Every fraction holds a window

Three essays here name the devil’s staircase. The satellites that need a second integer calls it a sequence of plateaux at rational wavevectors with the widest at the simplest fractions; the crystal that rounds τ off says an alloy locks into a sequence of rational values as it cools; and the extra dimension that makes it periodic names the competition that produces it and then stops, in as many words, because deriving a staircase needs an energy and none had been computed anywhere.

One is computed here. It is not a material’s energy and is not offered as one: it is a chain of particles on the integers with a stated repulsion between them. What makes it worth computing is that the staircase does not depend on which repulsion, and the thing it does depend on is a condition a reader can check on any potential in one line.

Every rational density holds an interval of the controlling parameter to itself, with no smooth stretch anywhere between them. For the shortest-ranged repulsion tried, the density one half takes 19 per cent of the whole range on its own; the densities with denominator up to 10 take 95.25 per cent of it; those up to 24 take 99.995 per cent; and the narrowest of the 709 computed is two parts in a million million. What remains after every rational has taken its window is the irrational densities — which are the genuinely incommensurate ground states, and which have no width at all.

The model, and the one condition it turns on #

Particles sit on the integers. Any two of them a distance $r$ apart cost $V(r)$, with $V$ positive, decreasing, and strictly convex: $V(r-1) - 2V(r) + V(r+1) > 0$ at every $r$. The density $\rho$ is the fraction of sites occupied, the energy per site of a ground state at that density is $u(\rho)$, and the parameter that controls the density is the chemical potential $\mu$ — the ground state at a given $\mu$ minimises $u(\rho) - \mu\rho$, which is the ordinary trade between packing more particles in and paying for their repulsion.

Convexity is doing all the work, and it is worth saying why before any number appears. Take a uniform arrangement and move one particle one step. It comes closer to a neighbour on one side by one and further from the other by one, and a convex $V$ makes the closer approach cost more than the retreat saves. So a convex repulsion punishes unevenness, always, at every scale — and the ground state is whatever the most even arrangement is.

The most even arrangement, and what it turns out to be #

At density $P/Q$ the whole numbers will not let the particles sit at equal spacings unless $P$ divides $Q$, so the most even arrangement uses the two spacings nearest $Q/P$ — and it uses them in a particular order. Putting the $n$-th particle at $\lfloor nQ/P \rfloor$ gives exactly that, and the gaps take only two values, never three, at every one of the 276 densities checked.

That sequence is not a new object. It is a cut through a square lattice at slope $P/Q$, which is the cut-and-project construction with a rational slope in place of an irrational one, and the two-gap structure is the three-distance theorem arriving from the other side. The Fibonacci chain is the same construction at the golden slope. So the commensurate ground states of this model are the rational members of a family whose irrational members are the quasicrystals, and the staircase below is a statement about how much of the parameter axis each member gets.

The claim that the most even arrangement is the ground state is Hubbard’s, and it is checked here rather than quoted: at every density with denominator at most 14 — 63 of them — every arrangement of the particles was built and its energy computed, and the most even one won every time. The search is exhaustive rather than clever: at density 5/13 it compares 792 arrangements, and the point of doing it that way is that a clever search can only find what it was told to look for.

There is a further thing the two-gap structure settles, and it is the reason the arrangement is unique rather than merely optimal. Two gap lengths in some order is not enough — the order matters, and almost every order costs more. At density 2/5 the gaps are 2 and 3 and the only arrangements are 2 3 and 3 2, which are the same sequence shifted; at density 3/8 the gaps are 2, 3, 3 and there are three orders, of which the most even is the one that separates the two threes. The number of orders grows quickly with the denominator and exactly one of them wins, which is the same statement as saying the ground state is a cut at slope $P/Q$ and not merely a sequence with the right gap counts.

A curve with a corner at every rational #

The grand potential to be minimised is $u(\rho) - \mu\rho$, and minimising it over $\rho$ is a Legendre transform: the ground state at a given $\mu$ is the density where the tangent to $u$ has slope $\mu$. So the chemical potential is the slope of $u$, and where $u$ has a corner the density stops moving over a whole interval of $\mu$ — every slope between the two tangents at the corner picks out the same density. The curve above has a corner at every one of the 177 points on it, and the check that it does is that $u$ is convex — verified at every point, for all three repulsions used.

