Which way the order parameter points

A one-component order parameter has only a size, and once it is non-zero the surviving symmetry is settled: the kernel of the sign it carries, and nothing else to decide.

A two-component one has a direction. The crystal’s symmetry operations rotate that direction, and which operations leave it alone depends on where it points — so a single representation offers several low-symmetry phases, and which of them a crystal takes is decided by something symmetry does not contain.

That is the situation for every plane class with a rotation of order three or more, since each of them has a two-dimensional representation. This essay enumerates what each offers.

The stabiliser of a direction #

For a two-dimensional order parameter, the value is a point of a plane and the parent group acts on that plane by matrices. The operations leaving a given value fixed form the isotropy subgroup of that value, and the surviving symmetry of the crystal is exactly that subgroup.

Computing it needs no theory. For each candidate direction — taken over integer vectors, so that the arithmetic is exact — check which operations fix it, collect them, and name the group they form. The naming is settled by three numbers, since in the plane the order, the reflection count and the highest rotation order determine the class with nothing left over — which is not true in space, where two classes can share all three.

Doing that over a window of directions gives the whole list of isotropy subgroups a representation offers, with a representative direction for each. For 4mm’s two-dimensional representation the answer is: m along each of four special directions, and 1 everywhere else.

Two low-symmetry phases, one representation #

That answer is the point of the essay. A single order parameter offers two different low-symmetry phases, of different symmetry and different index, and they are not alternatives at different temperatures or in different materials — they are alternatives available to the same order parameter in the same crystal.

For 4mm the two are:

• the parameter along an axis or a diagonal, leaving m — index four in the parent;
• the parameter in a general direction, leaving 1 — index eight.

Both are legitimate; both are reached by the same representation; and the symmetry contains nothing to prefer one. What decides is the quartic term of the free energy, whose coefficient can favour either, and its sign is a fact about the material.

This collection’s position on such questions is stated elsewhere and applies here without change: symmetry supplies the list and never the choice. What is new is how short the list is.

Why two orbits are needed #

A subtlety appears here that the one-component case does not have, and the machinery ran into it before it was noticed.

A two-dimensional order parameter at the zone centre is a vector, and the obvious way to realise it is to displace every atom by that vector. But displacing every atom by the same vector is a rigid translation: the crystal is unchanged and its symmetry is untouched. The first version of this computation did exactly that and reported an index of one for every group — a symmetry-lowering routine that lowered nothing.

The realisation that works needs two sets of atoms moving against each other: one orbit displaced by +v and another by −v. Then the structure genuinely changes, and its symmetry is the isotropy subgroup of v.

That is not a modelling convenience. It is why a crystal with one atom per cell has no optic mode at the zone centre and cannot undergo this kind of transition at all: the only zone-centre vector distortion available to it is a translation. A crystal needs at least two atoms per cell for a polar mode to exist, and the requirement is visible in the arithmetic rather than imposed on it.

Where the directions are #

The special directions of a two-dimensional order parameter are not arbitrary; they are the directions the group’s own reflections fix.

For 4mm the four special directions are the two axes and the two diagonals — the fixed lines of the four mirrors. For 6mm there are six, and the isotropy subgroup along each is m, but they fall into two classes of three: the mirrors of one kind and of the other, which are not related by any operation of the group. So 6mm’s order parameter offers two distinct m phases as well as the general one.

For a rotation group with no reflections — 3, 4 and 6 — there are no special directions at all: no operation other than the identity fixes any direction, so the isotropy subgroup is trivial wherever the parameter points, and the representation offers exactly one phase.

That is a clean statement worth keeping: the number of phases a two-dimensional order parameter offers is decided by the reflections of the parent, and a chiral parent offers exactly one.

The domains a choice creates #

Once the parameter has chosen a direction, the operations that were lost are not gone from the argument: they carry the chosen direction to the others, and each image is an equally good choice.

