The insulator band theory cannot see

The one-electron account of solids makes a claim it cannot compute. A half-filled band is not always a metal sets out the argument in words — that a material whose electrons repel each other strongly enough can decline to conduct even when the one-electron picture says it must — and records plainly that a one-electron model has no term for electron repulsion and therefore no way to produce the effect.

This is the computation. It is done on four sites, because four sites can be solved exactly.

The system, and why it is the awkward case #

A ring of four sites has one-electron levels at $-2t$, $0$, $0$ and $+2t$. Put four electrons in and the lowest level takes two, leaving two electrons for a degenerate pair at zero.

That is the definition of a metal at this size. There is no gap between the highest occupied and lowest empty level, because they are the same level; an arbitrarily small excitation is available; and the one-electron picture has nothing further to say. The site established the large-system version of that in what a metal actually is, where a half-filled ring’s cheapest excitation is shown to fall towards zero as the ring grows.

The four-site ring is the smallest system with that property, which makes it the right test case: any gap it acquires cannot come from finite size, because at $U = 0$ it has none.

What a charge gap is #

The quantity that decides whether a material conducts is not the HOMO–LUMO gap. It is what it costs to move an electron from one copy of the system to another:

$$\Delta_c = E(N+1) + E(N-1) - 2E(N)$$

Take one electron off one copy and put it on another. If that costs nothing, charges move freely and the material conducts; if it costs something, they do not.

That definition needs three many-electron energies, and there is the whole difficulty. A one-electron model computes $E(N \pm 1)$ by adding or removing an electron from an orbital, so the expression collapses to the HOMO–LUMO gap and can never be anything else. A model in which the energy is not a sum over orbital energies gives something different, and this one does.

The computation is three exact diagonalisations per point: the half-filled ring, the ring with one electron more, and the ring with one fewer.

The numbers #

The exact gap is zero at $U = 0$ — to twelve decimal places, which is the check that the calculation is describing the same system band theory is — and rises monotonically, reaching six times the hopping amplitude at $U = 8$ and continuing.

The mechanism is legible in the arithmetic rather than assumed. Adding an electron to a system where every site already has one means putting two on some site, which costs $U$; removing one leaves a hole, which recovers some kinetic energy but not the whole of it; and the net is a gap that approaches $U$ minus the band width at large $U$.

So the material does not conduct, not because there is nowhere for an electron to go, but because getting there costs a fixed amount that does not go away as the system grows.

What a one-electron model does with size — rings from two sites to forty, every level crowding into a fixed interval — is correct and is about a different question. The four-ring at the left of that sequence is the system the exact calculation here is done on.

The two ways an insulator can happen #

It is worth setting this mechanism beside the other one already computed, because they are genuinely different and are often blurred together.

A Peierls insulator opens a gap by distorting. A chain cannot stay even computes it: a half-filled chain alternates its bonds, the alternation splits the levels at the Fermi energy, and a gap of $4\delta$ appears. That is a one-electron effect throughout — the matrix changes, and the new matrix has a gap. Band theory sees it perfectly well, provided it is applied to the distorted structure.

A Mott insulator opens a gap without moving anything. The structure is unchanged, the one-electron matrix is unchanged, and the gap comes from the cost of double occupancy. No band calculation on the undistorted structure can find it.

The two are distinguishable experimentally, and the distinguishing observation is a structural one: a Peierls insulator has a distorted structure with a doubled repeat, and a Mott insulator does not. Polyacetylene is the first; nickel oxide is the second.

They can also both happen, and in one-dimensional materials they compete — which is part of why one-dimensional conductors are such awkward objects and why so much effort has gone into deciding, for each of them, which mechanism has opened the gap.

Why the two answers cannot be reconciled #

The difference here is not a matter of accuracy, and it is worth being precise about why.

Band theory computes a difference between one-electron levels. That difference is a property of a matrix in which $U$ does not appear, so it is zero for this system at every $U$ — not approximately zero, and not zero within some error that a better basis or a finer grid would reduce. There is no parameter to adjust.

The exact charge gap is a difference of three many-electron energies, and those energies are not sums over orbital energies. At $U = 8$ the exact ground state has a double occupancy of 0.031 per site against the independent-electron 0.25, so the electrons have rearranged in a way no assignment of electrons to orbitals describes.