Why a corner rather than a smooth bend is the heart of it, and the reason is the two-gap structure. Approach $P/Q$ from below and the extra particles have to go in somewhere: each one is a defect in an otherwise perfect $P/Q$ arrangement, and it costs a definite amount however far apart the defects are, because the repulsion is short enough that two defects far apart do not know about each other. Approach from above and the missing particles do the same. The two costs are different numbers, and the difference is the corner.

A convex curve without corners would give a staircase with no steps at all — the density would move smoothly with $\mu$ and no rational would be preferred. So it is the corner and not the convexity that produces the plateau, and convexity is what produces the corner.

That reading gives the width a physical name. The right-hand slope is the cost of the cheapest way to put one more particle into a perfect $P/Q$ arrangement, per unit of density; the left-hand slope is the saving from taking one out. The width of the step is the energy of a defect, and it is positive exactly because a defect in a most-even arrangement costs something. In the language a crystallographer would use, the defect is a discommensuration — a place where the two-gap sequence slips by one — and the step is wide when discommensurations are expensive and narrow when they are cheap. A discommensuration in a fine arrangement is cheap because the arrangement is nearly uniform already, which is why the steps shrink so fast with the denominator.

The staircase #

Every step is flat, every step has positive width, and the risers between them are not smooth stretches — they are where the densities with larger denominators sit, and a finer computation fills them with more steps.

The widths are not equal and their ordering is the one a reader would guess: one half is widest, then one and two thirds, then the quarters, then the fifths. That is the Farey ordering, and it is the same ordering that makes an epitaxial mismatch climb in steps rather than smoothly, for the same reason — the simplest fraction available in an interval is the one with the most room around it.

The ordering is worth stating exactly, because it is a construction rather than a tendency. Between any two fractions that are neighbours in a Farey sequence, $P/Q$ and $P'/Q'$ with $|PQ' - P'Q| = 1$, the fraction with the smallest denominator is their mediant $(P+P')/(Q+Q')$ — and it is the widest step in that gap. Applying that recursively from the two ends of the axis builds the whole staircase in order of width, which is the Stern–Brocot tree. So the answer to “which locked phase appears next as the parameter is tuned” is not a matter of energy at all once the two neighbouring phases are known: it is the mediant, and the arithmetic settles it.

The fall is steep. The step at one half is nearly ninety thousand times wider than the widest step at denominator 24, and the narrowest step computed is five parts in a million of the axis. That is what makes the staircase look like a staircase rather than like a fog: at any finite resolution only the simple fractions are visible and the rest of the axis looks continuous while being nothing of the kind.

It is also a caution about experiments. A measurement that sweeps a parameter and reports four or five locked phases has not shown that the others are absent; it has shown that the others are narrower than the sweep’s step size. The steps are all there, and almost all of them are too narrow to find.

One consequence of the fall is worth extracting before leaving it, because it bears on how a phase diagram is read. The steps at denominators 2 through 6 account for 76 per cent of the axis, and every remaining step put together accounts for the other 24 — but there are infinitely many of the latter. So the axis is not “mostly the simple phases with a few exotic ones at the edges”: it is mostly the simple phases, with the rest of it divided among an unbounded number of increasingly narrow ones, each of which is a perfectly good crystal with a perfectly good unit cell. A material sitting in the 24 per cent is in a locked phase with a large cell, and whether the experiment calls it locked or incommensurate depends only on how large a cell the measurement can resolve.

The steps are the arithmetic; the widths are the material #

Three repulsions were used: two exponentials with ranges differing by more than a factor of two, and an inverse sixth power. The set of steps is identical for all three — every rational density holds a window, in the same place, for every one of them. The sizes are not remotely alike.

The inverse sixth power, which is very short-ranged, puts 96 per cent of the axis on the single step at one half and essentially nothing anywhere else. The long exponential spreads it: one half takes 7 per cent, and the steps up to denominator 20 are needed to account for 95 per cent. So a system with a long-range interaction has many accessible locked phases and one with a short-range interaction has essentially two, and that is a statement about the potential rather than about the arithmetic.