So the number of domain states — regions of the crystal that made different choices — is the index of the isotropy subgroup in the parent, which this collection has computed before for descents in general and which appears here as a property of a direction. For 4mm’s parameter along an axis, the index is four and there are four domains: the parameter pointing along +x, −x, +y, −y.

The domains of the general phase are eight, and here is the part worth noticing: the general phase has more domains and less symmetry, and both facts are the same fact. Index and symmetry trade off exactly, since their product is the order of the parent.

What the picture cannot say about the choice #

The figures draw the order-parameter plane and mark each direction by what survives. What no figure here can show is which direction a crystal takes, and it is worth being precise about why.

The free energy of a crystal near the transition is a series in the order parameter’s components, and its terms are the invariants of the representation. The quadratic invariant is the one whose coefficient changes sign; it does not depend on direction. The quartic invariants are two in number for these representations, and their combination does depend on direction — so the direction is settled at fourth order, by a coefficient this collection does not compute and cannot.

The link to the invariants ladder is exact rather than an analogy. The number of quartic invariants a representation admits is a Molien coefficient, and it is two for the two-dimensional representations of 4mm and 6mm. Two invariants means one ratio, and that ratio’s value chooses between the phases.

The Lifshitz condition, computed and empty #

There is a second condition on a continuous transition, and running it here gives a negative result worth recording.

A term in the energy that is linear in a gradient of the order parameter and antisymmetric in its components — the Lifshitz invariant — makes the uniform state unstable: the energy is lowered by letting the parameter rotate as it goes, which produces a modulated structure with a period nothing fixes. A transition whose order parameter carries one does not go to a commensurate phase at all.

The count is a character computation: the antisymmetric square of the representation, tensored with the vector representation, and the multiplicity of the identity in the result. Run over every order parameter of every plane class at the zone centre, it comes out zero in every case.

That is a real result and a limited one. It says the Lifshitz condition never bites at the zone centre in the plane, so no plane-group transition of this kind is forced to be incommensurate by that argument. It does not say incommensurate structures do not occur — they do, and this collection has essays about them — but their order parameters live at wavevectors away from the zone centre, where the little group is smaller and the count is different.

Chirality, and a phase with no symmetry at all #

The general phase of any of these representations has trivial isotropy: no operation survives at all, and the crystal’s group is p1 or the corresponding translation-only group.

That is not a degenerate outcome. A structure with no point symmetry is perfectly ordinary, and reaching one in a single transition from a highly symmetric parent is exactly what a general-direction order parameter does. What is worth noticing is that such a phase is chiral: with no reflection surviving, the structure and its mirror image are different structures, and the transition has produced handedness from a parent that had none.

The count of domains says how many of each hand: half the images of the general direction are related to the original by proper rotations and half by reflections, so the domains come in two handednesses in equal numbers. That is the arithmetic behind a familiar observation about twinned crystals — a transition through a general direction produces both hands, in equal measure, in one specimen.

The direction is not a direction in the crystal #

A warning about reading these figures, because the two planes involved look alike and are not the same plane.

The plane in the figures is the space of order-parameter values, not the crystal. A direction in it is a ratio of two amplitudes — how much of one basis mode and how much of the other — and it has no geometric meaning until the modes are named. Two crystals with the same order parameter pointing the same way can have distortions that look nothing alike, because the basis modes they carry are different.

What is geometric is the relation between the two planes: the group acts on both, and the action on the order-parameter plane is the same representation as its action on the crystal for the vector case, which is why the pictures can be read together at all. For a two-dimensional representation that is not the vector one — and in three dimensions there are many — the two planes are related by nothing but the group, and a reader who reads a direction in the order-parameter plane as a direction in the crystal will be wrong.

This collection has the same warning about a reciprocal space drawing and about a stereogram: each is a plane the group acts on, and none of them is the crystal.

Counting the phases across the plane groups #

Gathering the results gives a short table of what the plane offers.