The two curves in the hero figure therefore do not converge for any value of anything. They describe different quantities computed from different objects, and the only reason they are drawn together is that one of them is routinely used to predict the other.

What is measured #

The compounds this argument was invented for are the first-row transition-metal monoxides, and their behaviour is the reason the problem was noticed at all — the same compounds whose magnetic coupling what couples two spins computes, which is not a coincidence.

Manganese oxide, iron oxide, cobalt oxide and nickel oxide all have partly filled d bands. Band theory, applied to them in the 1930s, predicted metals in every case. All four are insulators, with gaps of two to four electronvolts — nickel oxide is a pale green transparent solid, which is not what a metal looks like.

The discrepancy is not marginal. A predicted metal with a measured four-electronvolt gap is a prediction wrong by the whole of the quantity, and it stood as the standing embarrassment of band theory for two decades.

Two further observations complete the picture. Doping such a material — changing the electron count away from half filling — makes it conduct, because there are then sites without an electron and moving one costs no $U$. And some of these materials undergo a transition to a metal under pressure or with temperature, as the hopping is increased relative to the repulsion; vanadium sesquioxide is the standard case, and its transition is sharp.

Both are consequences of the same competition between $t$ and $U$ that the four-site ring shows, and neither is available to a theory in which $U$ does not appear.

What four sites cannot tell #

The honesty this site owes here is considerable, because a four-site ring is a molecule and the claims being illustrated are about materials.

There is no thermodynamic limit. A four-site ring has a charge gap; it does not have a phase transition, a critical $U$, or a metal–insulator transition temperature. Whether the transition in a real material is sharp or gradual is a question about a limit this calculation does not take.

There is no lattice, and deliberately so. This site’s solids field is built on the principle that a chain of two hundred atoms is a large molecule, computed with no periodicity assumed, no wavevector and no Brillouin zone — the boundary set out in a band with no structure in it. The present essay stays inside it: four sites in a ring, a matrix, three diagonalisations.

There is one orbital per site. Real transition-metal oxides have five d orbitals per metal, oxygen orbitals in between, and a gap that in several cases is better described as charge transfer from oxygen to metal than as the on-site excitation modelled here. The distinction between a Mott–Hubbard insulator and a charge-transfer insulator is real and this model contains only the first.

The size is chosen by exactness. The solver refuses more than six sites rather than approximating, which is the trade made here: a small system solved completely, instead of a large one solved with assumptions that would have to be argued for separately — the trade the smallest many-electron calculation sets out in full.

What a one-electron calculation can do that this one cannot is follow a half-filled ring’s cheapest excitation to two hundred sites. Size and exactness are the two things no calculation of this kind can have at once, and the trade is stated rather than hidden.

What the four-site answer is evidence for #

A sceptical reading of all this is available and should be answered: four sites is not a material, so what has been shown?

What has been shown is that the mechanism exists and is not a finite-size artefact. The gap here is zero at $U = 0$ by construction — the ring was chosen precisely because its one-electron gap vanishes — so every wavenumber of the gap that appears is attributable to the repulsion term and to nothing else. That is a cleaner attribution than a larger calculation would provide, because a larger calculation would have a finite-size gap to disentangle.

What has not been shown is that any particular material is an insulator for this reason, or at what value of $U/t$ the transition happens, or whether there is a transition at all rather than a smooth crossover. Those are questions about a limit, and this calculation does not take one.

The relation between the two is the ordinary relation between a mechanism and a material. A four-site ring establishes that a half-filled system with strong on-site repulsion has a charge gap; whether nickel oxide’s four electronvolts come from that mechanism, from charge transfer between oxygen and metal, or from a combination, is settled by spectroscopy on nickel oxide.

The right habit is to compute the mechanism and quote the material, and to say which is which. The gap in the table is computed. The four electronvolts is quoted.

Where the two halves meet #

Both halves of the comparison are now available, and it is worth putting them side by side.

The one-electron calculation goes to two thousand sites. It produces a density of states, a band width, the way a gap closes with size, the Peierls instability and the difference between a chain and a ring — all the band results, from a solid is a molecule that did not stop onwards, and none of them available to a four-site calculation.

