Two glides always share a line

A screw keeps a line and a glide keeps a plane took a question about phase transitions to every space group at once. A crystal that orders at a wavevector on the edge of its Brillouin zone can lock into a commensurate structure, doubling its cell, or it can drift into a long incommensurate modulation, and which happens depends on whether the free energy holds a Lifshitz invariant: a term linear in a spatial gradient of the order parameter, $\eta_1 \partial \eta_2 - \eta_2 \partial \eta_1$ along some direction. Where such a term is allowed, the commensurate structure is unstable to a slow twist. Where a gradient term survives at the edge of the zone had set out the condition for the plane groups.

The census in space found a rule for the monoclinic groups. At a point of the zone where a screw or glide is active, meaning that its square is a lattice translation on which the wave has phase minus one, a term exists along what the operation keeps: a screw keeps its axis, a glide keeps its plane. With one active operation there was always a term, with none or two there was not. The orthorhombic groups did not obey that count. Of 104 points where two operations are active, 87 carry a term and 17 do not, and the essay proposed a rule to separate them. If both active operations keep some common direction, the term should lie along it. If they keep nothing in common, there should be no term. It asked whether that rule is necessary and sufficient, and left the cases with more operations as the place to look for exceptions.

Two active operations, settled #

The rule holds exactly at all 104 points with two active operations. At the 87 with a term, the two operations keep one common axis and the term lies along it. At the 17 without, they keep no common axis. There are no exceptions in either direction. Every order parameter at those points is two-dimensional, which matters for what follows.

The rule is checked in the strongest form available. For each point the census recomputes, from the characters of each order parameter, the directions along which a term exists, and compares them with the directions kept by both active operations. They agree as sets, not only as “some” against “none”. And it checks the shared axis against the number of active operations directly: the operations are found by their linear parts and translations, and the kept axes are read off their matrices.

Why: planes meet, lines do not #

The rule reduces to geometry once the two kinds of operation are separated. In an orthorhombic group every screw lies along one of the three axes, and every glide plane is normal to one of them. A screw along an axis keeps that axis and reverses the other two. A glide normal to an axis keeps the two axes in its plane and reverses the third.

Two glides normal to different axes therefore always keep a common axis: the third one, where their planes intersect. The census finds forty such points and all forty have a term. Two screws along different axes keep nothing in common, since each keeps only its own line; there are ten such points and none has a term. A glide and a screw share an axis exactly when the screw lies in the glide’s plane. That happens at forty-seven points, all with a term, and fails at seven, where the screw is normal to the plane, all without one.

So the 87 and the 17 are a count of planes and lines. Planes in three dimensions always meet in a line; two distinct lines through the origin meet only in a point; a line meets a plane in a line only if it lies in it. The question the earlier essay posed, which direction a second active operation leaves for a term, has the answer a child could draw with three pencils. What remains is to say why the shared axis is the right object at all, and why the picture breaks where it does.

The monoclinic count, read again #

The monoclinic census that started these questions said something that now looks less like a separate fact. In the monoclinic groups a term existed at every point with one active operation and at no point with two. A monoclinic group has a single special axis, $b$, and its screws all lie along it while its glides are all normal to it. So two active operations in a monoclinic group are always a screw along $b$ and a glide in the plane normal to $b$: a line and a plane that meet only at a point. They keep no common axis, and the shared-axis rule says there is no term. The monoclinic “none with two” is the orthorhombic glide-and-screw case in its one unlucky arrangement, with the screw normal to the plane, and the census confirms that all four two-dimensional order parameters at those monoclinic points behave so.

The orthorhombic groups simply have room for the other arrangements. With three mutually perpendicular axes, a glide and a screw can be arranged so that the screw lies in the glide’s plane, and then they do share an axis. Two glides can be normal to different axes, which the monoclinic system cannot offer. The count of 87 against 17 is the census of those arrangements over every orthorhombic group and every special point, and the rule that sorts them is the same one that made the monoclinic count come out at zero.

One determinant #

The Lifshitz invariant of a two-dimensional order parameter $(\eta_1, \eta_2)$ is the antisymmetric product $\eta_1 \partial_d \eta_2 - \eta_2 \partial_d \eta_1$ along an axis $d$. Under an operation $h$, the antisymmetric pair transforms by the determinant of the matrix $\rho(h)$ that $h$ applies to the order parameter, and the derivative transforms by the component $R(h)_{dd}$ of the operation’s rotation part. So the invariant survives exactly when the two agree for every operation: $\det \rho(h) = R(h)_{dd}$. The determinant can be read from the characters the census already has, as $\tfrac12(\chi(h)^2 - \chi(h^2))$, the same formula what a group does to a function uses for any antisymmetric square.

