An order parameter is a representation

A crystal that loses symmetry does not choose a subgroup at random. Something in it becomes non-zero — a displacement, an ordering of two kinds of atom, a tilt — and the symmetry that survives is exactly the set of operations that leave that something alone.

Which is to say: the thing that becomes non-zero decides everything, and what it is is not a number but a set of numbers the parent group mixes among themselves. It carries a representation. That single observation turns the question of which subgroups a crystal may fall to into a question about representations and their kernels, and it makes the answers computable in exact arithmetic.

This essay computes them, and then checks every one by building the distorted structure and handing it to the detector that knows nothing about representations.

What makes a set of numbers an order parameter #

Three conditions, and they are all about the parent group.

The components transform among themselves. Applying an operation of the parent group to the crystal takes one value of the order parameter to another, linearly. So there is a matrix per operation, and the assignment is a homomorphism: the order parameter carries a representation.

The representation is irreducible. If it were not, the order parameter would split into two independent quantities that could become non-zero at different temperatures, and the transition would be two transitions. Irreducibility is what makes it one.

It is not the trivial representation. A quantity every operation leaves alone breaks nothing; it can grow without changing the symmetry at all, and that is not a transition but a change of a free parameter — which this collection has already met as the free coordinate of a Wyckoff position.

With those three, the list of available order parameters for a given parent group is finite and computable: it is the list of non-trivial irreducible representations, and for the plane point groups it runs to twenty across the ten classes — fourteen one-dimensional characters and six two-dimensional representations.

The kernel is the surviving group #

For a one-dimensional order parameter the arithmetic is immediate. Each operation acts by +1 or −1 — a sign — and the operations that act by +1 leave the value alone. Those form a subgroup of index two, the kernel, and it is the symmetry the crystal keeps.

So each non-trivial one-dimensional representation names exactly one subgroup, and the transitions it can drive are the descents to that subgroup. For 6mm there are three such representations, and their kernels are 6, 3m and 3m again — the two 3m subgroups being genuinely different subgroups with the same name, distinguished by which mirrors they contain.

That last point is the reason this computation is done on the group’s own operations rather than on a class symbol. p3m1 and p31m have the same point class and different mirrors; a character table indexed by symbol would attach the signs to the wrong operations, and the kernels would come back describing a group that is not there. The machinery here refuses a representation whose length does not match the group it is handed.

From a representation to a displacement #

A representation is an abstraction; a crystal is atoms. Turning one into the other is the step where a claim becomes checkable, and it is short.

Pick a general position and let its orbit be the atoms. Choose a displacement u₀ for the representative atom. The displacements of the others are then forced: the atom at g·p₀ moves by χ(g)·M_g u₀, where M_g is the operation’s linear part and χ(g) is the sign the order parameter attaches to it.

That is the mode, and nothing about it is free once u₀ is chosen. Applying another operation g' to the whole field multiplies it by χ(g') — so the field is carried to itself exactly by the operations with χ(g') = 1, which is the kernel again, arrived at by transporting arrows rather than by reading a character table.

Move every atom by a small multiple of its displacement and the result is a structure. Its symmetry is a fact about a point set, and the detector settles it.

Two computations that must agree #

The check this ladder is built on compares two answers that come from different objects.

The prediction asks whether an operation carries the displacement field to itself: for every atom, is the displacement at the image of the atom equal to the image of the displacement? That is a test on arrows and involves no point matching.

The detection asks which operations map the displaced point set onto itself. That is the round trip this whole site is built on, and it knows nothing about modes, characters or kernels.

The two lists of operations must be identical. Over one hundred and eighty modes — every order parameter of all seventeen plane groups, at the zone centre and at three zone-boundary wavevectors — they are, in every case.

Agreement is not automatic and the essay’s next two sections are about the two occasions this computation reported a disagreement, both of which turned out to be findings rather than bugs.

The first disagreement: a motif that was too symmetric #

The first version of this machinery displaced a single point per orbit, and reported that freezing an odd mode into pm produced a structure with a two-fold rotation — a symmetry the mode cannot supply and the parent does not have.

