A band that is a hundred and seventy decades of nothing

The chain length at which a scattering of identical defects stops being a set of levels and becomes a band is 985 sites for runs of four low sites and 81,510 for runs of eight. The criterion is the one every account of a band uses — a set of levels is a band when the coupling between neighbours exceeds the spacing between them — and it works because the two quantities move in opposite directions as the chain grows.

A band has a width as well as an onset, and that is the thing left over. The criterion above uses the coupling at the closest pair, because that is what crosses the spacing first. A width is set by the coupling at the typical spacing, which is a different average over the same distribution.

The two averages are not close. For runs of eight at half concentration they differ by a hundred and seventy-three orders of magnitude, and every conclusion in this essay follows from that.

Two exponentials, compounded #

Runs of a given length occur with a density that falls exponentially in the length: a run of exactly $L$ low sites at concentration $x$ needs $L$ lows bounded by two highs, so its density is $x^L(1-x)^2$. At half concentration that is one in 64 sites for runs of four and one in 1,024 for runs of eight, and the typical gap between consecutive runs is 60 and 1,016 high sites.

The coupling between two runs falls exponentially in that gap, with a decay length of 1.44 to 2.56 sites depending on how deep the level sits below the barrier top.

Those two exponentials compound. The exponent of the coupling is the reciprocal of the run density divided by the decay length, so it is exponential in $L$ inside an exponential — and the result is the column above: $1.1\times10^{-19}$, $6.2\times10^{-33}$, $2.4\times10^{-56}$, $2.0\times10^{-98}$, $3.6\times10^{-175}$.

A width computed from those numbers is not a small width. It is a number below the level spacing of every finite chain, below anything a spectrum could resolve, and below the precision the eigensolver works at. The picture in which a defect band is a set of levels broadened by their mutual coupling has, at this concentration, no content at all.

The width has a ceiling #

What that criterion actually says, read carefully, is that the closest pair in the chain hybridises. That is a statement about a pair rather than a band — and it is the only statement available, because the typical pair does not.

So the honest question is: how wide can a defect band get? And that has a finite answer, because there is a closest possible separation. Two runs cannot be nearer than one high site apart without being one longer run, so the largest coupling any chain can produce is the splitting at a separation of one. It is 0.0877β for runs of four, falling to 0.0158β for runs of eight.

No chain length raises that ceiling. A longer chain contains more pairs at the closest separation, not closer ones — which is why the onset moves with the chain and the width does not.

The bands never touch #

With a ceiling on the width, the question of merging has a definite answer. Runs of different lengths put their levels at different energies — $2\cos(\pi/(L+1))$ — and those levels crowd together as $L$ grows, approaching 2 from below. If the widths fell more slowly than the spacings, the bands would eventually merge into a continuum, and the gap between the defect states and the barrier band would close from the inside.

They do not merge. The ratio runs 1.68, 1.62, 1.62, 1.63, 1.65, 1.67, 1.69, 1.70, 1.72, 1.74, 1.75 and 1.76 for run lengths three to fourteen — a shallow minimum of 1.6187 at $L = 5$, and rising at both ends.

The reason is that both quantities fall as the same power.

The level spacing between adjacent box levels is the derivative of $2\cos(\pi/(L+1))$, which goes as $2\pi^2/(L+1)^3$; the widest splitting goes as the same cube, with a coefficient of about 11.2 that is constant to a fifth across a factor of five in the run length. So their ratio is a constant with a slight curvature, and it never comes near one.

A defect band is narrower than the space between it and its neighbour, at every run length, and by a factor that barely changes.

Nor does one reach the barrier band #

The other way a gap could close is from the top: a defect band broad enough to overlap the band of the barrier sites themselves.

At a barrier of four the margin is 8.7 half widths for runs of four and 22.7 for runs of twelve. It grows rather than shrinking, and it grows for a reason worth stating: as the run gets longer its level rises towards 2 — closing on the barrier band — but its band narrows as the cube, which is faster. The numerator falls as $(L+1)^{-2}$ and the denominator as $(L+1)^{-3}$.

