The ice rule is a conservation law

The arrangements a crystal keeps at absolute zero counts what the ice rule allows. Every oxygen holds two protons close and two far, the count of arrangements is a number with more digits than there are atoms, and its logarithm is an entropy a calorimeter reads. The essay ends with a paragraph it could not compute: the rule is a conservation law, so the disorder it leaves has correlations that fall as a power of distance, and the diffuse scattering has points where it approaches different values from different directions.

A facet with no energy in it then counted piles of cubes and found that a count alone makes a shape — flat where a height has no room to move, rounded where it has plenty. A height function did the work there, and one turns up here too.

The count says how much freedom the rule leaves; the scattering says what the rule is. This essay computes the second. The picture at the head of it is the scattering of square ice, averaged over a hundred sampled arrangements, and its most visible feature — a bright band crossing a dark one at the origin — is the rule written in reciprocal space.

Two in, two out, and a move that keeps it #

Square ice is the two-dimensional version of the rule. Put an arrow on every edge of a square lattice, and require exactly two arrows into every vertex and two out. It is the six-vertex model, and its count is known exactly: the number of arrangements grows by Lieb’s factor, the three-halves power of 4/3, about 1.54 a vertex.

To sample the arrangements rather than count them, the lattice is wrapped into a torus thirty-two vertices a side and started from the arrangement with every arrow pointing up or right, which obeys the rule. Then it is shuffled by a move that cannot break the rule. Start at a vertex, leave along one of its two outgoing arrows, arrive at the next vertex, leave along one of its outgoing arrows, and continue until the walk comes back to a vertex it has already visited. The part of the walk between the two visits is a closed loop of arrows all pointing the same way round it. Reverse every arrow on the loop.

Every vertex the loop passes through had one of its arrows entering along the loop and one leaving along it; after the reversal it still has one of each, so its count of two in and two out is untouched. A thousand and twenty-four such moves make one sweep of the lattice. Four hundred sweeps were discarded to forget the starting arrangement, and then an arrangement was recorded every six sweeps until there were a hundred. Every recorded arrangement was checked vertex by vertex, and none broke the rule.

A conservation law in the language of scattering #

Count an arrow pointing out of a vertex as plus one and an arrow pointing in as minus one, and add them up. The total is the divergence of the arrow field at that vertex, and the ice rule is the statement that the divergence is zero everywhere. That is the form a conservation law takes on a lattice: the arrows are a flow, and nothing is created or destroyed at any vertex.

A diffraction experiment does not see vertices; it sees the Fourier transform of the arrows. So the rule has to be translated. The divergence at a vertex is the horizontal arrow leaving it, minus the one arriving from the left, plus the vertical one leaving it, minus the one arriving from below. A difference of neighbours becomes, after Fourier transformation, a multiplication by a factor that depends on the wavevector. So the ice rule in reciprocal space is one linear relation between the horizontal arrows’ transform and the vertical arrows’ transform, at every wavevector separately.

Any field of arrows can be split, at each wavevector, into the part the divergence sees — the longitudinal part — and the part it cannot see. The rule says the longitudinal part is zero. Not small, not zero on average over arrangements, but zero in each arrangement at each wavevector, because it is the transform of a divergence that is zero at every vertex. Measured on the sampled arrangements, the longitudinal intensity at the smallest wavevector along the horizontal axis is 2.9 × 10⁻³², which is the rounding of the arithmetic.

What survives is a field whose arrows at each wavevector are forced to point across that wavevector rather than along it. For the horizontal arrows alone, that means their intensity must vanish where the wavevector itself is horizontal and need not vanish where it is vertical.

The pinch #

That is exactly what the map shows. Along the horizontal axis through the origin, the horizontal arrows scatter nothing; along the vertical axis they scatter strongly, with an intensity near 1.75 at the smallest wavevector. The two axes meet at the origin, and approaching the origin along one gives zero while approaching along the other gives 1.75. The intensity has no single value there. That discontinuity is a pinch point: bright and dark arms meeting at a point, like a bow tie drawn in intensity.

A pinch is not what disorder usually looks like, and the comparison makes the point.

Arrows drawn at random, each pointing either way with no rule at all, scatter a level grey with no feature: at the smallest wavevector the horizontal arm is 1.089 of the vertical, which is one up to sampling. Ice scatters the pinch, with a ratio of zero. The difference between the two is not in how disordered they are — both have enormous numbers of arrangements — but in whether a conservation law holds. Random disorder has short-range correlations and a smooth diffuse intensity; a local constraint that conserves something has long-range correlations and a singular one.