The mode was not at fault. The midpoint of any point and its own lattice translate is an inversion centre, so a one-point orbit is always more symmetric than the group that made it. This is the site’s founding finding, met again in a routine that has nothing to do with drawing patterns: a single dot verifies only eleven of the seventeen groups, and a single dot per orbit verifies no transition correctly either.

Three points in no particular arrangement remove it, and the machinery now uses a triple per orbit chosen so that the two triples share no coordinate. The disagreement disappeared and has not returned.

The lesson is worth stating plainly because it generalises past this collection: a test of a symmetry-lowering computation needs an object that is not accidentally symmetric, and the accident is easiest to make in the simplest case.

The second disagreement: a mode that was a translation #

The other case is p2’s odd order parameter at the zone centre, and it is a genuine feature rather than a defect.

For p2 the two operations are the identity and the half-turn. The odd character attaches +1 and −1; the half-turn’s linear part is −I; so the displacement at the rotated atom is (−1)·(−u₀) = u₀. Every atom moves by the same vector. The field is uniform, the structure is the parent structure moved bodily across the page, and its symmetry is untouched.

The prediction disagreed with the detector because the two ask subtly different questions. The equivariance test asks whether each operation survives where it was; a rotation about a point that has moved does not. The detector asks whether any operation with that linear part survives anywhere, and finds the rotation at its new centre.

So the machinery now names a uniform field as what it is, and reports the parent’s operations about the moved origin. One of the one hundred and eighty modes is a rigid translation, and it is the only one whose index is one.

Which subgroups are reachable, and which are not #

Each one-dimensional order parameter reaches a subgroup of index two. That is a strong restriction, and it says something about the descent graph this collection built earlier.

A crystal cannot fall to an arbitrary subgroup in one transition. It falls to one that is the kernel of an irreducible representation, and for a one-dimensional order parameter that means index two — so a descent of index four happens either in two steps, or through a two-dimensional order parameter, which is the next essay’s subject.

That is a prediction with content, and it is one of the few places where symmetry says something a material must obey rather than something it may. A single continuous transition driven by one order parameter cannot produce an arbitrary subgroup: the subgroup has to be an isotropy subgroup of an irreducible representation of the parent.

The converse — that every such subgroup is reached by some material — is not a symmetry statement at all, and nothing here claims it.

Translations are part of the group #

One thing this treatment does that a point-group treatment cannot: it keeps the translations.

The order parameters above are those of the point group, and their kernels are subgroups of the plane group containing every translation. So the cell does not change, and the descent is what the literature calls translationengleiche — the same lattice, fewer operations.

The other kind exists, and it is where the cell doubles. An order parameter that alternates from cell to cell has a kernel containing only some of the translations, and the frozen structure has a larger cell. That is a mode at a wavevector on the zone boundary, and this ladder’s third rung computes those.

Keeping the translations in view is also what makes the machinery’s arithmetic exact at the zone boundary: a wavevector that is half a reciprocal lattice vector has phases ±1 on every lattice vector, so the displacement field stays rational and the frozen structure is an exact point set like any other on this site.

The seventeen, and what each of them offers #

Running the enumeration over every plane group gives a list short enough to describe.

The groups with a point class of order two — pm, pg, cm, p2 — offer one non-trivial order parameter each. Those with 2mm offer three, since 2mm has three non-trivial one-dimensional representations, and their kernels are the three subgroups of index two. The classes 4 and 6 offer one apiece plus a two-dimensional one; 4mm and 6mm offer three plus a two-dimensional one; 3 and 3m offer, respectively, none of dimension one and one.

The last of those is worth a sentence, because a class with no non-trivial one-dimensional representation cannot lose symmetry through a one-component order parameter at all. The threefold group is that case: its three operations are cyclically permuted by conjugation, and any sign assignment consistent with the group structure is trivial. A p3 crystal has no index-three descent driven by a single number, and every transition available to it needs the two-dimensional order parameter.

Reading a mode from the picture #

The middle panel of each figure is the mode, and it can be read directly once the convention is known.

Every arrow is the displacement of one atom. Arrows on atoms related by an operation of the kernel are related by that operation — rotated or reflected, with no change of sign. Arrows on atoms related by an operation outside the kernel are related by that operation and then reversed. So the pattern of arrows carries the character: an operation is in the kernel exactly when applying it to the picture leaves the arrows alone.