Now the active operations become transparent. An active operation’s square is a translation on which the wave has phase minus one, so $\rho(h)^2 = -1$ and its eigenvalues are $\pm i$. If they are $i$ and $-i$, its determinant is $+1$ and the condition says it must keep the axis. If both were $i$, the matrix would be a scalar, its determinant $-1$, and it would have to reverse the axis instead. The census looked for the second case and did not find it: all 586 actions of active operations on two-dimensional order parameters have determinant $+1$. So in two dimensions every active operation demands that the axis be kept, which is the earlier essay’s rule, and two of them demand a shared axis.

The determinant condition is an identity, not a hypothesis, and it is checked as one. Applied to every element of every group, it reproduces the counted directions of all 314 two-dimensional order parameters at the monoclinic and orthorhombic points with an active operation. What it adds is an account of which operations matter and why. An operation acting trivially on the order parameter, or as minus the identity, has determinant $+1$ and also demands the axis be kept; a pure translation keeps everything. Only an operation that swaps the two components, with eigenvalues $1$ and $-1$, makes a different demand.

The operation that takes away #

That different demand is what explains the cases with one active operation, which the shared-axis rule says nothing about.

With a single active operation, the monoclinic census found a term along everything the operation keeps, which for a glide is a whole plane. In the orthorhombic groups the same is true at 55 of 82 order parameters. At the other 27 the term lies along fewer axes than the active operation keeps, and in every one of the 27 the lost axis is kept by some operation of determinant minus one, an operation that exchanges the two components of the order parameter and so demands that the axis be reversed. It keeps the axis instead, and the term along it dies.

The example in the figure is the plainest. In the glide-mirror group $Pma2$ at the point $(0, \tfrac12, 0)$, the active operation is the glide normal to $a$, which keeps $b$ and $c$. The two-fold rotation along $c$ and the mirror normal to $b$ both act by exchanging the components. The two-fold keeps $c$, so $c$ is lost. The mirror reverses $b$, which is what an exchange demands, so $b$ survives. The term lies along $b$ alone.

So the full rule for two-dimensional order parameters has two halves. Active operations keep; exchanging operations reverse. The shared-axis rule is the first half, and it is the whole story at points with two active operations only because no exchanging operation there ever keeps the shared axis. The census confirms that last fact without needing it: at all 232 two-dimensional order parameters at points with more than one active operation, the terms lie along exactly the shared kept axes.

Where the dimension doubles #

The rule fails sixteen times, at points with three or four active operations, and every failure has the same cause. The order parameters there are four-dimensional. Fifteen of the failures are in the centrosymmetric groups $Pbcm$, $Pnma$, $Pcca$, $Pnna$, $Pbcn$, $Pccn$ and $Pbca$, at the points where two or three of the half-integer coordinates combine, and one is in $P2_12_12_1$ at the corner of its zone. At several of them the term lies along all three axes at once, although no axis is kept by every active operation.

One case shows the shape of the failure. $Pnma$, written here in the setting $Pnam$, at the point $(\tfrac12, 0, \tfrac12)$ has four active operations: a screw along $c$, which keeps $c$; a glide normal to $b$ sliding along $a$, which keeps $a$ and $c$; a screw along $a$, which keeps $a$; and a diagonal glide normal to $a$, which keeps $b$ and $c$. No axis is kept by all four, since the screw along $a$ reverses $c$ and the screw along $c$ reverses $a$. Yet the single four-dimensional order parameter there has three independent Lifshitz invariants, along $a$ and along $c$. Each screw forbids the other’s direction for a two-dimensional order parameter, and in four dimensions neither prohibition binds.

A four-dimensional order parameter has six antisymmetric products of its components, not one. They do not transform by a single determinant, and different operations can act on different products in different ways. An active operation still squares to minus one on the whole space, but one product can be kept by it while another is reversed, and a term along an axis needs only some combination of the six to survive. The two-dimensional argument, which turned one antisymmetric product into one sign per operation, has nothing to say here. The census does not replace it with a rule; it records where the old one stops.

