The amplitude the collapse left behind

The exponent was the windows warmed five rings until their Peierls distortions went away, and found that the five curves are one curve: divide each ring’s alternation by the value it settles at when cold, plot against the reduced temperature, and the five agree to 3.41 per cent across a factor of two in size and a factor of two in stiffness. That is what explains the stability of a fitted exponent across systems without the exponent meaning anything — a slope taken over a stated window is the same number for every case because there is only one curve to take it on.

It also leaves something behind. If the scaled curves are identical, then the cold alternation is the whole of what distinguishes one ring from another — and that number was obtained five times by a golden-section search over diagonalisations, with no expression for it anywhere.

There is one. It is exact, it is a sum of square roots, and it reproduces all five searched values to better than one part in three million. What it then says about the five is not what it was written down to say.

A ring of n is n/2 two-level problems #

A dimerised ring alternates its hopping between $1+\delta$ and $1-\delta$, so its repeating unit is two sites and it is a circulant on $n/2$ cells. Each cell contributes a two-by-two problem whose levels are

$$\pm\sqrt{t_1^2 + t_2^2 + 2t_1t_2\cos k},$$

and at half filling the whole positive branch is occupied. The electronic energy per site is therefore the mean of $n/2$ square roots, evaluated at the $n/2$ wavevectors a ring of that size allows.

That is a closed form for the object, not an approximation to it. Against the diagonalisations it replaces it agrees to $10^{-11}$, which is the residual the eigensolver leaves rather than any error of the formula.

Letting the ring grow turns the sum into an integral, and the integral is a complete elliptic integral of the second kind:

$$E_{\text{el}}(\delta) = \frac{4}{\pi}\,\mathrm{E}(1-\delta^2).$$

At $\delta = 0$ that is $4/\pi$, which is the half-filled chain’s energy per site and has been on this site since its first essays. The elliptic integrals themselves are computed by the arithmetic–geometric mean, which doubles its correct digits every pass, and are checked against the published $\mathrm{K}(\tfrac12) = 1.8540746773$ and $\mathrm{E}(\tfrac12) = 1.3506438810$ to twelve figures — a special function has no chemistry to catch it, so it has to be checked against a table.

The amplitude, in closed form #

The free energy per site at zero temperature is that energy minus the elastic cost $K\delta^2$. Setting its derivative to zero and using $\mathrm{d}\mathrm{E}/\mathrm{d}m = (\mathrm{E}-\mathrm{K})/2m$ gives the condition the alternation satisfies:

$$\frac{2}{\pi}\,\frac{\mathrm{K}(m) - \mathrm{E}(m)}{1-\delta^2} = K, \qquad m = 1 - \delta^2.$$

One transcendental equation in one unknown, solved by bisection on the logarithm of $\delta$. The same number found by maximising the free energy directly agrees to five figures, which is as much as a golden section on a smooth maximum can give and is the check that the differentiation is right — a sign error would not produce a plausible number twice by two different routes.

As $\delta \to 0$ the first integral goes as $\ln(4/\delta)$ and the second to one, so the condition becomes $\ln(4/\delta) = \pi K/2 + 1$ and

$$\delta_0 = \frac{4}{e}\,e^{-\pi K/2},$$

which is the expression every account of a Peierls distortion quotes.

The received expression is an asymptote in the wrong variable #

It is quoted as an expression in the elastic constant, and it is an expansion in the alternation. Those are not the same statement, because the thing being expanded in is the thing being predicted.

Measured against the exact condition, the exponential form is 10.46 per cent low at $K = 1.0$, 6.49 per cent at 1.2, 2.42 per cent at 1.6, 0.30 per cent at 2.4 and 0.0034 per cent at 4.0. Fitted across that whole range the error falls as $\delta^{1.72}$.

This is not a complaint about a textbook formula. It is exactly right in the limit it was derived in. The point is about where a chemist stands when using it: a distortion that can be seen in a bond length or a gap is a distortion of a few hundredths at least, and at a few hundredths the expression is already a fraction of a per cent out; at the tenth or so that a real polyacetylene-like chain shows, it is several per cent out. An expression whose error is largest exactly where the phenomenon is observable is a poor instrument for the observable case, however good its asymptotics.

