The residual was the stiffness

Matching five rings at three values of n·δ∞ left a residual of 2.52, 2.07 and 1.62 per cent, falling with the target, and ruled out one reading of it: the collapse was tighter than the even-site rounding at the small end, so the rounding could not be translating one-for-one into spread. What that essay could not say was how much of the rounding did get through. It needed a number it had not measured — how far a ring’s scaled curve moves when its n·δ∞ is moved — and it offered an estimate in its place: that near the matched value the curves are flat in n·δ∞, so a mismatch of ε moves them by something like $\varepsilon^2$.

The number is measurable and has now been measured. It is small, it is linear rather than quadratic, and the rounding contributes about a hundredth of the residual through it.

Which leaves the residual unexplained by the thing it was compared with — and explained by something the matching was never able to reach. At a fixed number of coherence lengths round the ring, the curve still depends on the stiffness.

Twenty-five rings #

The matching idea is that the scaled alternation — the distortion at a temperature divided by the ring’s own cold distortion, plotted against how far below the transition the ring is — is a function of the reduced temperature and of n·δ∞ alone, where δ∞ is the bulk amplitude at a stiffness and its inverse is the coherence length in bonds. If that were exactly true, rings at five stiffnesses with the same n·δ∞ would lie on one curve and the collapse would have no residual at all.

Testing it needs more than five rings. The grid here is five stiffnesses, K = 1.2 to 2.0, each at five sizes centred on the ring that holds seven coherence lengths and stepping two sites either side — 26 to 34 at the softest, 106 to 114 at the stiffest. Stepping by two alternates the ring’s class: a half-filled ring of 4m sites and a ring of 4m + 2 are different objects, because the undistorted 4m ring has a degenerate pair of levels at its Fermi energy and the 4m + 2 ring has a closed shell — the same count that separates 4n + 2 aromatic rings from 4n ones. So the twenty-five rings cover both classes at every stiffness.

Each ring’s curve is the one every essay on the collapse used: the transition temperature by bisection, then the distortion at six reduced temperatures from 0.3 to 0.001 below it, each by golden-section minimisation of the ring’s free energy. Nineteen of the twenty-five were new.

At each reduced temperature the twenty-five values are fitted, by ordinary least squares, to four terms: an intercept, a slope in n·δ∞ about seven, a term linear in δ∞ itself, and a step for rings of 4m sites. The slope is the response the rounding acts through. The other two are what the matching assumes away.

A ring’s size barely moves its curve #

The response is the first answer, and it is plain on one stiffness before any fit. Nine rings at K = 1.6, from 46 to 70 sites — n·δ∞ from 5.6 to 8.6, half again either side of seven — move their scaled curve by under half a per cent end to end, most of it at the small end. Near seven the curve moves by 0.04 per cent per unit of n·δ∞ at a reduced temperature of 0.03, and the fit over all twenty-five rings puts the slope at 0.033 to 0.087 per cent per unit across the six temperatures, largest nearest the transition.

At a target of seven the even-site rounding leaves the five rings spread over 0.11 units of n·δ∞. Through the measured slope that is at most 0.010 per cent of spread — against a residual of 0.88 to 2.07 per cent. At 9.6 it is at most 0.009 per cent, and at five, where the rings are smallest and the rounding worst, 0.028. The rounding is a hundredth of the residual everywhere it was measured.

So the earlier essay’s conclusion — the supposed floor was a bound and never a floor — stands, and more strongly than it was stated. Its mechanism does not. The curves are not flat in n·δ∞ at the matched value, with a response that starts at second order. They respond linearly, with a slope that is simply small, because by seven coherence lengths a ring is already most of the way to the long-chain limit in this variable. A small first-order response and a vanishing one give the same verdict here and would give different ones wherever the slope is larger, and the test at five below is where it is.

The zigzag in the figure is the other thing size does. Rings of 56 and 60 sites sit a seventh of a per cent below their 4m + 2 neighbours, which is about thirty times what n·δ∞ does across the rounding’s whole range.

The step between two kinds of ring #

Every stiffness shows the same step. Across the grid, rings of 4m sites sit 0.13 to 0.15 per cent below rings of 4m + 2 at every reduced temperature from 0.3 to 0.01, and the fit finds it to the same size whether the ring has 28 sites or 112.

It is the degenerate pair again. A 4m ring’s undistorted levels include a pair exactly at the Fermi energy, and the distortion’s first job is to split it; a 4m + 2 ring has nothing there to split and distorts only because the whole band gains. The carriers a distortion was hiding found the pair’s consequence at the Fermi level — one pair of thermal carriers pinned at every temperature in a rigid 4m ring — and here the same pair shows up one level removed, in how fast the distortion comes back as the ring is cooled from its transition. Near the transition the step closes: 0.10 per cent at 0.003, and nothing measurable at 0.001, where the distortion is small enough that the splitting of one pair is no longer distinguishable from the splitting of the band.

