The cubic term that forbids a continuous change

A crystal that changes structure has two ways to do it. The new arrangement can grow out of the old one continuously, with the quantity that measures the difference rising from zero as smoothly as anybody could ask; or it can arrive all at once, at a definite temperature, with that quantity jumping from nothing to something.

Which of the two is possible is decided before any material is named. The deciding fact is a count of polynomials, and the count is one that this ladder has already computed for a reason that had nothing to do with matter.

The quantity that measures the loss #

Symmetry does not disappear from a crystal by degrees: a mirror is either a symmetry of the arrangement or it is not. What varies continuously is the size of the distortion that breaks it, and that size is the order parameter — zero in the symmetric phase, non-zero in the other, and a number a measurement can return.

An order parameter is not a single number in general. It is a set of them, and what makes the set behave itself is that the parent group mixes its components among themselves: applying an operation of the high-symmetry group to the crystal takes one value of the order parameter to another. So the order parameter carries a representation of the parent group, and its components are as many as that representation’s dimension.

That is the whole of the structure Landau’s argument needs. Everything below is about polynomials in those components.

The energy is an invariant, so it is a polynomial in invariants #

Near the point where the change happens, the free energy is written as a series in the order parameter — nothing else being small enough to matter. The energy of a crystal cannot depend on which of several equivalent orientations was chosen for the coordinate axes, so the series must be invariant under the parent group.

Every term in it is therefore an invariant polynomial in the order parameter’s components. The quadratic term is the one whose coefficient changes sign at the transition; there is exactly one of those, because every order parameter used here has exactly one quadratic invariant, which is checked rather than assumed. And the question is what else the series is allowed to contain at low degree.

If there is an invariant of degree three, the series has a cubic term. Then at the moment the quadratic coefficient reaches zero, the energy is dominated near the origin by a cubic, and a cubic has a minimum away from zero on one side. The order parameter does not grow from nothing; it appears at a value already finite, at a temperature above the one where the quadratic coefficient vanishes. The change is discontinuous — first order, in the vocabulary of the subject.

If there is none, the leading term after the quadratic is quartic, which is even and can hold the parameter at zero until the quadratic coefficient turns. Then the parameter grows continuously from nothing, and the transition can be second order.

The count is a Molien coefficient #

The number of independent cubic invariants of a representation is the degree-three coefficient of the Molien series of its image group. That identification is what makes Landau’s condition computable here without new machinery: the sequence was already being computed for the invariant ring, and its third entry answers a question about phase transitions.

For a one-dimensional order parameter no computation is needed. Such a parameter is a single number η on which each operation acts by +1 or −1, and η³ is invariant only if every operation acts by +1 — which is the trivial representation and breaks nothing. So no one-dimensional order parameter carries a cubic invariant, and every transition driven by one may be continuous as far as this argument is concerned.

For a two-dimensional one the answer is not free, and in the plane it comes out as follows: the two-dimensional representation of the threefold group 3 has two cubic invariants, that of 3m has one, and those of 4, 4mm, 6 and 6mm have none.

Why three and not four or six #

The threefold cases stand out and the reason is worth stating in terms of the group rather than the answer.

An invariant of degree three in a two-dimensional order parameter is a cubic form the group leaves alone. Write the order parameter as a complex number η; a rotation by 2π/n multiplies it by a root of unity, and a cubic monomial η³ picks up the cube of it. That cube is one exactly when 3 · 2π/n is a whole turn — when n divides three.

So n = 3 admits a cubic, and n = 1 trivially. For n = 4 and n = 6 the phase does not return, and the cubic is forbidden. The sixfold group contains a threefold rotation and still has no cubic invariant, which is the part of the pattern that catches a reader out: the extra operations of 6 are precisely what kill the cubic that 3 permits.

That is also the answer to why the condition is a statement about the order parameter rather than about the parent group. A hexagonal crystal has plenty of threefold rotations in it; what matters is how the order parameter transforms, and a hexagonal crystal has order parameters of several kinds.

Landau’s condition is necessary, not sufficient #

A condition that is checked and passed does not make a transition continuous. It removes one obstruction.