A ring of forty is not always a long chain #

The closed form’s other use is that it can be evaluated at any ring size for nothing, which makes a question askable that was previously five hours of searching: how far is each of the collapse’s five rings from the infinite chain?

At $K = 1.2$, a ring of forty is within $5.5\times10^{-3}$ per cent of the limit. At $K = 1.6$ it is 0.83 per cent above it. At $K = 2.4$ — one of the five cases the collapse was measured on — it is 36.37 per cent above it, on the same forty sites. At $K = 3.0$ it is 127.58 per cent above.

The reason is the one thing the model has that a count of sites does not know about. A stiffer spring gives a smaller alternation, a smaller alternation gives a smaller gap — the gap is $4\delta$ here — and a smaller gap is a longer coherence length. A ring is long when it holds many of those, not when it holds many atoms.

That reading is checkable and it collapses.

A ring of four hundred at $K = 3.0$ and a ring of forty at $K = 2.0$ sit at the same place on this axis and are the same distance from the limit — 0.53 and 0.63 per cent — at ten times the site count. The number of sites is not the size of the system.

What this does and does not do to the collapse #

It does not break it. The collapse divides each curve by that ring’s own cold alternation, and the closed form says what that number is rather than saying it was wrong. The five scaled curves still lie on one another to 3.41 per cent, and the argument built on them — that a fitted exponent stable across systems is evidence the number belongs to the window — is untouched.

What it does is remove a reading that the five cases invite. Three of the five differ only in size at fixed stiffness, and their agreement looks like evidence that the size no longer matters. It is: at $K = 1.6$, rings of forty, sixty and eighty are 0.83, 0.061 and 0.0046 per cent from the limit, so the sequence has converged and the collapse is a bulk statement for those three. The other two vary the stiffness, and one of them — $K = 2.4$ at forty sites — is more than a third away from its own bulk value. Its scaled curve is a perfectly good curve for that ring. It is not a curve for a long chain, and nothing in the collapse could have said so.

The distinction matters because a whole line of argument about Peierls chains is built on rings of forty. What a metal is was argued on one, the filling that chooses a distortion on the same, and the carriers a distortion hides on the same again. Every one of those was run at a stiffness where forty sites is genuinely large — the closed form now says which stiffnesses those are, which is a thing none of them could check.

This is the same failure found in a different instrument. A healing length fitted over a window belongs to the window unless something outside the fit says the window was long enough; here an amplitude measured on a ring belongs to the ring unless something outside the measurement says the ring was long enough. In both cases the thing that says so is a closed form, and in both cases the collapse or the fit looks equally good either way.

What a formula is worth here #

An amplitude that had to be searched for could only be read forwards: pick a spring, get a distortion. A closed form can be read backwards, and the interesting direction is the backwards one, because a distortion is measurable and an elastic constant is not.

The sensitivity is the derivative of the logarithm of the alternation with respect to the elastic constant, and the asymptotic form says it should be exactly $-\pi/2$. Measured on the exact condition it is $-1.738$ at $K = 1.2$, $-1.633$ at 1.6, $-1.579$ at 2.4 and $-1.572$ at 3.0 — arriving at $-\pi/2$ from above, on the same schedule as everything else here.

Read forwards that is bad news: a one per cent error in the spring moves the alternation by two and a half per cent at $K = 1.6$, so a distortion computed from an assumed force constant carries several times that constant’s uncertainty. Read backwards it is good news, and it is the direction anybody actually has data for. A measured alternation known to one per cent fixes the elastic constant to about six thousandths in absolute terms, whatever the value — because the sensitivity is nearly the same number everywhere, which is what an exponential dependence buys.

This is the shape priced for a pair of bands: what an inversion is worth depends on which quantity is being recovered from which, and the answer can be very different in the two directions. Here the asymmetry is a factor of a few hundred, and it is the reason a Peierls elastic constant quoted from a measured bond alternation is a much better number than a bond alternation predicted from a quoted elastic constant.

What was computed, and how #

The searched amplitudes are golden-section maxima of the free energy per site of a ring at a temperature of one thousandth, with the levels from the same eigensolver used for every band here and every eigenpair required to satisfy $Av = xv$. They are the numbers the collapse actually divided by, taken from the same calculation that produced it rather than recomputed by a route that might differ.

