The distortion the filling chooses

A chain cannot stay even is the classic argument, and it is a good one. Take a chain of equal bonds with one electron per site, alternate the bonds slightly — one a little stronger, the next a little weaker — and diagonalise again. The electrons gain more than the springs lose, and they do so for every spring constant whatever, because the gain is steeper than a parabola near the origin and the elastic cost is exactly a parabola.

That essay was about a half-filled chain, and did not say so in its title. This one is about what happens at every other filling, and the answer is that the alternation stops being the right distortion almost immediately.

Alternating is one pattern among several #

A distortion of a chain is a pattern of bond strengths. Alternating — strong, weak, strong, weak — is the pattern that repeats every two bonds. There is no reason a chain should prefer it over a pattern that repeats every three, or four, or seven, except that at half filling it happens to be the best one available.

To compare periods honestly, the comparison has to be fair, and the obvious way of setting it up is not.

Write the modulation as a cosine of period p and give it an amplitude δ. A cosine of period two takes the values +1, −1, +1, −1 on successive bonds, and the mean of its square is one. A cosine of period three takes 1, −½, −½, and the mean of its square is a half. So at the same δ, the period-two pattern is a distortion twice as large in the sense the springs care about — the elastic energy is the sum of the squared displacements — and a comparison at equal δ hands period two a head start and then reports that it wins.

Here the amplitude of each pattern is scaled so that the sum of squared bond changes is identical whatever the period. The springs pay exactly the same in every case, and what varies between the bars in every figure below is only what the electrons give back.

Take the same ring and impose a period of four instead, and the bond pattern changes without the arithmetic changing: the same twenty-four sites, the same one number per bond, and a different repeat imposed on them.

Half filling picks two, and picks it decisively #

What decides between them is not the pattern but the filling, and the comparison is made by computing the gain for every period at a fixed one.

This is the established case, recovered. The margin is worth noticing: six to one is not a preference, it is a landslide. Any chain whose spring constant is small enough to permit a distortion at all will take this one.

One third filling picks three #

Change the filling and the whole ranking is redrawn, which is the essay’s claim in the form a reader can check: nothing about the ring, the springs or the arithmetic has moved.

Nothing about the ring has changed. The same 120 sites, the same hopping, the same elastic cost, the same set of candidate patterns. Forty electrons have been removed, and the answer is a different distortion.

A quarter picks four, and the runner-up is not the neighbour #

The ordering in this panel is the one that would be hardest to guess and is the clearest evidence that the mechanism given below is the right one. Period three and period five are equally far from the winner as patterns; as gap-openers they are not equally far at all, because five straddles the boundary from one side and three from further away on the other.

A pattern-similarity account — the best period is four, so three and five should be roughly equal seconds — predicts the wrong ordering. The account in terms of where the gap opens predicts the observed one.

The rule, and where it comes from #

The pattern is that the period is 1/f, where f is the fraction of the band that is filled. The mechanism can be stated in the finite language this field uses, without a wavevector anywhere.

A modulation of period p couples levels whose index round the ring differs by n/p. That coupling pushes the two levels apart — one down, one up — and the system gains energy only if the lower of the two is occupied and the upper is empty. Every other pair is either both occupied, in which case the two shifts cancel, or both empty, in which case neither is felt.

So the useful couplings are the ones that straddle the boundary between occupied and empty. At a filling f, the highest occupied level sits at index fn round the ring, and the modulation whose index difference lands exactly there is the one with p = 1/f. Every other period couples pairs that are on the same side of the boundary and gains almost nothing from them.

That is the whole of it, and it explains the two facts the figures show that a cruder account would miss. The winner wins by a margin because it is the only period that opens its gap in the right place. And the second best period is not the one nearest in number to the winner — at quarter filling period five beats period three — because what matters is where the gap opens rather than how similar the pattern looks.

Two thirds gives the same answer as one third #

The fourth panel in the grid above is the one that makes the rule precise. A ring filled two thirds of the way chooses period three, not the period one-and-a-half that a literal reading of “one over the filling” would give.