Every class with a rotation of order three or more has exactly one two-dimensional representation. Of those, the chiral ones — 3, 4 and 6 — offer a single phase, of index equal to the order of the class. The reflection ones offer two or three: 4mm gives m and 1; 3m gives m and 1; 6mm gives two distinct m phases and 1.

Reading down that list, the number of phases is the number of orbits of directions under the group, which is a Burnside count in disguise — the same counting of what a group cannot tell apart applied to directions rather than to colourings. It is a small number in every case, and it is finite for a reason worth stating: the isotropy subgroups form a finite list because the group is finite, even though the directions form a continuum.

That the continuum of directions collapses to two or three cases is what makes the whole framework usable. A physical theory in which every direction gave a different symmetry would have no predictions in it at all.

What is checked #

Every isotropy subgroup named here is checked twice, as everywhere in this ladder.

The prediction is the stabiliser computation: which operations fix the direction, as exact integer arithmetic on a rational vector. The detection builds the distorted structure — two orbits displaced against each other by the chosen vector — and asks the site’s detector what its group is — the same round trip every pattern figure here makes. The two agree on every direction of every group tried.

There is a third check specific to this essay, and it is the one that would catch a wrong convention. The index of the descent must equal the number of distinct directions in the orbit of the chosen one, since orbit and stabiliser multiply to the order of the group. That is checked as part of the direction census, and it is the reason the figures can print an index beside each subgroup without computing it separately.

The sign that makes the choice #

The essay says symmetry produces the list of phases and says nothing about which one occurs. What decides it is one number, its form is fixed by the invariants, and setting the two together shows exactly where symmetry stops.

Write the order parameter in polar form — a magnitude and an angle in its own plane. The free energy is a series in the components, and every term must be an invariant of the group. For 4mm the invariants are generated in degrees two and four, so to fourth order the energy is

$$F = a\,\eta^2 + b\,\eta^4 + c\,\eta^4\cos 4\varphi,$$

with η the magnitude and φ the direction. The first two terms do not depend on the direction at all — they are functions of the quadratic invariant — and the whole of the choice is in the third.

So the phase is decided by the sign of c. Negative, and the energy is lowest where the cosine is one, which is along the axes. Positive, and it is lowest along the diagonals. Zero, and the two phases are degenerate to this order and the choice is made by the next term in the series.

That is the cleanest example this collection has of the division it keeps drawing. Symmetry fixes which terms may appear, exactly, by a character computation with no physics in it. One measured coefficient chooses between the phases the terms permit. And the coefficient’s sign is enough — its magnitude sets no part of the answer, which is why the prediction is a discrete choice rather than a continuum.

When a continuous transition is forbidden outright #

There is a second thing the invariants decide, it is a prohibition rather than a choice, and it is checked by the same computation.

Landau’s argument for a continuous transition needs the energy to have no third-order invariant. If a cubic term exists, then arbitrarily close to the transition the energy has a minimum at a non-zero order parameter as well as at zero — so the parameter jumps rather than growing from nothing, and the transition is first order.

Whether such a term exists is a character sum: how many invariants of degree three the representation admits. For most of the plane classes the answer is none, because a rotation of even order sends the parameter to minus itself and kills every odd-degree invariant. For the three-fold classes it is not: 3 admits two cubic invariants and 3m admits one, which are exactly the ones the syzygy essay counted while computing something else.

So a transition into a three-fold order parameter cannot be continuous, by symmetry alone, before any material is named. That is a prediction of a different logical kind from the one above — a prohibition rather than a selection — and it is the direction in which this collection’s arguments are always strongest.

Where this leaves the ladder #

One representation, more than one phase, and the choice made outside symmetry. That is the shape of every two-dimensional case, and the enumeration above is complete for the plane.

What has not yet been touched is the lattice. Every order parameter here is at the zone centre, so every frozen structure has the parent’s cell, and the descent keeps all the translations. Letting the parameter alternate from cell to cell changes that, and it changes the count of available transitions by an order of magnitude — from twenty to a hundred and thirty-five. That is the next rung, and it is where a transition starts to make a structure a diffractometer would notice.