The closed forms evaluate $n/2$ square roots and one elliptic integral respectively, and neither builds a matrix. The comparison between them is the point: the two agree to a few parts in ten million on every case, which is the precision of the golden section rather than of either formula.

Everything here is at half filling with one electron a site, the elastic cost is $K\delta^2$ per site, and there is no electron repulsion anywhere — so this is the one-electron Peierls picture and not a statement about any material. The temperature enters only through the searches; the closed forms are at zero temperature, which is where the amplitude the collapse scales by is defined.

The refusal is the pairing of routes. If the stationarity condition and the free-energy maximum disagreed, or if the discrete sum and the diagonalisation disagreed, no number here would mean anything — and all three pairs agree to the precision each is capable of.

Where the model stops #

The closed form also changes what a disagreement between two rings would mean. Before it, five curves that failed to collapse could always be blamed on the searches: a golden-section search over diagonalisations returns a number with a tolerance on it, and five numbers each good to a few parts in a thousand will not lay one curve on another exactly. Now the amplitude each curve is divided by is exact, so a residual spread is a fact about the physics and not about the arithmetic that found it — and the spread that remains is entirely the ring of four, which the formula itself explains.

That is the ordinary payoff of replacing a search with an expression, and it is worth naming because it is not the obvious one. The expression is faster, which nobody needed; it is exact, which matters a little; and it removes an alternative explanation for every discrepancy measured against it, which is what actually advanced the argument here.

The elliptic form is a statement about an infinite chain with no ends, and it is already established that a chain with ends distorts hardest where it stops. Every ring here is wrapped, so nothing above says what an oligomer with two ends settles at — and the end effect and the finite-size effect measured here are different quantities with different size dependences.

The wrapped ring also has no place for the two dimerisation patterns to differ, which is what makes it the clean object here: an open chain’s two dimerisation patterns are different molecules and its energy carries a term that is a property of the ends alone. A closed form for the open chain would have to carry that term, and it is not the same closed form.

The elastic term is quadratic and the hopping is linear in the displacement, which is the standard caricature. A real bond’s energy is not quadratic to a tenth of a bond length, and at $K = 1.0$ the alternation is 0.342, which is far outside where either linearisation deserves to be believed. The numbers at the soft end are properties of the model rather than predictions about anything.

And a half-filled band being unstable to dimerisation is a one-electron statement. Repulsion can open a gap without any distortion at all, and nothing here can distinguish the two.

Who found it, and when #

The arithmetic-geometric mean that evaluates the integral is Gauss’s, and it is the reason the closed form is worth having numerically as well as algebraically: it converges quadratically, so a dozen iterations give every digit a double holds. That is why the expression can be checked against the searches to five figures rather than to two.

Peierls’s argument is from the nineteen-thirties and the exponential form is its standard closing step. The elliptic form of the band energy is older than the argument it is used in: it is what the integral of $\sqrt{1 - m\sin^2 u}$ is, and Legendre had it in 1811. That a dimerised chain’s energy is a complete elliptic integral is remarked in most careful treatments and then dropped in favour of the logarithm, because the logarithm is what makes the exponential dependence visible.

The finite-size behaviour is the same object as the crossover in the Su–Schrieffer–Heeger model’s soliton width, from 1979, where the width goes as the reciprocal of the gap — the length being compared against the ring here is that same length, arrived at from the other direction.

Still open: the closed form at finite temperature, and matched sizes #

The obvious open question is the temperature. Everything above is at zero, and the collapse is a statement about a curve running from zero up to the point where the alternation vanishes. The free energy at finite temperature is the same sum of square roots with Fermi factors and an entropy term, which is still a closed form and still avoids every diagonalisation — so the whole collapse could be recomputed as a formula, and the shape of the universal curve could be asked for in the same way the amplitude has been here.

The nearer question is the one the third case now poses. If a ring of forty at $K = 2.4$ is thirty-six per cent from its bulk amplitude, its whole scaled curve is a curve for a small ring, and whether the collapse holds because the scaling is a good scaling or despite one case being finite is a question with a definite answer: run the collapse again at sizes chosen so that every case sits at the same value of the product the departure turns out to be a function of. If the five curves lie closer together than 3.41 per cent, the residual spread was finite size; if they lie no closer, it was not, and the collapse is better than the reason given for it.