The matched sets at each target mix the two classes by accident — three 4m rings of five at 9.6, one at seven — which is a spread of a seventh of a per cent that nobody chose.

The term the matching could not reach #

The third term is the residual. Plotted against δ∞, each stiffness’s five rings form a tight cluster — size barely moves them — and the clusters lie on a straight line. At 0.03 below the transition the softest stiffness sits 1.63 per cent above the stiffest, and the line through them is straight to the width of a cluster: the fit’s misfit over all twenty-five rings is 0.032 per cent.

That spread is not finite size in any sense matching can repair. The rings compared hold the same number of coherence lengths; what differs is how many bonds a coherence length is. At K = 1.2 the bulk amplitude is 0.239 and a coherence length is about four bonds; at K = 2.0 it is 0.064 and a coherence length is about sixteen. The scaling argument treats the chain as a continuum in units of the coherence length, and a coherence length of four bonds is a coarse continuum — the same coarseness that makes a ring’s middle a poor stand-in for the bulk at the soft settings. The term linear in δ∞ is that coarseness — the lattice showing through — and it is a property of the stiffness, not of the ring.

The refusal makes the point without the picture. Fitted without the δ∞ term — intercept, slope and step only — the same twenty-five rings are missed by 0.32 to 0.60 per cent rms from 0.3 to 0.01, seven to fourteen times the four-term fit’s 0.023 to 0.047. Nothing else in the fit can stand in for it.

What the matched residual is made of #

Put side by side, the parts account for the whole. At a target of seven the matched residual is 0.88, 1.41, 1.60, 1.68, 1.80 and 2.07 per cent from 0.3 to 0.001 below the transition. The δ∞ term between the softest and stiffest settings is 0.89, 1.44, 1.63, 1.70, 1.76 and 1.89. The two lines lie on each other. The step is a tenth of a per cent and the rounding a hundredth, and both are below the residual by the factors those numbers imply.

It also explains why the residual did not fall between seven and 9.6 where it was most carefully measured. From 0.3 to 0.01 below the transition it is 0.88, 1.41, 1.60 and 1.68 at seven and 0.86, 1.40, 1.58 and 1.62 at 9.6 — the same to a few hundredths, because the δ∞ term does not depend on the ring. The fall the earlier essay reported, 2.07 to 1.62 per cent in the worst case, is confined to the two temperatures nearest the transition, where the fit is loosest and single rings scatter most.

The correction, where it was fitted and where it was not #

A decomposition fitted to twenty-five rings can explain twenty-five rings by construction, so the test that matters is the rings it was not fitted to. The matched sets at 5 and 9.6 are those: sizes of 20 to 78 and 40 to 150, none of them in the grid.

At seven, in sample, taking the δ∞ term and the step off each measured curve shrinks the spread from 0.88–1.68 per cent to 0.027–0.043 from 0.3 to 0.01 — about forty times. That is the fit reproducing itself.

At 9.6, out of sample, with rings up to 150 sites, it shrinks from 0.86–1.62 per cent to 0.13–0.16 across the same range and 0.15 at 0.003 — seven to eleven times, on rings the fit never saw.

At five it fails, and the failure is informative. The spread shrinks only from 1.27–2.52 per cent to 0.80–1.89, because at five coherence lengths the softest ring has twenty sites, and the curve’s response to n·δ∞ there is no longer the small slope measured at seven. That is finite size, genuinely — the effect the earlier essay attributed to the whole residual turns out to live at the small end, below about six coherence lengths, and to be absent above it.

The published unmatched collapse makes the same point from the other side. Its five cases span n·δ∞ from 1.36 to 9.77 and spread by 3.41 per cent; the same four terms predict 1.62 of it. The other 1.8 is the cases below five coherence lengths, where size is doing real work — which is why matching them helped at all.

The curve the rings were collapsing onto #

The intercept of the fit — the value the line through the clusters reaches at δ∞ = 0 — is a curve of reduced temperature with no stiffness in it. It is what the scaled alternation would be for a chain whose coherence length is infinitely many bonds: the continuum limit the scaling argument always meant.

That limit has a name elsewhere. When the band is wide against the gap, the mean-field Peierls problem’s self-consistency condition becomes the gap equation of BCS theory, with the alternation playing the part of the superconducting gap; the identification is standard, and Lee, Rice and Anderson’s 1973 treatment of the Peierls transition starts from it. So the intercept should be BCS’s weak-coupling gap curve, Δ(T)/Δ(0) at reduced temperature 1 − T/Tc, which is a universal function computed from one integral equation with no parameters at all.