The reason is that the argument is about the boundary between occupied and empty, and a boundary two thirds of the way up is the same distance from the top as a boundary one third of the way up is from the bottom. What decides is the smaller of the electron count and the hole count — the same pairing that makes an alternant system’s levels symmetric about the middle.

The two cases do not merely agree about which period wins. They agree about the energy gained by every period, to the last bit a double holds — a difference of less than 10⁻⁹ across all six patterns. That is not an approximate correspondence between two similar situations; it is an exact symmetry of the matrix, the same electron–hole symmetry that makes a bipartite structure’s spectrum symmetric about its centre.

The margin is what makes it a prediction #

An arrangement that wins by two per cent is a curiosity; one that wins by six to one is a prediction. It is worth setting the margins out, because they are what decides whether this is a rule a material could be expected to obey.

The margin narrows as the band empties, and the reason is visible in the shapes of the last essay: where the states pile up decides how sharply the boundary between occupied and empty is defined. At half filling the chain’s density of states is at its flattest and the boundary is crisp; at a quarter filling the boundary sits further towards the edge, where the density is rising, and more of the neighbouring periods get some of the benefit.

So even the size of the preference is a fact about the density of states rather than about the chemistry, which is the pattern this whole field keeps producing.

What the distortion opens, and what it does not #

A distortion is worth having because it opens a gap, and the gap is where the electrons’ gain comes from. But the relation between the size of the distortion and the size of the gap is not the relation between the distortion and the width, and the gap is not the band width is the essay that separates the two.

Why the distortion is worth anything at all is a statement about two sequences rather than about one structure: the gap of a uniform chain shrinks as the chain grows and the gap of an alternating one does not. A uniform chain of any length is a metal in the limit and a distorted one is an insulator at every length, and that distinction is what the whole argument turns on.

The width is set by how many neighbours a site has and the gap by how unequal they are.

The competition itself is easiest to see on the smallest ring that has one, where both sides can be drawn against the size of the distortion.

A period-three modulation makes the bonds unequal in a different pattern from a period-two one, and the gap it opens is at a different place in the band — but the width of the band is very nearly unchanged in both cases, because the mean of the squared bond strengths is what fixes it and that has been held constant by construction.

The same argument in a molecule #

This field’s rule is that a solid is a molecule that did not stop, and the reverse reading is available here. A distortion that removes a degeneracy at the top of an occupied set, at a cost that is quadratic and a gain that is not, is the Jahn–Teller argument, and copper is never quite octahedral is its molecular member.

The correspondence is close but not exact, and the difference is instructive. In a molecule the degenerate set is small — two orbitals, or three — and which distortion removes the degeneracy is settled by a direct product in the point group: there is usually one answer and no competition. In a chain the “degeneracy” is a band of closely spaced levels, the candidate distortions are a whole family indexed by their period, and choosing between them is an arithmetic competition rather than a group-theoretical deduction.

So the molecular case has a rule and the extended case has a calculation. What this essay adds is that the calculation has a rule after all, and the rule is the filling.

Why the elastic side has no say in which period wins #

The elastic cost has been held equal between periods deliberately, and it is worth being explicit about what that assumes and what it buys.

It assumes that the springs holding the structure together do not care about the pattern of the displacements, only about their sizes — that the cost of moving a set of atoms is the sum of a cost per atom. That is the harmonic, nearest-neighbour picture, and it is the same assumption the original argument makes when it writes the elastic cost as a parabola.

What it buys is that the whole competition happens on the electronic side. Every bar in every figure above is what the electrons gained; the springs’ contribution is a constant that has been subtracted from all of them equally. So the answer to which period is decided entirely by the band, and the spring constant decides only whether any distortion happens at all.

That separation is what makes the rule clean, and it is also where a real material could depart from it. A structure whose springs are stiffer against one pattern than another — because of a second-neighbour interaction, or because the atoms are not all alike — has a competition with two sides to it, and the period it settles on need not be the one the band alone would choose.