The third panel is ice with the rule broken now and then, and there the dark arm has begun to fill in near the origin: its ratio at the smallest wavevector is 0.520. The pinch has acquired a width.

What a defect does #

A defect is a vertex where the rule fails — three arrows in and one out, or the reverse — and it carries a charge of plus or minus two in the divergence. To admit them, the sampling above was given a second move: reverse a single arrow, and accept the change with the probability a cost μ for each unit of squared charge allows. At μ = 3.5 the sampled arrangements carry defects at 3.28 per cent of their vertices, and at μ = 4 at 2.02 per cent. The densities were measured, not set.

A defect switches on the longitudinal part. The divergence is now the pattern of charges, so the longitudinal intensity is the charges’ own scattering, and a charged fluid has a known form for that. In a salt solution the ions arrange themselves so that each is surrounded, on average, by a cloud of opposite charge, and beyond a screening length the cloud cancels it. Debye and Hückel’s theory of 1923 gives the longitudinal intensity a Lorentzian form, $\kappa m^2/(s^2 + m^2)$, where s is the wavenumber, κ is the stiffness of the field and m is the inverse screening length — and it predicts $m^2 = 4\rho/\kappa$ for charges of ±2 at a density ρ. Nothing in the sampling knows about that theory; the charges have no energy of interaction at all, only a cost for existing, and any screening has to come from the loop moves’ reshuffling of the arrangements around them.

The fit is made over every wavevector near the origin, not read off one arm, because $1/(\kappa m^2)$ and $1/\kappa$ appear as the slope and the intercept of a straight line when the reciprocal of the intensity is plotted against $s^2$. At 3.28 per cent defects the fitted $m^2$ is 0.0823 and Debye and Hückel’s prediction is 0.0718; at 2.02 per cent the fit gives 0.0405 against 0.0433. The stiffness κ that goes into the prediction is read independently, from the part of the intensity the divergence cannot see, and comes out near 1.85 at both densities. The width of the pinch is the screening length of the defects, and the salt-solution formula predicts it to within a sixth.

Below about one per cent of defects the measurement fails, and the reason is worth stating. The screening length grows as the density falls, and once it is longer than the torus, or the charges are too few for a hundred arrangements to average their scattering, the fitted $m^2$ wanders to nonsense — at 1.3 per cent, in a trial run on the same torus, it came out thirty-four times too small. The agreement is established at two densities and nowhere else.

How far one arrow reaches #

The pinch in reciprocal space is a statement about correlations in real space, and the two are transforms of each other.

For ice, two horizontal arrows one step apart along a row point the same way a little more often than not: their average product is 0.244. Two steps apart it is 0.088, three steps 0.039, four steps 0.024. From two steps to four the correlation falls by a factor of 3.7, where a fall as 1/r² would give four. Along a column the same numbers appear with the opposite sign, −0.284, −0.088, −0.033, −0.020. A correlation that falls as a power and changes sign with direction is the shape of the field around a dipole, which is what a divergence-free field looks like from far away, and it is the real-space face of the pinch.

With defects the correlations begin to fall faster: 0.086 at two steps, 0.034 at three, 0.020 at four. The difference from ice is small at these distances, because the screening length at 3.3 per cent defects is about three and a half steps, and it would grow with distance; at the sizes this torus reaches, sampling noise of about a thousandth sets the limit on following it.

The height function the rule makes #

In two dimensions there is a second way to say that the divergence vanishes, and it connects the pinch to the pile of cubes.

A field of arrows on a square lattice with zero divergence at every vertex is the rotated gradient of a function on the lattice’s faces. Give every face a height, and let each arrow record the difference between the heights of the two faces it separates, with a sign fixed by its direction. Going round any vertex crosses four arrows, and the four differences add to nothing exactly when two point in and two point out. So the ice rule is the condition that a height function exists, as the lozenge tilings’ stacking rule was the condition for a pile of cubes to have one, and a defect is precisely a vertex around which the height changes by two and is not a function at all.