It is. Solved from its own gap equation, BCS gives 0.82877 at 0.3 below the transition and 0.17295 at 0.01; the intercept gives 0.82798 and 0.17249. The intercept sits 0.10, 0.17, 0.21 and 0.27 per cent below BCS from 0.3 to 0.01, and 0.46 and 0.99 at the two deepest points, where the fit misses its own rings by 0.13 and 0.39. The softest ring, by contrast, sits two per cent above BCS and the stiffest 0.4 — the lattice term, read from the other end.

Nothing in the chain’s model refers to BCS. The rings are diagonalised, the free energy minimised, the transition found by bisection; the intercept is a least-squares fit to what those return; and BCS is a separate integral solved separately. Agreement to a fifth of a per cent is the strongest check available that the decomposition has found the physics rather than a way of describing noise — and it is the surprising connection in this essay. The residual of a collapse on a chain of carbon-like bonds was a superconductor’s gap curve with the lattice left in.

What the intercept’s remaining 0.1 to 0.3 per cent is, is not settled here. It is evaluated at seven coherence lengths and for rings of 4m + 2 sites, so it carries whatever size dependence remains at seven and whatever the closed shell adds; both push it the same way, below BCS, and both are small.

The four terms at every temperature #

The table is the whole of the measurement. Two of its columns matter most for anyone reusing it. The response to n·δ∞ is what turns a size mismatch into spread, and at hundredths of a per cent per coherence length it says that at seven or more coherence lengths the size of a ring can be chosen loosely. The δ∞ term is what does not go away, and it says that a collapse across stiffnesses is limited at the per-cent level by the stiffnesses themselves.

Where the model stops #

The grid is centred on seven. Its slope in n·δ∞ is a local slope, and the failure at five shows that it grows below about six coherence lengths. A grid at five or at twelve would measure the response there rather than extrapolating it.

The two deepest temperatures are the weakest, which is also where a fitted exponent turned out to be its window’s. At 0.003 and 0.001 below the transition the fit misses its own rings by 0.13 and 0.39 per cent, the 4m step closes, and individual rings scatter by amounts that do not follow n·δ∞ or class — the ring of 82 sites at K = 1.8 is the worst, 1.4 per cent from its fit at 0.001. Each is a converged minimisation, so the scatter belongs to the rings: near the transition the distortion is small enough that the discreteness of each ring’s levels matters to it individually. Nothing above is claimed at those two points except that the pattern loosens there.

The model is mean-field. A classical lattice with the electrons’ free energy minimised at each temperature has a sharp transition and BCS’s curve in its continuum limit; a real quasi-one-dimensional chain has fluctuations that smear the transition, and the rule about what counts as a metal needs a second limit once temperature is involved, as a thermometer on a finite ring shows. The agreement with BCS is a statement about the model’s continuum limit, not about any material.

The δ∞ term is linear because it was fitted linear. Five stiffnesses on a straight line to the width of a cluster is good evidence for linearity over this range, and a quadratic term would be the first thing to add if the range were extended to stiffer settings — where the line would have to be extrapolated a shorter distance to reach zero, and the intercept would be better determined.

Who found what #

The Peierls instability is Peierls’s, from the 1950s. The mapping of its mean-field theory onto BCS — the alternation as a gap, the transition temperature as the BCS ratio times the gap — underlies Lee, Rice and Anderson’s work of 1973, and the weak-coupling gap curve is from Bardeen, Cooper and Schrieffer’s 1957 paper. Finite-size scaling and the use of a scaling collapse to test it are standard in statistical mechanics; the correction linear in a lattice spacing over a correlation length is the ordinary leading correction to scaling on a lattice.

What is computed here is the specific decomposition of one published residual: the response, the step and the lattice term, measured on twenty-five rings; the out-of-sample test at two other targets; and the intercept against BCS.

Still open: the intercept’s last fifth of a per cent, and a stiffer lever #

The obvious open question is what separates the intercept from BCS. It sits 0.1 to 0.3 per cent below from 0.3 to 0.01 below the transition, and it was evaluated at seven coherence lengths on closed-shell rings. The same grid at twelve coherence lengths — the stiffest ring would be about 190 sites, affordable at the cost measured here — would say whether the gap shrinks with size, which would make it finite size in the continuum, or stays, which would make it something about the closed shell that no size removes. The 4m class’s intercept is already in the fit, a seventh of a per cent further below, and it is the other half of the same question.

The nearer question is the lever in stiffness. The line through five stiffnesses is extrapolated from δ∞ = 0.064 to zero, a quarter of its own length. Adding K = 2.4 and 2.8 — δ∞ about 0.03 and 0.015 — would halve that extrapolation, test whether the δ∞ term stays linear as the coherence length grows past thirty bonds, and give an intercept that owes less to the fit. The rings would be large, about 220 and 440 sites at seven coherence lengths, and the cost measured here grows quickly with size; the stiffer of the two is a budget decision rather than a formality.