The third way to be an insulator

This field has now found two ways for a material to refuse to conduct.

The first is the one band theory is built on: fill a band completely and there is nothing at the Fermi level to move. That is the counting argument, and it goes wrong in known ways — a full band is not always an insulator when two bands overlap, and a half-filled band is not always a metal.

The second is repulsion. Put a large enough on-site repulsion into a half-filled band and the electrons stop each other moving, which is the insulator band theory cannot see and needs a many-electron calculation to produce at all.

There is a third, it needs neither a filled band nor any repulsion whatever, and it is the one this essay is about. Make every site a little different from its neighbours.

What is changed, and what is not #

Every site of a ring is given an energy drawn from a flat distribution of width $W$, and nothing else is touched. Every hopping is still exactly one. Every site still has two neighbours.

That last point deserves emphasis, because it disposes of most of the ways one might expect the argument to work. The connectivity is untouched, so the second moment of the levels is still exactly the coordination, every counting argument in this field returns exactly what it returned for the clean chain, and no bond has been broken or weakened anywhere.

The band widens a little, from 4.00 to 6.80 in units of the hopping, because the site energies spread the levels. It does not split, no gap appears anywhere, and the density of states at the middle stays finite. By every measurement that counts levels, this is a metal at every disorder here.

Where the states are #

The measurement that separates the cases is not about energies at all. It asks, for a given state, how many sites it lives on — the participation ratio, which is the number of sites a uniformly spread state would need in order to have the same fourth moment of its amplitudes.

For a clean ring the answer is two thirds of the ring, whatever the ring is: a plane wave has amplitude on every site, and two thirds is what the ratio gives for a sinusoid. Fifty sites gives 33.3, a hundred gives 69.0, two hundred gives 126.7.

The convergence on the right is the whole argument. A quantity that stops depending on the size of the system has stopped being a property of the system and become a property of the disorder — which is what the word localised means and is the only evidence a calculation on a finite ring can give for it.

Why that makes an insulator #

The step from localised to insulating is the one place this essay goes beyond its own calculation, and it is worth marking clearly.

What is computed here is a set of stationary states and how far each one extends. Conduction is not computed anywhere in this collection: it needs a current, a field, a scattering time, and a way of talking about transport that a diagonalisation does not supply.

What is quoted is the standard reading, and it is short. An electron in a state that decays exponentially away from some point cannot be found far from that point; a set of such states, however dense in energy, provides no way of moving charge from one end of a sample to the other at zero temperature. The material has states at the Fermi level and no way to use them.

This is Anderson’s argument, from 1958, and the localisation it describes is named after him. In one dimension the result is stark: any disorder at all localises every state, so the model here has no transition in it — only a length that gets shorter as $W$ grows. In three dimensions there is a genuine transition, with extended states in the middle of the band and localised ones at the edges, separated by an energy that moves as the disorder is turned up.

The length that is not in the level spacing #

Two lengths are in play and the essay is about the difference between them.

The level spacing is set by the size of the sample: $n$ levels in a band of fixed width means a spacing of order $1/n$, and it goes to zero however disordered the sample is. That is what the first figure shows.

The localisation length is set by the disorder: at $W = 2$ it is about thirty sites, at $W = 4$ about ten, and it does not care how large the sample is.

A sample smaller than the localisation length looks metallic — its states reach the ends, and its levels crowd. A sample larger than it does not. So the same material, measured on two sizes, gives two answers, and the crossing point is a length rather than an energy. That is a very different kind of statement from a gap, and it is why the criterion that works for the first two kinds of insulator says nothing here.

The first of those two lengths is drawn above and needs no second picture: what a metal actually is plots the clean chain’s spacing at five sizes falling as 1/n, and every disordered ring in this essay falls the same way. The second length is the one worth seeing, and the way to see it is to make it much shorter than the ring.

Two numbers, 2.59 and 100.14, on one ring at one energy, and the level spacing at that energy is the same in both to within the width of the band. That is the whole of the essay in two figures and a ratio.

What a spectroscopy would see #

The measurements that count levels cannot tell these cases apart, and it is worth listing them because they are the usual ones.

A density of states is a histogram of levels, and the disordered ring’s is a slightly wider version of the clean one’s with the same finite value at the middle. A density of states is not a spectrum and it is not a map of where anything lives.

A photoelectron spectrum measures the energies electrons come out at, which is again a distribution of levels — and what a photoelectron spectrum measures is not where the electron was.