The height difference across r steps of a row is minus the sum of the vertical arrows crossed, and its mean square can be measured on the same arrangements. For random arrows, where no height exists and the sum is a random walk, it grows in proportion to the distance: 2.00, 3.99, 8.00 and 16.11 at two, four, eight and sixteen steps. For ice it grows far more slowly — 1.43, 1.90, 2.46, 3.33 — and at the first doublings of the distance, from one step to two, two to four and four to eight, it gains 0.43, 0.47 and 0.56: close to a fixed amount per doubling, which is growth in proportion to the logarithm, where a random walk’s gain doubles each time. The largest distance on a torus thirty-two across also carries the torus’s net flow of arrows, which a surface without edges does not have, so the last step is larger. With defects the height stops being a function wherever a defect sits, and the sum grows faster again: 1.49, 2.13, 3.22 and 5.29.

A height whose mean square difference grows as the logarithm of distance is a rough surface, only just, and it is the reason for everything above. The surface’s fluctuations at wavenumber q have a size that grows as 1/q² towards long wavelengths; the arrows are that surface’s slopes, rotated, so each horizontal arrow’s intensity is the vertical slope’s, which is $q_y^2/q^2$ times a constant. That fraction is zero along the horizontal axis and one along the vertical axis, and it has no limit at the origin. The pinch point is the logarithmic roughness of the height, seen through a Fourier transform.

So the facet in the pile of cubes and the pinch in the scattering of ice are one object in two conditions. A height function with a local rule is flat where the rule leaves it no room, as at the pile’s corners, and rough where it leaves room, as everywhere in square ice; the flat part scatters nothing diffuse, and the rough part scatters a pinch.

What a diffraction experiment would see #

The average of every arrangement puts every arrow at half pointing each way, which is a structure with more symmetry than any arrangement has — the symmetry of an average — and its Bragg reflections are identical for ice and for random arrows. A structure refinement cannot tell the two apart, for exactly the reason what diffraction cannot tell apart gives: the average is the same.

The difference is entirely in the diffuse scattering, and there it is not subtle. A random arrangement’s diffuse intensity is level; a rule-bound arrangement’s has pinch points, and their widths measure how often the rule is broken. The pinch is the rare case where a feature of diffuse scattering is a singularity rather than a smooth bump — a sharper cousin of the peaks that are neither Bragg peaks nor diffuse bumps, with the sharpness in the dependence on direction rather than in the dependence on distance.

The same shape of argument appeared in a quite different place: the hat’s corners scatter an average crystal plus a remainder, and the remainder is where the aperiodicity is. Ice’s arrows scatter an average structure plus a remainder, and the remainder is where the rule is.

What the sampling does not settle #

This is square ice, not ice. The ice in a glass of water is three-dimensional, its oxygens sit on a hexagonal diamond-like lattice, and its protons have four bonds each in a tetrahedral arrangement. The conservation law is the same, and the pinch points are expected at reciprocal lattice points of that lattice; their measurement in real materials has been made in spin ice, where magnetic moments obey the same rule. None of that is computed here.

The defect model is a choice. Defects cost μ for each unit of squared charge and do not otherwise interact. Real charged defects in ice do interact, and the screening computed here is the entropic part alone. That the Debye–Hückel form fits at all with no interaction energy is a statement about the rule, and it is established at two densities.

The power law is not fitted. The correlations fall from two steps to four by 3.7 against the four a 1/r² fall would give, and the statement that they fall as 1/r² is the known theory, consistent with the numbers and not derived from them.

The torus has a net flow. On a surface without edges the total flow of arrows round each direction is conserved by loop moves that do not wrap, and wrapping loops change it; the height’s roughness at the largest distances includes it.

The checks, and what they refuse #

The first refusal is the defect, recognised by its pair of opposite charges. The second is a matter of method rather than physics: a fast Fourier transform needs a side that is a power of two, and a lattice of twelve is refused rather than padded to sixteen, because padding would place every wavevector somewhere other than where the arrows put it.

Who found the pieces #

Linus Pauling estimated ice’s residual entropy in 1935 by treating the vertices as independent, and Elliott Lieb solved square ice exactly in 1967, which is where the factor of 1.54 a vertex comes from. Peter Debye and Erich Hückel gave the theory of screening in electrolytes in 1923. The recognition that an ice-rule system is an electrostatics without charges — a field with zero divergence, whose correlations are dipolar and whose scattering pinches — was assembled over the following decades, and in 2009 Tom Fennell and his collaborators measured the pinch points directly by polarised neutron scattering from the spin ice holmium titanate. The loop move is the standard way such constrained arrangements are sampled.

Where this goes: the entropy a defect carries #

The defects above were counted and their screening measured, but their effect on the quantity the count of square ice began with, the entropy, was not. 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, and the height function says where to look for the answer: in how much freedom a defect gives the surface around it.