Five rings that were five different sizes

The amplitude the collapse left behind did two things that did not quite meet. It confirmed a collapse — five rings warmed until their Peierls distortions vanish, each ring’s alternation divided by its own cold value, plotted against the reduced temperature, and the five curves agreeing to 3.41 per cent. And it found, separately, that a ring’s departure from the infinitely long chain’s amplitude is not a function of the ring’s size or of its stiffness but of the two together: the ring measured in its own alternations, $n\delta_\infty$.

Put those side by side and a question appears. If that product is what decides how far a ring is from being a chain, then five cases at five different values of it are five different systems, and a collapse across them is being asked to hold across a variation nobody controlled.

The five sit at 9.56, 4.89, 1.36, 7.33 and 9.77. A factor of 7.2.

That essay could see it. What it could not do was act on it, because the two findings arrived in the same essay: the scaling variable was identified in the course of explaining why one of the five cases sat so far from its own bulk amplitude, by which point the collapse had already been measured on the cases it had. Recomputing it on matched cases is not a repair of an oversight — it is the second half of a finding that could only be made in that order, and it needs rings the first half had no reason to build.

What the collapse is claiming #

It is worth restating what is being tested, because the claim is stronger than an exponent and that is the whole reason it matters.

A ring warmed from cold loses its alternation, and it does so along a curve. The collapse says that if each ring’s alternation is divided by the value it settles at when cold, and the temperature is measured as a fraction of that ring’s own transition, then every ring gives the same curve — whatever its size, whatever its spring. If that holds, then a slope fitted over any stated stretch is the same number for every case, which is exactly why the exponent was the windows found a fitted exponent stable across systems without the exponent meaning anything. The stability was the collapse, not the physics.

So the residual on the collapse is load-bearing. It is the number that says how much of that explanation is real, and a residual that is mostly finite size is a different statement from one that is a genuine failure of the scaling.

Three rings of forty are three different rings #

The reason the published set is spread out is not carelessness — it is that the natural way to pick cases is to vary the two knobs the model has. Three of the five are rings of forty at stiffnesses of 1.2, 1.6 and 2.4, which looks like holding size fixed and varying stiffness.

In the units that decide finite-size behaviour it is nothing of the kind. The bulk alternation falls from 0.239 to 0.034 across those stiffnesses, so the same forty sites is 9.56 alternations at the first and 1.36 at the third. The third case is a very small ring; the measurement of the amplitude said so, and put a number on it — thirty-six per cent from its own bulk amplitude.

That number — thirty-six per cent — is what makes this worth an essay rather than a footnote. A case a third of the way from its own asymptote is not a slightly imperfect member of the set; its whole scaled curve is the curve of a small ring, and it is being averaged with four others as though the five were the same object seen five times.

Holding the product fixed instead means growing the ring as the spring softens: 40, 56, 78, 108 and 150 sites at stiffnesses of 1.2, 1.4, 1.6, 1.8 and 2.0. That is the comparison the amplitude measurement asked for, and the reason it was not made there is visible in the last number — a ring of 150 warmed through its transition is a considerably larger calculation than a ring of 40, and the collapse it belongs to needs five of them.

What matching does #

The matched set agrees to 1.62 per cent where the published set agrees to 3.41 — a factor of 2.10 tighter. So the answer to the question is the one suspected: the residual was finite size, and the collapse is better than the 3.41 per cent first quoted.

Where the improvement sits is worth reading. The two sets are closest near the transition and furthest apart away from it, which is what a finite-size explanation predicts: close to the transition the alternation is small and every ring is large compared with it, and far from it the alternation is at its cold value and a ring of forty at a soft spring is genuinely short. The published set’s worst point is its coldest, and that is where the small case does its damage.

One more thing the matched set settles, cheaply, because it was computed anyway. Three of the published cases share a stiffness and differ only in size — rings of 40, 60 and 80 at K = 1.6 — and the spread between the largest two was reported separately, as the place a size dependence would show. In the matched set no two cases share a stiffness at all, so a residual that survives there cannot be a size effect hiding behind one spring. It is a statement about the whole family.

The floor the lattice imposes #

The obvious next sentence would be that the collapse is exact to 1.6 per cent. It is not available, and the reason is arithmetic rather than physics.

A dimerised ring needs an even number of sites. The sizes the matching asks for are 40.2, 56.5, 78.6, 108.7 and 149.7, and rounding those to even numbers leaves the product spread over 1.09 per cent rather than zero.

So the experiment has a floor built into it that is the same order as its result. A test whose control variable cannot be set exactly cannot claim a residual smaller than the error in setting it, and 1.62 against 1.09 is not a comfortable margin — it is a factor of 1.5. The honest statement is that the residual was finite size, that matching removes at least half of it, and that whether it removes all of it is not decidable here.

That limit is not an artefact of a bad choice of target. A larger target would need proportionally larger rings — the fractional rounding error falls as $1/n$ — so the floor can be pushed down, at a cost that grows with it. At $n\delta_\infty = 9.6$ the softest case is already 150 sites; halving the floor means doubling every ring.

What is left, and what it is not #

Two readings of the remaining 1.62 per cent are available and nothing here separates them, which is worth saying plainly rather than leaving to the reader.

It may be the rounding, in which case the collapse is exact and every number in this essay is a measurement of even-site arithmetic. Or it may be a genuine departure from a single scaling function — corrections to scaling, which are ordinary and expected, and which would show up as a residual that survives however carefully the sizes are matched.

What distinguishes them is how the residual moves when the mismatch moves, and that is a measurement has not been made here. There is one target and one mismatch, so one point on a curve that needs several. The claim that survives is the weaker and more useful one: the residual was at least half finite size, which is enough to say the collapse is better than 3.41 per cent and not enough to say how much better.