That gives a reader a way to check the claim without the arithmetic. Take any mirror of the parent group, reflect the middle panel in it, and see whether the arrows come back to themselves or to their negatives. The mirrors that survive are the ones under which they come back unchanged, and the group of the right-hand panel is generated by those.

The convention that makes this readable is the one this collection uses everywhere: reflected copies are drawn in the second colour, so handedness is visible without counting. A figure that drew every arrow the same colour would hide the one thing the reader most needs to see.

The same arithmetic under other names #

This construction has been rediscovered in several fields and carries a different name in each. Naming them is useful, because the literature a reader will meet is not written in one vocabulary.

In the ferroics literature the subgroup is the isotropy subgroup and the list of them is a table of species, written parent-to-child with the index between. In spectroscopy the same representations sort the vibrational modes and are labelled by Mulliken symbols rather than by kernels. In the mathematics of group actions the kernel is the stabiliser of a point in a representation space, and the fact that it decides everything is the orbit–stabiliser theorem applied to a space of order parameters rather than to the plane.

The arithmetic is identical in all three. What differs is which part is treated as given: the physics takes the modes as measured and asks their symmetry, the crystallography takes the subgroup as given and asks which order parameter reaches it, and this collection computes both ends and checks that they meet.

What the amplitude is not #

The distortion in every figure here is drawn at a definite size, and the size means nothing.

An order parameter’s magnitude is settled by an energy — how far the atoms actually move at a given temperature — and no energy appears anywhere in this computation. What is computed is the direction the crystal moves in, in the space of possible distortions, and the group that direction leaves alone. The amplitude is a parameter chosen small enough to draw.

That is not a limitation of this treatment so much as its whole point. The symmetry of the low-symmetry phase is the same for every amplitude, however small, and it changes discontinuously at zero. That is what makes symmetry breaking a sharp statement in a subject otherwise full of continuous quantities: the group is not a matter of degree.

What the pictures cannot show #

A transition is not drawn. There is no temperature in any of these figures and no sequence in time. What is drawn is a pair of structures and a field relating them, and the claim is about the relationship rather than about a process.

The choice between phases is not drawn either. Where several order parameters are available — and every group here has at least two — nothing in the symmetry says which becomes non-zero. That is decided by which mode’s restoring force falls to zero first, which is a question about a material.

And the figures are of a plane. In three dimensions the same construction runs with the same steps, over two hundred and thirty groups rather than seventeen, and the representations of a space group at a general wavevector are objects this collection has only begun to build. What changes is the size of the enumeration; the argument does not.

Where the ladder goes #

Three rungs follow. The next asks what happens when the order parameter has two components rather than one, so that where it points decides which subgroup survives, and one representation offers more than one phase.

After that comes the zone boundary, where the cell doubles and the count of modes goes from twenty to a hundred and thirty-five. And then the consequences: the shape change a descent brings with it, the walls between the domains it creates, and the polarisation that appears in a crystal whose transition was about something else entirely.

Irreducibility is an assumption about genericity #

The second condition — that the representation be irreducible — is stated above as a requirement, and it is worth saying what kind of requirement it is, because it is not a theorem and it can fail.

Nothing forbids two independent order parameters from becoming non-zero at the same temperature. If they do, the transition is one event and the quantity measuring it is reducible, so the framework’s list of available descents does not contain the group that results.

What makes the assumption safe is counting parameters. Two quantities become non-zero together only if two coefficients in the free energy vanish at the same temperature, which is one condition too many for a transition driven by temperature alone. So it does not happen unless something is tuned — a pressure, a composition — and the point at which it does is a multicritical point rather than an ordinary transition.

That is the same codimension argument this collection makes about accidental degeneracies, and it has the same force: the exclusion is generic rather than absolute. A phase diagram with two variables in it can have such a point, and materials with them are studied precisely because the ordinary framework does not describe them.

So the enumeration here is a list of the transitions a single parameter can drive, which is what happens on a line through a phase diagram, and it is complete for that.