Nothing in this argument fixes the sign of the quartic term. If that term is negative, the energy is again unbounded near the origin at the transition, higher terms take over, and the parameter jumps — a first-order transition in a case where the cubic invariant is absent. Such transitions are common, and the symmetry argument has nothing to say about them: the sign of a quartic coefficient is a fact about the material.

Nor does the argument say anything about whether a transition happens at all. It says which kind is permitted, and that is a statement of the same kind as what a class permits a property to do — permission is not presence, and this collection has an essay about the gap.

There is a further limitation worth naming, and it is a limitation of the whole framework rather than of the computation: Landau theory assumes the free energy is an analytic function of the order parameter near the transition. At a genuine critical point it is not, and the exponents a continuous transition actually shows are not the ones this expansion gives. What survives the criticism is the classification — which transitions may be continuous — because that part turns on the presence of a term rather than on its size.

The other condition, and where it bites #

Landau’s is not the only condition on a continuous transition. A second one concerns gradients rather than uniform values: if the energy admits a term linear in the derivative of the order parameter and antisymmetric in its components — a Lifshitz invariant — then the uniform state is not the best one, and the crystal prefers an order parameter that rotates as it goes. The result is modulated rather than commensurate.

Computing that count in the plane at the zone centre gives zero for every order parameter, and the reason is structural rather than accidental: the antisymmetric square of a two-dimensional representation is one-dimensional, and its product with the vector representation contains no invariant for any of these groups. The condition bites at wavevectors away from the zone centre, where the little group is smaller and the counting changes, and this collection’s incommensurate essays are where that story lives.

What the count is checked against #

The number of cubic invariants is computed here twice, in the sense that matters: the Molien coefficient comes from a trace recursion, and the averaged-monomial route recomputes it from the polynomials themselves. Both are exact integer arithmetic, and they agree at degree three on every class.

Two further checks constrain the result from outside.

Every order parameter has exactly one quadratic invariant. If a representation had two, there would be no single amplitude for the order parameter to be the amplitude of, and the whole framework would need rewriting. The count is made for all twenty order parameters of the ten classes, and it is one in every case.

The cubic count is zero for every one-dimensional order parameter, which the general machinery must reproduce and which the hand argument above settles independently. A discrepancy would mean the machinery had mis-identified a representation.

What the condition does not decide, in a crystal that has one #

A transition forced to be discontinuous is not thereby a small effect. It is the opposite: the order parameter arrives at a finite value, so the structure changes by a finite amount at a definite temperature, and the crystal has to accommodate that change all at once.

Two consequences follow that symmetry does not predict and that are worth separating from what it does. Hysteresis: a discontinuous transition can be overshot in both directions, so the temperature at which it happens on cooling need not be the one at which it happens on heating. And coexistence: the two structures can be present in one specimen at once, separated by a boundary, because neither is a small distortion of the other.

Neither is in the arithmetic. What the arithmetic gives is the permission, and the physics of how a particular material takes it up — how much it overshoots, how the boundary moves — is a question about energies, kinetics and defects that this collection does not enter. The line between the two is the line this site keeps everywhere: what a symmetry argument decides is which quantities may be non-zero, never how large they are.

The domains a discontinuous transition still makes #

One thing the cubic term does not change is the count of arrangements the crystal has to choose between. That count is the index of the new group in the old, and it is settled by the subgroup rather than by the shape of the energy.

So a first-order transition produces domains exactly as a continuous one does, and in the same number. What differs is the size of the difference between them: at a continuous transition the domains differ by a distortion that starts at nothing, so the walls between them are cheap and mobile just below the transition; at a discontinuous one they differ by a finite amount from the start.

The threefold case is the clearest illustration, because it produces three domains related by the rotations the crystal has lost, and the cubic invariant is what makes the three of them distinguishable at every temperature below the transition rather than only well below it. The invariant that forbids the gradual change and the invariant that distinguishes the three domains are the same polynomial evaluated at different points.

The transition the condition is usually illustrated with #

The condition is stated here in the plane, where the arithmetic is short enough to check completely. The case it is famous for is three-dimensional and is worth naming, because it shows the argument being used the way it was meant to be.