That is consistent with where four-dimensional order parameters come from. They appear at points of the zone where the group of the wavevector forces a degeneracy greater than two, which in the orthorhombic groups happens on the faces and corners of the zone in groups with several glides and screws. A glide sticks two levels together found the same kind of forced degeneracy in electronic bands at the zone boundary. The points where the rule breaks are the points where symmetry already sticks two two-dimensional order parameters together. And $Pnma$, the commonest space group among perovskites, is one of them, which matters for anyone modelling an incommensurate phase there.

What the shared axis predicts #

A Lifshitz term along an axis tells a crystal which way to modulate. Where the term exists, the free energy is lowered by twisting the order parameter slowly along that axis, so the ordered phase that would have doubled the cell instead appears at a wavevector shifted off the zone boundary in that direction, and the structure becomes incommensurate along it. The cell a zone-boundary mode doubles described the commensurate structure that the term destroys, and which way the order parameter points described the choices within it that the twist then rotates through.

So the geometry above is also a prediction about diffraction. At a point where two glides are active, the satellites of an incommensurate phase should be displaced from the zone-boundary position along the line where the two glide planes meet, and nowhere else. At a point where a glide and a screw are active with the screw in the plane, along the screw. At a point where two screws are active, or a glide and a screw normal to it, there should be no such displacement from symmetry alone, and an incommensurate phase there would need a different mechanism, a competition between interactions rather than a Lifshitz term. That is the distinction between the two routes to incommensurability that the theory of modulated phases draws, and here it is decided by whether two planes meet.

The prediction is only as strong as its premise. A term allowed by symmetry can be small, and a crystal can lock into the commensurate structure anyway if the energy of locking wins. What symmetry decides is the direction a modulation will take if one appears by this route, and that a modulation by this route is impossible where the active operations keep nothing in common.

What the census does not say #

The figures show counts, matrices and tables. They cannot show a modulated crystal, which is what a Lifshitz term ultimately produces: a structure that would lock into a doubled cell, twisted slowly along the term’s direction into an incommensurate one. Whether a given crystal actually does that depends on the size of the term’s coefficient against the energy that favours locking, and symmetry says only whether the coefficient can be non-zero.

The convention that the counts depend on is the choice of order parameter. The census uses physically irreducible representations: complex representations paired with their conjugates where they are not real, as an order parameter is a representation sets up. Where a four-dimensional order parameter is a complex two-dimensional representation joined to its conjugate, a calculation that used complex representations would see a two-dimensional object there, and the terms would be distributed differently between it and its partner. The statement that the shared-axis rule fails only on four-dimensional order parameters is a statement in the real convention.

Who worked out the pieces #

Lifshitz gave the condition for a gradient term in 1941, and Landau’s theory of phase transitions, in which it sits, is from 1937. The analysis of Lifshitz invariants at the special points of the space groups was made systematically by Stokes and Hatch, and by Tolédano and Tolédano, in the 1980s, with tables of which representations allow them. The sorting of the orthorhombic points by the kinds of their active operations, the reduction of the shared-axis rule to the geometry of planes and lines, the determinant account of which operations keep and which reverse, and the location of every exception on a four-dimensional order parameter are what this essay adds.

What the census must refuse #

The checks are independent of each other where they can be. The shared-axis rule is tested against counted directions, not against the determinant identity, so the identity’s success does not prop up the rule. The single-active losses are explained one by one, by finding the exchanging operation that keeps the lost axis, rather than by the identity in bulk. And the claim that a shared kept axis decides every orthorhombic point is put to the census and refused, with sixteen counterexamples, all four-dimensional.

Still open: the four-dimensional terms #

The sixteen exceptions call for their own rule. A four-dimensional order parameter at these points can be written, at least where it is reducible over the complex numbers, as a pair of two-dimensional ones that an operation of the group exchanges, and then its six antisymmetric products split into the two internal ones, each governed by a determinant as above, and four cross products between the halves. The terms found along axes no active operation keeps must come from the cross products. Writing the representation in a basis adapted to the pair, and asking which cross products each operation keeps or reverses, would turn the table of sixteen into a rule of the same kind as planes and lines. The groups one hand may sit in is a reminder that $P2_12_12_1$, the one non-centrosymmetric exception, is also the commonest group for chiral molecules, where an incommensurate phase would have a handedness of its own. Which cross products survive there has not been worked out.