What a defect costs the count

The ice rule is a conservation law admits defects into square ice, measures how they screen one another and finds the width they give the pinch point. It ends by naming what it did not do:

Each defect relaxes the rule at one vertex, and so each adds arrangements; the residual entropy per vertex must rise with the defect density, from Lieb’s value for pure ice towards the value for arrows with no rule at all. How it rises — whether each defect adds a fixed amount of entropy, or whether the screening cloud around it costs some back — is a count rather than a correlation.

It is a count, and what it takes is already here. Counting the arrangements a purely local rule permits is where such counts start, and the answers there are enormous, exact, and useful as a growth per site rather than as a total. The arrangements a crystal keeps at absolute zero counts square ice on a torus with a transfer matrix that walks along a row choosing, at each vertex, the one horizontal arrow the rule permits. Enumerating both choices instead, and carrying with them a polynomial whose powers record how many vertices end up wrong, counts every defect number at once.

The paragraph’s expectation turns out to be wrong in two places, and both are worth more than the number it asked for.

There is no arrangement with one defect #

The second column of the census is empty, and it is empty for a reason that has nothing to do with how hard it is to make a defect.

Add the divergence over every vertex of the torus. Each arrow leaves one vertex and arrives at another, so it contributes plus one to one sum and minus one to another, and the total is nought however the arrows are drawn. So the charges always cancel, and a single defect would leave plus or minus two over. The column at one defect is empty, the column at two holds 397,296 arrangements at this size, and the emptiness is a fact about a closed surface rather than about ice.

That is the conservation law the arrow field obeys, used as a counting argument rather than as a statement about scattering. There it said the longitudinal part of the arrow field vanishes; here it says the census has a hole in it at k = 1 and nowhere else.

The columns at odd k above one are not empty, and the reason is the second kind of defect. A vertex with all four arrows in carries minus four, so three defects can cancel as +4, −2, −2. Nothing forbids an odd number of broken vertices; what is forbidden is an odd number of them all carrying two.

The entropy does not rise to the free value; it passes it #

The question assumed the entropy runs from Lieb’s number up to the value for arrows with no rule. It does not.

The entropy at a fixed defect density is the logarithm of that column divided by the number of vertices. It starts at 0.4998 on a four-by-four torus, rises steeply, peaks at a density of 0.625 and comes back down to 0.919 when every vertex is defective.

The peak is where the argument gets interesting, because “arrows with no rule at all” is not a defect density of one. Arrows drawn at random break the rule at some vertices and not at others.

Ten of the sixteen states at a vertex break the rule, so a lattice of arrows drawn with no rule whatever has about ten sixteenths of its vertices defective — a density of 0.625. That is exactly where the exact count peaks, to the nearest whole number of vertices, at every size checked. Summing every column and taking the logarithm gives two log two, which is 1.386 per vertex, and that is the free value; the peak column alone gives 1.287, because fixing the defect number exactly is itself a constraint.

So the free state is five-eighths defective, not wholly defective, and the census runs past it. The question’s framing had the destination in the wrong place, and the peak sitting exactly at Pauling’s own probability is the clue to where it should have been.

Almost all of the rise is deciding where the defects go #

The rise from 0.4998 to the peak is steep and most of it is not about ice at all.

If a fraction ρ of the vertices are to be defective, there are as many ways of choosing which ones as there are ways of choosing ρN things out of N, and the logarithm of that is the familiar mixing term −ρ log ρ − (1 − ρ) log(1 − ρ). Its slope at ρ = 0 is infinite. The first defect can go anywhere, and the logarithm of “anywhere” grows faster than any fixed amount per defect.

So the answer to the first half of the question is no, and for a reason that is not physics: no quantity whose slope is infinite at the origin rises by a fixed amount per defect. What the question was reaching for is what is left after that freedom is taken out.

Subtracting the mixing term leaves a quantity that falls before it climbs. It starts at the ice value, drops to its lowest at a defect density near a fifth, and only passes the ice value again some way further along. Past the middle of the range it rises steadily to the value for wholly defective arrows.

So the second half of the question has the answer it guessed, and with a sign: the cloud of rearrangement around a defect does cost entropy back, and at low densities it costs more than the defect’s own freedom buys. The count says so without any correlation being measured — which is what the question asked for, a count rather than a correlation, and it is the first quantity in this series of counts whose sign was the thing in doubt.

Pauling’s estimate is low at one end and high at the other #

The same census gives Pauling’s approximation a test the pure ice count cannot.

Pauling’s estimate treats the vertices as independent: there are 2 raised to the number of edges arrangements in all, each vertex obeys the rule with probability six sixteenths, so the ice count is estimated as that product. It is famously low, by about two and a half per cent in the count. Extending it to a fixed defect number is immediate — multiply by the binomial and by ten sixteenths for each defective vertex — and then it can be compared at every density rather than at one.

At no defects the estimate is low, as it always is. Everywhere else in the range it is high — by about a tenth of a unit at the middle — and it comes back to roughly the right answer only when every vertex is defective. The crossing is not a small effect and it is in the direction that should be expected once it is stated: the independent-vertex estimate ignores the constraint that the charges cancel, which forbids some defect configurations outright, and forbidding configurations makes the true count smaller than the estimate.

The one thing the estimate gets exactly right is the position of the peak, because that is a statement about a single vertex and single vertices are what the estimate is made of. It places the maximum perfectly and gets its height wrong, which is a fair summary of the whole approximation.

The dip, read on the surface the arrows are #

The field description of the arrows is the one that explains the sign.