The contrast is sharper than a reader might expect from three potentials that all look much alike when plotted. What the range of the repulsion controls is how far a defect’s influence reaches, and a defect that is felt by only its nearest neighbours costs the same whether the arrangement around it is coarse or fine — so the steps at large denominators get nothing. A long-range repulsion lets the defect be felt across the whole repeat, so a fine arrangement resists it too, and the fine steps stay open.

This separates two things that the phrase “devil’s staircase” runs together. That every rational locks is a theorem about convexity and is the same for every material. Which rationals are wide enough to see is a question about the interaction, and it is the question an experiment is actually answering when it reports a phase diagram.

What a well does to it #

The negative test is the one that says the convexity is load-bearing rather than decorative. Give the repulsion a shallow well — make $V$ smaller at two sites’ separation than a convex interpolation would allow — and at 12 of the 45 densities tried, some arrangement beats the most even one. The winners are clustered rather than spread: pairs of particles sitting at the well’s own separation, with wider gaps between the pairs.

With that potential the whole argument collapses. The ground state is no longer the two-gap sequence, $u$ is no longer convex, and there is no reason for any particular rational to hold a window. A real interaction with a minimum at a preferred distance is exactly this case, which is why the staircase belongs to systems where the frustration is between a repulsion and a lattice rather than between two preferred lengths — and why an alloy with a strong preferred bond length shows a phase diagram that looks nothing like this one.

What is computed, and what is not #

The energy is a model’s and not a material’s. Nothing here computes a binding energy, a phonon spectrum or an electronic structure; the interaction is postulated and its only stated property is convexity. What that buys is generality rather than accuracy — the conclusion applies to every convex repulsion at once, so it needs no particular material to be right about and makes no prediction about any particular material’s numbers. The place this can be checked against something real is the shape of a measured phase diagram rather than any of its values.

The widths are upper bounds that fall as the density list is refined. A plateau’s width is read off as the jump in the slope of $u$ between its neighbours in the list computed, and a finer list puts more corners in between, so every number here is slightly too large. Measured at denominators up to 16, 24 and 48, the width of the step at one half is 0.457681, 0.457569 and 0.457568 — so the convergence is fast, and it is reported rather than assumed.

The irrational densities cannot be reached by this computation at all, and they are the interesting set. What the numbers say about them is indirect: the rational steps with denominator up to 24 account for all but a thousandth of the axis for the shortest-ranged repulsion, and the remainder shrinks as the list grows. The irrational densities are what is left, they are dense, there are uncountably many of them, and they occupy no length. That is the precise sense in which an incommensurate ground state is both ubiquitous and improbable — and it is the same statement the modulated crystal makes about a wavevector that drifts with temperature and locks when it can.

The set of irrational densities deserves a sentence of its own, because it is easy to say two wrong things about it. It is not small in the sense of being rare — it is dense, and between any two rationals there are uncountably many of them. It is not large in the sense of taking up room: the rationals take all of it. Both statements are true at once, and the second is what the staircase measures. An incommensurate ground state exists for every irrational density and is reached at exactly one value of the chemical potential, so it is stable in the sense of being a genuine minimum and unstable in the sense that any perturbation of the field lands somewhere else. That is the precise content of the observation that incommensurate phases are hard to hold and easy to find.

And this is one dimension with one interaction and no temperature. A real modulated crystal has three dimensions, a wavevector rather than a density, and a free energy rather than an energy; the parameter swept is temperature or pressure rather than chemical potential; and at any temperature above zero the narrow steps are washed out by entropy long before they are washed out by the experiment’s step size. None of that changes the shape of the argument, and all of it changes the numbers.

Still open: how wide a step has to be to be a phase #

The computation gives every step a width in units of the chemical potential, and an experiment measures widths in kelvin or in gigapascals. Converting between them needs the relation between the controlling field and the model’s $\mu$, which is material-specific and is exactly the part nothing here supplies.

What could be supplied and is not is the other end: the width at which a step stops being a phase at all. A locked state whose window is narrower than the fluctuation in the controlling field is not observed, and a locked state whose coherence length is shorter than the sample’s domain size is not distinguishable from an incommensurate one. Both are thresholds with a number in them, both are computable from the widths above plus one more measured quantity, and putting them on the same axis as the step widths would turn a staircase into a prediction of how many steps a given experiment can see. That is the calculation this account stops short of.