Raising the barrier moves the band edge away and narrows the defect band at the same time, so the margin grows there far faster: 8.7, 88.1 and 224.9 at barriers of four, six and eight for runs of four. A barrier of four is the tightest case there is, because at four the barrier band’s bottom and the low sites’ band top coincide exactly and the two touch.

The one case where it is close, and why it is not a case #

Every margin above is comfortable except one, and it is worth looking at rather than passing over: a barrier of four with runs of four is 8.7 half widths, and a barrier of four is where the two bands of the underlying chain touch. Site energies of 0 and 4 give bands of $[-2, 2]$ and $[2, 6]$, so there is no gap between them at all, and the “defect levels” are levels of the low runs sitting inside a range the barrier band’s edge also reaches.

That is not a marginal case; it is a degenerate one. Below a barrier of four there is no gap, so there is nothing for a defect state to be in, and what a defect level even means requires the gap to exist first. The tightest margin in the table is therefore at the edge of where the question can be asked, and the answer everywhere the question is well posed is a margin of tens or hundreds.

The same reading applies to the run length. Longer runs put their levels closer to 2, which is where the barrier band’s bottom is when the barrier is four — so the natural worry is that a long enough run closes the margin. It does the opposite, because the width falls one power faster than the distance does, and the run census says how rare a long run is anyway: a run of twelve at half concentration is one site in sixteen thousand.

What this leaves of the onset #

Nothing in it is wrong, and one thing in it should be read differently.

The onset is real: at 985 sites a chain of runs of four does contain a pair close enough that its splitting exceeds the mean level spacing, and at 81,510 the same is true of runs of eight. Those numbers were computed from a coupling and a spacing that were both correctly identified.

What that onset is the onset of is a pair, not a band. Past it the spectrum near a box level is not a broadened band of hybridised states; it is a set of nearly degenerate localised levels with a handful of resonant pairs among them, and the number of those pairs grows with the chain while the rest do not. That is the picture the localisation measurements already imply, arrived at here from the width instead.

It also explains why the two onsets are so far apart — 985 sites and 81,510 — for run lengths differing by four. The onset is where the closest pair’s coupling reaches the spacing, and the closest pair is $1/(N\rho^2)$ sites away, so the onset goes as the reciprocal square of a density that itself falls exponentially in the run length. The same compounding appears wherever a rare configuration carries a state, and it is the reason nothing in this field has a scale.

The distinction has a consequence for anything read off such a spectrum. A band conducts because its states are extended; a set of resonant pairs does not, however dense the levels are. So the onset is not the point at which a doped chain becomes a conductor, and the criterion that located it cannot be used for that.

What was computed, and how #

Every splitting is a difference between two eigenvalues of a real symmetric tridiagonal matrix, from implicit QL, checked against a general eigensolver to $10^{-14}$. Runs are placed rather than found — a chain of two hundred with two runs of stated length at a stated separation — so what is measured is the splitting at that separation and not a statistic over a random chain.

The typical separation comes from the closed form for the run density, $x^L(1-x)^2$, which is checked against counts in generated chains rather than assumed. The coupling at that separation comes from an exponential fitted over separations of two to fourteen, where the decay is clean and the splittings are large enough to measure. That is an extrapolation of eighteen to a hundred and seventy-three decades, and it is the one number here that is not measured — which is why the conclusion is stated as a bound rather than as a width: whatever the true coupling at 1,016 sites is, it is not larger than the fit says, because the fit’s exponential is the slowest decay the profile shows.

The refusal is a pair at no separation. Two runs with nothing between them are one run of twice the length, whose level is a different level entirely, so the ceiling has to be taken at a separation of one.