An arrow field with no divergence is the rotated gradient of a height on the faces, so square ice is a surface whose slopes are the arrows — the same construction a pile of cubes uses, where the stacking rule is what lets a height exist at all. A defect is precisely a vertex around which the height changes by two on going once round, so it is not a bump in the surface; it is a place where the surface stops being a function.

That is a dislocation, and a dislocation has a cost that a bump does not. Inserting one forces the height to be multivalued in a whole region around it rather than at one point, and the arrangements in that region are the ones consistent with the mismatch — fewer than the arrangements consistent with nothing. The freedom the defect buys is local and the freedom it destroys is not, and at low density the second outweighs the first. That is the dip, stated on the surface instead of in the count.

It also says why the dip ends. Once the defects are dense enough that their regions overlap, there is no coherent surface left to disrupt, nothing further to lose, and the count rises on the freedom alone. The density where the excess passes the ice value is where the surface stops existing as an object, and reading a number off the curve for that is a measurement this page has and does not know what to do with.

What an experiment would see of any of this #

Nothing, which is worth saying because the count is otherwise so specific.

A diffraction experiment measures an average over the arrangements, and the average of every arrangement at any defect density puts each arrow at half pointing each way — the symmetry of an average, with Bragg reflections that do not move as the defects are added. The entropy above is not in the sharp scattering and no refinement reaches it, which is the general difficulty an experiment measuring an average always has.

Where the defects are visible is the diffuse scattering, and there they are visible as a width rather than as a number: the pinch point acquires one, and that width is what a fit returns as a screening length. So the density is measurable and the entropy is not, which is exactly why a calorimeter and a diffractometer are asked different questions about the same crystal. The count on this page is what a calorimeter would integrate to, if the density could be held fixed while the temperature was swept, and that is not an experiment anybody can do either.

What the numbers are, and what they are numbers about #

The entropies quoted above are for a torus four vertices a side, and a torus that small counts too many arrangements: the exact value falls with size, 0.4998 at four and 0.4749 at five, heading for Lieb’s 0.4315 for an infinite lattice. The excess over Pauling falls with it — 0.094 at four, 0.069 at five — and would fall to the 0.026 the infinite lattice gives.

A torus is where every count here is made, for the reason a calculation over a crystal is made on a finite block with its edges glued, and the overcount is the standard cost of that. The same enumeration on an open square would undercount instead, which is what the choice of boundary decides.

The shape of the curve does not depend on the size. The peak sits at ten sixteenths at both; the dip in the excess sits near a fifth at both; the crossing of Pauling’s estimate is at a small density in both. Those are the statements this essay makes, and the absolute heights are the ones it does not.

There is a second boundary on the argument and it belongs to the counting rather than to the ice. The count at a fixed defect number is a count with a constraint in it, and constrained counts are smaller than unconstrained ones — which is why the peak column is below two log two. The quantity a thermodynamics would use is the free energy at a fixed chemical potential for defects, which is a Legendre transform of this curve rather than the curve itself. Nothing here computes that, and the sampling that fixes a cost μ and measures the density that results is working on the other side of the same transform.

Where the exactness stops #

Computed here. The number of arrow arrangements with exactly k defective vertices, for every k, on tori of side two to five, by a transfer matrix carrying a polynomial. Two checks that can fail: the coefficient at no defects must equal the ice count the earlier transfer matrix gives, and the coefficients must add to two raised to the number of edges. Both hold at every size, exactly, in integers.

Five is where it stops and the reason is arithmetic, not time. The counts are exact integers only while they fit inside a double-precision number, and the total at side six is two raised to seventy-two. The transfer matrix itself would run.

A defect here is a vertex where the rule fails, of either kind. A vertex with three arrows in and one out and a vertex with all four in are both counted as one defect, though they carry different charges. Separating them would need a polynomial in two variables and would answer a different question.

And the mixing term is exact rather than approximate. The number of ways of choosing which vertices are defective is a binomial coefficient, and subtracting its logarithm is subtracting something the count genuinely contains. What is not exact is the claim that the remainder is “what the arrows gain”: the two are not independent, since which vertices may be defective is itself constrained, and the dip measured above is partly that constraint.

The first refusal is Pauling’s estimate offered as the answer, which is the standing error these counts exist to correct. The second is a residual entropy quoted without saying which boundary it was counted on, and that one is what the choice of boundary decides.

Who counted what #

Linus Pauling’s estimate is from 1935 and Elliott Lieb’s exact solution of square ice from 1967, and both are in the count of pure ice. The extension to a fixed number of defects is not a famous computation and there is no named result to quote for it; the transfer matrix that does it is the standard one with one extra variable, which is the usual way such a census is taken.

What the defects themselves have a literature about is the other side of the transform. In spin ice the defective vertices are magnetic monopoles, and the entropy at a given density of them is what a measured heat capacity integrates to; the sampling with a cost per unit of squared charge is the standard way that is computed. The two approaches meet where the transform is taken, and neither this page nor that one takes it.

Still open: the two kinds of defect, counted apart #

Every defect above is one defect, whether it carries a charge of two or of four. Separating them needs a second variable in the polynomial, and the transfer matrix would carry it as readily as it carries the first.

Two questions would follow, and the second is the interesting one. How rare is the charge of four? In the sampling with a cost per defect they are effectively absent, because the cost was set proportional to the square of the charge and four costs four times what two does; in a census with no cost at all they are simply less numerous, and how much less is a number nobody here has.

And does the dip survive? The excess entropy falls at low density, and the account given for it is that the arrangements round a defect are constrained. If the two kinds of charge behave differently — if the charge of four costs more of the surrounding freedom than two charges of two do — then the dip is really two dips with different depths, and the single curve measured here is an average over a mixture whose composition changes along it. Nothing above can tell those apart, and a polynomial in two variables would.