The many-electron calculation goes to six sites. It produces the charge gap, the magnetic coupling, the double occupancy and the ground-state spin — all of them consequences of a term the other calculation has not got, and none of them available at any size to a model without it.

Two thousand levels binned into a density of states, against the closed form they converge on, is the scale the one-electron calculation reaches. Nothing in that figure knows about electron repulsion, and everything in it is correct about a material that has none.

Neither is a better version of the other, and the useful move is to know which questions belong to which. A question about a band width, a Fermi level or a density of states is a one-electron question and the answer scales. A question about whether something conducts at half filling, or how two moments couple, is a many-electron question and the answer is available only in miniature.

The gap that is measured, and the gap that is computed #

One more distinction, because the word “gap” carries at least three meanings in this subject and they are not equal.

The charge gap is the one computed here: the cost of separating an electron and a hole onto different copies of the system. It is what decides conduction and it is what a transport measurement or a photoemission-plus-inverse-photoemission pair reports.

The optical gap is the lowest energy at which the material absorbs, which creates an electron and a hole in the same system where they can attract each other. That attraction lowers the energy, so the optical gap is smaller than the charge gap by the binding energy of the pair — a distinction that matters a great deal in molecular and low-dimensional materials.

The one-electron gap is the difference between the highest occupied and lowest empty orbital energies, which is what band theory computes and what a Hückel calculation reports.

For a weakly correlated semiconductor all three are close, which is why they are often used interchangeably. For a Mott insulator the third is zero and the other two are electronvolts, so the vocabulary has to be watched. The same care is needed in a band gap is not a bond energy, where two quantities measured in the same units turned out to be answers to different questions.

The band as a limit is computed on rings of six, twelve and twenty-four against a ring of two thousand, and it is right about every system with no repulsion in it. What this essay adds is that the systems it is right about are not the ones anybody was asking about.

A gap that a few per cent of doping destroys #

There is a test that separates this gap from a band gap without measuring either, and it is the reason the whole subject acquired an industry.

A band gap belongs to the band. It exists because a set of levels is full and the next set is far above, and adding or removing a few electrons per thousand atoms changes nothing about that arrangement — the gap is still there, the doped carriers sit just inside one band or the other, and the material becomes a semiconductor with carriers rather than a metal.

The gap computed here belongs to the filling. It exists because there is exactly one electron per site and moving one means putting two somewhere, and that condition is destroyed by a small deviation from one. Remove five electrons per hundred sites and five per cent of the sites are empty; an electron can move into an empty site at no cost in repulsion at all; and the material conducts.

A few per cent of doping cannot close a band gap and does close a Mott gap. That is a sharp experimental distinction, it requires no spectroscopy, and it is what the copper-oxide superconductors are.

The parent compounds of those materials are insulators of exactly this kind — one electron per copper, antiferromagnetically ordered, a gap of about two electronvolts. Replacing a few per cent of the atoms in a neighbouring layer removes a corresponding fraction of an electron per copper, and the material becomes a metal and then, below a temperature that can exceed a hundred kelvin, a superconductor.

So the arithmetic on four sites has a consequence at the scale of a research field. The gap is not a property of a band that doping fills into; it is a property of a count that doping breaks, and the difference is between a material that becomes a better semiconductor and one that stops being an insulator altogether.

One more reading is worth having, because it prices the two axes against each other over the range that matters.

Who found it, and when #

Nevill Mott raised the problem in 1937 and returned to it repeatedly for thirty years: a lattice of hydrogen atoms at large spacing must be an insulator, because moving an electron means creating an ion pair, and yet band theory says a half-filled band conducts at any spacing. Rudolf Peierls made the same point in correspondence at the time.

John Hubbard’s model, in 1963, was written to make the argument computable, and the four-site calculation here is that model at the smallest size where the effect is not confused by finite-size gaps. Elliott Lieb and Fa-Yueh Wu solved the one-dimensional chain exactly in 1968 and found it insulating for every $U > 0$ — no threshold at all in one dimension.

The subject did not close there. The two-dimensional Hubbard model remains unsolved, it is the standard model of the copper-oxide superconductors, and the amount of computation directed at it since 1986 is enormous. What can be stated exactly is stated about clusters of a few sites, which is the position this essay occupies: a small system, solved completely, illustrating a mechanism whose consequences at scale are still being argued about.