A specific heat measures the density of states at the Fermi level, and finds it finite. Counting electrons in the structure — which this field has done carefully — gives the same answer for the clean and the disordered ring, because the counting never looked at the states.

The measurement that does see it is one that asks whether charge moves — conductivity against temperature, which distinguishes a metal from a localised system unambiguously because the two go in opposite directions as the sample is cooled.

What none of them has is a feature to point at. Where the states pile up draws the clean chain’s density of states, and the disordered chain’s is that shape with its edges smeared and its middle very slightly lowered — no gap, no shoulder, no dip at the Fermi level, nothing that would let a spectrum be classified by looking at it. A measurement that returns a distribution of energies is measuring the wrong object, and the disorder has been careful not to disturb it.

Why a wave stops being a wave #

The mechanism is worth stating, because it is short and because it explains why one dimension is so unforgiving.

A state in a clean chain is a wave with a definite wavelength, and it moves because every site looks like every other. Change one site and the wave is partly reflected there: a small amplitude comes back, and the rest goes on. Change every site and there is a reflection at every step.

Reflections from many sites add with phases, and in one dimension there is nowhere for a wave to go around an obstacle — the reflected amplitudes accumulate along a single line, and they interfere constructively often enough that the net transmission through a long chain falls off exponentially with its length. That exponential is the localisation length, and it is why any disorder at all suffices in one dimension: there is no path that avoids the obstacles.

In three dimensions a wave can go round, the reflected amplitudes arrive by many paths and no two of them in step, so they largely cancel; only when the disorder is strong enough do they stop cancelling. That is the difference between “always localised” and “localised above a threshold”, and it is a statement about the number of paths rather than about the strength of the scattering.

The simplest version of that is one changed site rather than all of them, and a defect is a level in the gap draws it: a single altered energy pulls out a state that decays exponentially away from the site it belongs to, with a decay length set by how far that energy was moved. Disorder is that picture at every site at once, with the amplitudes interfering — and the figure above is what the interference produces.

The three ways, side by side #

There are now three distinct reasons for a material not to conduct, and they need three different kinds of calculation.

A gap from the structure. Filled band, empty band, a gap between them — computed from a one-electron model with a distortion or two orbitals per site in it.

A gap from repulsion. Half-filled band, no distortion, and a gap that appears because moving an electron costs the on-site repulsion. This needs a many-electron calculation and is invisible to every one-electron method.

And no gap at all. Disorder, states at every energy, and every one of them stuck.

There is a fourth case and it runs the other way, which is worth keeping in the same list because it is the same kind of failure. Two bands that overlap in energy leave a count saying insulator about a material that conducts — a full band is not an insulator computes the overlap and the count side by side. Three insulating mechanisms and one metallic exception is the whole of what a band picture can and cannot decide, and only one of the four is the argument a first course gives.

The three are independent, they can occur together, and only the first is what an undergraduate account of conduction contains. The second and third are the two great twentieth-century additions to it, and both were resisted for years on the grounds that the band structure said otherwise.

Where the model stops #

One dimension is the easy case and the misleading one. Every state is localised for any disorder, so there is no transition to find and no mobility edge. The interesting physics — a critical disorder, a mobility edge, a scaling function — is three-dimensional and is well outside what a ring of two hundred sites can show.

Nothing here computes conduction. The reading from localised to insulating is quoted, standard, and not derived here.

The averaging is over four arrangements of the site energies, on stated seeds, which is enough for a participation ratio that varies by tens of per cent between samples and not enough for anything finer.

The disorder is diagonal only. Real disorder changes the distances between atoms as well as their energies, so the hoppings vary too, and a band width that is a count of neighbours stops being exactly that.

And there is no repulsion anywhere, so the interplay between the second and third mechanisms — which is the actual state of the art, and hard — is not touched.

And the defect picture meets this one in the middle. One defect is a level, many are a band changes a few sites rather than all of them, and watches the levels pulled out of the band become a band of their own as the count rises from one site to forty-eight out of a hundred and sixty. At around thirty per cent that calculation and this one are the same calculation approached from opposite ends — a few strong defects and a whole ring of weak ones — and neither account currently says where the two should be joined.

What the word metal has been doing #

One answer to what a metal actually is — a sequence of level spacings that goes to zero — is attractive because it is a property of the electronic structure rather than of a measurement.