The ordering of copper and gold in the three-to-one alloy is a transition from a disordered face-centred cubic solid solution to an arrangement with the gold on one of the four sublattices — the transition the antiphase domains essay describes. Its order parameter has several components, corresponding to the several ways the ordering can be arranged, and the representation it transforms as admits a third-order invariant.

So Landau’s condition fails, and the transition must be first order. It is, unmistakably: the alloy has a latent heat, a two-phase region, and a discontinuity in the long-range order parameter at the transition — none of which a continuous transition has.

The prediction was made from symmetry before the thermodynamics was measured carefully, and it is the standard illustration in the alloy literature for that reason. What is worth noticing is how little was used: the parent group, the child group, the representation relating them, and one coefficient of a series. No energies, no interactions between atoms, and nothing about copper or gold.

The condition as a procedure #

Stated as a test rather than as a theorem, the condition is four steps and every one of them is a computation this collection already performs.

Identify the parent and the child. Two space groups, with the second a subgroup of the first — which is a descent, and the diagram says which are available.

Find the representation. The order parameter transforms as an irreducible representation of the parent under which the child is the isotropy subgroup, and finding it is a character computation of the kind the mode essays run.

Count the cubic invariants. The degree-three coefficient of the Molien series of that representation’s image, which is a sum of traces over the group.

Read the verdict. Non-zero forbids a continuous transition; zero permits one and predicts nothing further.

Every step is exact and the last one is one-directional, which is the shape of nearly every result in this collection. A non-zero count is a prohibition that no measurement will contradict. A zero count is the absence of one prohibition among several, and a transition passing it can still be first order for reasons — the quartic coefficient’s sign, the fluctuations, the strain — that the arithmetic does not reach.

Reading the table the other way #

The census of cubic invariants can be read as a statement about groups rather than about transitions, and read that way it says something short: in the plane, a cubic invariant of a natural representation exists exactly where the rotation order is three.

That is a small enough statement to check against the whole of the rest of this ladder. The classes with a cubic invariant are 3 and 3m; the classes whose invariant rings are not free are 2, 3, 4 and 6; the classes whose second invariant degree is odd are 3 and 3m again. Three different questions, three different arithmetics, and the threefold groups singled out by two of them.

It is not a coincidence and it is not a theorem proved here. The common cause is that three is odd and small: an odd degree survives no group containing the half-turn, and a degree smaller than the rotation order cannot be reached at all. The plane’s rotation orders are 1, 2, 3, 4 and 6, and three is the only one of them that is both odd and greater than one.

A hundred and one years of the same sum #

Landau wrote this condition down in 1937, and the arithmetic it turns on is Molien’s from 1897, which was about counting invariants and had nothing to do with matter. The meeting of the two is the sort of thing this collection exists to point at: a question about how a solid changes when it is cooled is answered by a coefficient in a power series that was written to count polynomials.

There is a second meeting worth naming, in the other direction. The descent of symmetry — which classes a crystal can fall to — is a question about subgroups, and it produces a graph with an index on every edge. Landau’s condition does not act on that graph: it acts on the representations, and a single edge of the graph can be reached by several representations, some carrying a cubic invariant and some not. So the two classifications are not the same classification, and the essays on the order parameter as a representation take that difference as their subject.

Both classifications are read off the same series, and the series is worth seeing whole rather than one column at a time. Landau’s argument depends on three of its terms and they are usually met separately: the quadratic, whose coefficient changes sign; the cubic, whose presence forbids a continuous change; and the quartic, whose sign the argument cannot fix but whose existence it can check. Laying the low degrees side by side puts all three in one place, and it makes the pattern of the odd columns visible — which is the fact this essay has been circling.

What the pictures cannot show #

None of the figures here shows an energy, because none is computed. There is no coefficient in this essay whose value depends on a material: what is drawn is which terms the energy may contain, and that is a fact about the group.

Nor is a temperature drawn. The vocabulary of the subject — above the transition, below it, the coefficient changing sign — is the language a free energy is written in, and every statement here is about the form of that energy rather than about its behaviour. A crystal that never reaches the temperature in question has the same permitted terms as one that does.

What the figures do show is the arithmetic: which counts are zero, which are not, and the fact that the ones that are not belong exclusively to representations with a threefold axis and no sixfold. That is a small, sharp result, and it is the kind that a computation can establish completely.