What was computed, and how #

Each case is a ring relaxed to its best alternation at a sequence of temperatures approaching its own transition, with the transition located first by bisection. The alternation at each temperature comes from a golden-section search over the ring’s free energy, which is a closed form — a sum of square roots over the ring’s levels — so no matrix is diagonalised anywhere in the collapse.

The published five are recomputed rather than quoted: the spread comes back at 3.41 per cent from the same function that produced it, which is the tripwire for the whole comparison. If that number had moved, the matched set would be being compared against something other than the published result.

Nothing here diagonalises anything, and that is the reason a ring of 150 is affordable at all. The same collapse computed by diagonalising each ring at each temperature would be a different order of work — which is the trade the carriers a distortion was hiding had to make in the other direction, where the quantity wanted was a population and no closed form was available.

The refusal is the quantisation itself, and it is checked rather than mentioned: the residual after matching must be within a small factor of what the rounding leaves. If a future version of this ever reports a matched spread far below the mismatch in its own control variable, it is measuring its own arithmetic and the check is there to say so.

Where the model stops #

Five stiffnesses at one value of the product is one line through a two-dimensional space. The collapse could be tested at a second target — 5, say, and 20 — and the prediction is that the residual falls as the target rises, since every case is then closer to its own bulk. That is the same calculation at different sizes and it is bounded work, though the largest ring at a target of 20 is over three hundred sites.

What a metal actually is draws the distinction this whole argument rests on, and none of these rings is near the thermodynamic limit in the sense that essay needs — the largest is 150 sites. Everything here is a statement about how a finite ring approaches a chain, which is the useful question precisely because the chain itself is not computable.

The temperature range is also fixed by the original choice of reduced temperatures, and the spread is a worst case over those points. A collapse that is excellent over most of the range and poor at one end reports the poor end, which is the right convention and is worth remembering when comparing to anything quoted as an average.

And the whole construction is mean-field: the ring’s free energy is minimised over one alternation amplitude, with no fluctuations in it. The exponent was the windows established that the exponent near the transition is one half, which is the mean-field value and is what this calculation is built to give. Nothing here tests whether a real chain does that.

It is also worth being clear that none of this touches whether the chain is a metal. The alternation is what opens the gap, and a half-filled band is not always a metal is where that argument lives; this essay is about how accurately a scaled curve of that alternation collapses across sizes, which is a question about the instrument rather than about the conductor.

The generalisation #

The lesson is about what “holding a variable fixed” means, and it is not confined to rings.

Three of the published cases hold the site count fixed. That is a real thing to hold fixed and it is the wrong thing, because the quantity that governs the behaviour under study is the site count in units of a length the system itself sets — and that length varies by a factor of seven across the cases. Fixing the raw number is fixing the label rather than the quantity.

This is a common shape. A series of molecules at the same temperature spans a range of $k_BT/\Delta E$; a set of calculations with the same grid spacing spans a range of points per feature; a set of chains with the same number of units spans a range of units per persistence length. In each the natural knob is not the governing one, and a trend measured along the natural knob is partly a trend in the governing one.

The failure is quiet in a specific way: nothing about a set of cases at the same site count looks uncontrolled. The label is identical across them, the variation is in a quantity nobody printed, and the residual it produces is a smooth contribution rather than scatter — so it reads as the method’s accuracy rather than as a spread in the inputs. That is the same shape the metal a thermometer cannot find is about, where a quantity varied systematically under a measurement that reported only its average.

The repair is cheap when the governing quantity is computable, which it is here: the bulk alternation is a closed form, so choosing the sizes was arithmetic. What made it worth doing was knowing which quantity governs — and that was the earlier finding about the amplitude rather than this one.

A last remark on why the governing quantity was available to be fixed. It is a closed form — the bulk alternation solves a stationarity condition written in elliptic integrals, which the amplitude the collapse left behind derived precisely so that it could be had without a search. Had it still been something to be measured ring by ring, choosing sizes to match it would have required knowing the answer before designing the experiment, which is the ordinary reason this kind of matching does not get done. The closed form is what turned a circular requirement into arithmetic.

That is worth carrying beyond this model: a scaling variable is only usable as a design constraint once it is cheap to evaluate, and making it cheap is often the whole contribution.

Who found it, and when #

Finite-size scaling in the form used here — that a system’s departure from the thermodynamic limit is a function of its size divided by a correlation length rather than of its size — is Fisher’s, from the early 1970s, and is one of the standard tools of critical phenomena. Choosing system sizes so that the scaling variable matches across a series is the ordinary practice that follows from it.

What is new here is only the application: that the original five cases did not do it, that doing it halves the residual, and that on a lattice the matching has a floor which turns out to be the same size as the answer.

Still open: whether the residual is rounding or a real correction #

The obvious open question is the second target. Running the same matched collapse at $n\delta_\infty = 5$ and at 20 gives three points on a curve of residual against target, and the shape of that curve says whether what is left is finite size that keeps shrinking or a genuine departure from a single scaling function. The rings at the largest target are the expensive part and the answer is bounded by them.

The nearer question is the quantisation itself, and it can be turned from a limit into a measurement. The rounding error is known exactly for every case — it is the difference between the even size used and the size the target asked for — so the residual spread can be plotted against it across many random targets rather than one. If the spread tracks the mismatch, then the collapse is exact and everything measured here is the rounding; if it is flat, then 1.6 per cent is real and the rounding is beside the point. That is a hundred small collapses rather than five, on rings no larger than the ones already computed, and it would settle in one sweep what can only be bounded here.