Where the model stops #

The width is measured on a sweep of couplings rather than on a real defect, and the two differ in a way worth naming. A sweep moves one number over decades and reports where the band stops being distinguishable from the host’s; a chemical defect sits at one coupling, decided by what the atom is and how it bonds, and the question a chemist asks is which decade that coupling falls in. Nothing here answers that, because nothing here is an atom.

What the sweep does supply is the shape of the answer, and that shape is what the onset calculation assumed rather than measured. A band whose width falls off over a hundred and seventy decades is not a band with a threshold; there is no coupling at which the defect stops having a level and starts having a band, only a coupling below which the width has fallen under whatever resolution is being claimed. A statement about a defect band is therefore a statement about a measurement, and it has to name the resolution to mean anything.

Everything here is at half concentration and one barrier height, and the density of runs is exponentially sensitive to both. At a concentration of 0.8 the density of runs of eight is 0.0067 rather than 0.00098, so the typical gap is 141 sites rather than 1,016 and the typical coupling is $10^{-25}$ rather than $10^{-175}$. That is still nothing, and the qualitative conclusion is unchanged; but the arithmetic is a strong function of a number that a real doped material chooses for itself.

This is a one-dimensional model, and the difference is not a detail. In three dimensions the number of runs within a given distance grows as the cube of that distance, so the typical separation between neighbouring defects goes as the cube root of the reciprocal density rather than as its reciprocal — 10 sites where one dimension gives 1,000. An impurity band in three dimensions is a real object for exactly that reason, and nothing above should be read as a claim about one.

And there is no repulsion anywhere. A half-filled defect band in a real material is the situation in which a band can be an insulator without any of this arithmetic.

The generalisation #

The useful statement is about criteria rather than about defects. A criterion written with “the coupling” in it is not a criterion until the average is named, and when the coupling distribution is broad the choice of average is the whole answer.

Here the distribution is as broad as a distribution gets: separations are geometrically distributed and the coupling is exponential in the separation, so the coupling itself follows a power law with no scale in it. Its maximum, its mean and its typical value are three quantities that share no digits. The onset takes the maximum because that is what crosses the spacing first; a width takes the typical value; and a conductivity would take something else again.

That is the same shape as a decay length that was a property of the window it was fitted over and a shape departure whose order came from the lattice rather than the arithmetic — a quantity established by one reading of an expression, used under another.

Who found it, and when #

The practical form of that is a rule about reporting rather than about physics. A defect band’s width should be quoted with the coupling it was computed at and the resolution it is being called a band at, because neither number is recoverable from the other and the two together are the whole claim. A width alone is a number that could have come from anywhere in the sweep.

Impurity bands and the criterion that forms them are Mott’s, and the version used here — coupling against spacing — is the standard argument for the metal–insulator transition in doped semiconductors, where it gives the Mott criterion $n^{1/3}a \approx 0.25$ and works very well in three dimensions. The one-dimensional case is different for the reason above and has been known to be since Mott and Twose in 1961: in one dimension any disorder localises every state, so a one-dimensional impurity band is never a conduction band whatever its width.

The specific arithmetic here — that the level spacing between adjacent box levels and the widest possible splitting both go as the cube of the run length, so their ratio has a floor — does not appear to be a standard remark, and it is what turns “the bands are narrow” into “the bands cannot merge”.

Still open: the same comparison in two dimensions #

The obvious open question is the dimension. Everything above turns on the typical separation being the reciprocal of the density, which is a one-dimensional statement; in two dimensions it is the reciprocal square root and in three the cube root. Running the same comparison on a square net of low and high sites would put a number on how much of the hundred and seventy decades is dimension and how much is the model, and those nets are already built here.

The nearer question is the distribution rather than its averages. If the coupling follows a power law, then the number of pairs coupled more strongly than a stated amount is a computable function of the chain length, and that count is what decides how much of a spectrum near a box level is resonant pairs and how much is isolated levels. It needs no new calculation — the run census already generates the chains and counts the separations — and it would turn “a pair rather than a band” from a reading into a fraction.