changes have an experimental signature that requires no measurement of a gap, a carrier count or a conductivity. It is worth naming, because it is how the effect is actually identified in materials.

Alternating the bonds of a chain doubles its repeat distance. Two atoms that were equivalent are no longer, and a unit cell that contained one now contains two. A diffraction pattern responds to that immediately: new reflections appear, at positions halfway between the existing ones, because the reciprocal lattice has been halved along the chain direction.

Those extra peaks are not weak versions of the existing ones. They are absent above the transition and present below it, so their appearance is a yes-or-no observation rather than a shift to be measured — and their intensity, which grows from zero, is proportional to the square of the distortion amplitude the calculation here computes.

The materials this happens in are a well-defined family. Quasi-one-dimensional conductors — chain-like inorganic compounds and the organic charge-transfer salts — undergo exactly this transition on cooling, at temperatures from tens to a couple of hundred kelvin, and each is identified by superlattice reflections appearing below its own transition temperature. Above it the chain is uniform and conducts; below it the chain has alternated, a gap has opened, and it does not.

Polyacetylene is the case at the other extreme and is worth the contrast. Its gap is about one and a half electronvolts, which is fifty times room temperature, so the distortion is never undone by warming: there is no transition temperature to find, the alternation is present in every sample at every temperature, and the superlattice is simply part of the structure.

That gives the undoing temperature a physical meaning it would not otherwise have. The temperature at which the distortion is undone is a measurable transition temperature, it is what separates a material that conducts when warm from one that never conducts, and the quantity that decides which case a material is in is the size of the gap against room temperature.

The diffraction signature also supplies something the electronic measurements cannot, and it is the reason the family was sorted out this way rather than by resistivity. A resistance rising on cooling says only that carriers are disappearing, and three quite different mechanisms do that — a gap opening by distortion, a gap opening by repulsion, and states localising by disorder. Superlattice reflections say which: only the first changes the positions of the atoms, and only the first therefore has anything for a diffractometer to see. An electronic measurement identifies that something happened and a structural one identifies what, which is why the transition is named for a structural distortion rather than for the gap it opens.

Still open: repulsion against distortion, and the transition’s order #

The obvious open question is the competition the model has no term for. A repulsion and a distortion are two ways of opening a gap at the same degeneracy, and they are not additive: a lattice that has already distorted has less to gain from correlating, and a strongly correlated ring has less to gain from distorting. Each of them can be computed separately on rings of four to eight sites, and computing them together — one Hamiltonian with both a repulsion and an alternation in it — would say which wins where, and whether there is a region in which neither is worth paying for.

The nearer question is about the transition’s order. The exponent came out at 0.44 over the last decade before the alternation goes, which is near the mean-field half and is measured on one ring at one stiffness. Whether it is the same on rings of eighty and a hundred and sixty, and at other stiffnesses, would say whether the number is a property of the transition or of a forty-site ring — and a finite ring is exactly the system where a sharp transition ought not to survive, so an exponent that is stable across sizes would be the more surprising of the two answers.