The one-electron caution, and it is sharper here than usual #

Every energy above is a sum of one-electron levels with no repulsion between the electrons at all. In this particular argument that omission has a name and a direction.

A distortion of period 1/f competes, in a real material, with an arrangement of the spins of period 1/f — the electrons alternating in orientation rather than the bonds alternating in strength — and which of the two wins depends on the repulsion that this model does not have. The insulator band theory cannot see is the essay about the limit where the repulsion wins outright, and there the distortion is not needed to make an insulator because the repulsion has already made one.

So what is computed here is real and is one of two mechanisms. Where the repulsion is weak the distortion is what happens; where it is strong the model above is describing a competitor that has already lost.

Reading it backwards: a period is a measurement #

The rule runs in both directions, and the direction that is useful to somebody looking at a material rather than at a model is the reverse one.

A structure that has spontaneously modulated with a period of three carries, in that period, a statement about how many electrons are in its band: a third of a band, or two thirds of one. That is a quantity nothing about the structure’s composition makes obvious and which no measurement of a bond length gives directly. The period is a count of electrons, read off a geometry.

This is the same kind of inference that runs through the whole collection — the octet rule, Hückel’s 4n + 2 and the 8 − N rule are one rule counted three ways — and it is one of the few cases where the count is recoverable from a picture of where the atoms are.

The inference has one leg to stand on, and it is the exactness of the electron–hole pairing above. A period of three is consistent with a third of a band and with two thirds of one, and nothing about the distortion distinguishes them, because the two cases are the same distortion in the model. Deciding between them takes a different measurement.

The period at a filling that is not a simple fraction #

The rule that the period is the reciprocal of the filling is tested here at fillings that give whole numbers, and its most striking consequence appears where it does not — because nothing in the argument requires the period to be a whole number of lattice spacings.

Fill a chain to some fraction that is not a ratio of small integers and the favoured distortion has a period that does not fit the lattice. The pattern of long and short bonds never repeats in step with the atoms: it drifts, slowly, and the structure has two lengths in it that are not commensurate with one another.

Such structures exist and are observed. A quasi-one-dimensional conductor cooled below its transition develops a periodic lattice distortion, and diffraction sees it as a set of satellite reflections displaced from the ordinary Bragg peaks by a vector that need not be a rational fraction of the reciprocal lattice. In many of these materials it is not, and the structure is described as incommensurate.

The displacement is measured, and it agrees with the rule. The satellite sits at twice the Fermi wavevector — which is the reciprocal-space statement of the period is one over the filling — and it moves when the filling is changed by doping.

That gives the rule an unusual status among structural predictions. The period is a measurement of the filling, obtainable by diffraction, on a material whose carrier concentration might otherwise be hard to determine — and it is a measurement of a purely electronic quantity made by looking at where the atoms are.

It also explains why some of these materials lock in and others do not. A period close to a simple fraction can be pulled onto it by the lattice, which prefers a repeat it can accommodate; a period far from any simple fraction cannot, and stays incommensurate. Both happen, sometimes in the same compound at different temperatures, and the essay’s arithmetic supplies the number the lattice is negotiating with.

What is left #

Two things this could not do, both of them stated rather than worked around.

The periods available to a ring of 120 sites are its divisors, so the comparison is over 2, 3, 4, 5, 6 and 8 rather than over every real number. A filling of, say, two fifths would want a period of two and a half, which no ring of any size can hold exactly; what a real material does in that case is take a period that is incommensurate with its own structure, and nothing in a finite matrix of this kind can represent one.

And the amplitude here is fixed rather than optimised. What is compared is what each period is worth at one stated distortion, not what each period is worth at its own best distortion, which is the quantity a full treatment would minimise. The margins are wide enough that the ordering is not in doubt — six to one at half filling, four to one at a third — but the numbers are comparisons rather than energies of any real material.