That choice has now failed twice. Repulsion produces a gap the one-electron levels do not have; disorder leaves the levels exactly where they were and takes away the states’ reach. In both cases the definition returns the wrong answer, and in both cases the reason is the same: a spectrum is a list of energies, and conduction is a statement about wavefunctions.

The definition that survives all three cases is about the states rather than the levels, and it is awkward to compute: a material conducts if its states at the Fermi level extend across the sample in the limit of a large sample. Awkward, because it requires a limit, and because the quantity that has to be evaluated in it — how far a state reaches — is exactly what the participation ratio above is a crude version of.

That is not a defect of this collection’s model. It is the reason the theory of conduction took most of a century after the theory of bands, and the reason what a metal actually is is a harder question than it first appears.

What is quoted, and what is computed #

The site energies are drawn from a stated distribution by a stated generator on a stated seed, so every number here is reproducible rather than typical.

Every level and every state is computed by diagonalisation; every participation ratio and spacing is computed from those states. The reading of localisation as insulation is quoted, and the one-dimensional result that any disorder localises every state is quoted — this calculation is consistent with it and does not prove it.

What the calculation requires #

A clean ring spreads its band-centre states over two thirds of itself at every size, and quadrupling the ring quadruples the number of sites a state occupies.

At the largest disorder the participation ratio is almost independent of the ring size — a factor of 1.27 against a fourfold change in size — and is a small fraction of the ring rather than most of it.

And the level spacing at the band centre goes on falling as the ring grows at every disorder, which is the check that makes the essay’s point: the criterion that separates the first two kinds of insulator is satisfied throughout.

What the ring’s dimension is doing to the answer #

The rings here are one-dimensional, and that is not a neutral choice of geometry — it is the case for which the general answer is known and is unusually strong. Knowing it changes how the measured numbers should be read.

The scaling theory of localisation says that in one and two dimensions every state is localised for any disorder at all, however weak. There is no threshold, no critical value, and no surviving extended state in the limit of a large system. In three dimensions the situation is different: below a critical disorder the states in the middle of the band remain extended, above it they localise, and the energy separating the two is a genuine boundary.

So the finding here — that at the largest disorder the states occupy a fixed number of sites whatever the ring size — is not the discovery of a threshold. It is the localisation length becoming shorter than the rings being computed.

That reading accounts for the small-disorder rows too. At weak disorder the states still spread over most of the ring and the participation ratio still grows with the size, which looks like extended behaviour. It is not: the states are localised there as well, over a length far longer than four hundred sites, so the calculation cannot see it. A one-dimensional system at weak disorder is an insulator whose insulating character is invisible at any size a diagonalisation can reach.

That is a real limitation and it is also the reason the essay’s other criterion goes on being satisfied. The level spacing falls as one over the size at every disorder, which the level-spacing criterion takes as the definition of a metal — and it does so even where every state is localised, because a level spacing counts states and says nothing about where they are. Two quantities, one continuing to say metal and the other saying insulator, on a system where the exact answer is known to be insulator.

The dimensional statement also says where to look for the interesting case. A three-dimensional system has a genuine transition at a critical disorder, and it has been observed — in doped semiconductors driven through it by concentration, and in amorphous alloys driven through it by composition. And the low-dimensional result is not academic either: a wire thin enough or a film thin enough behaves as one- or two-dimensional at low temperature, and its resistance duly rises as it is cooled rather than falling, which is the localisation showing itself in the one geometry where the theory says it must always be there.

None of that weakens the essay’s finding; it names what the finding is a finding about. The participation ratio measured here is a localisation length, read off a system large enough to contain it, and the disorder at which it stops depending on the ring size is the disorder at which that length falls below four hundred sites. It is a measurement of a length rather than the location of a transition — which is a more modest claim and one the one-dimensional case is entitled to make.

Still open: disorder and repulsion together #

The obvious open question is the one this essay had to leave out: what happens when disorder and repulsion are both present.

They do not simply add. Repulsion screens the disorder, so a system with both can be less localised than the same disorder alone would make it; and disorder breaks the symmetry that a Mott insulator’s argument relies on, so a system with both can be more insulating than either. The competition is one of the hardest open problems in the subject, and the smallest system in which it can be posed is a handful of sites with unequal energies and an on-site repulsion — which is a calculation that can be done exactly, on four sites, and which would say something honest about the smallest case even if it says nothing about the large one.