A glide sticks two levels together

Of the seventeen plane groups, four contain an operation whose translation part cannot be removed by any choice of origin: pg, pmg, pgg and p4g, the non-symmorphic ones. This collection has taken that fact apart from several directions — a glide is a reflection with half a translation attached, the half is intrinsic rather than a choice of origin, and it announces itself in a diffraction pattern as a systematic absence.

Here is a fourth consequence, and it is the sharpest. At the edge of the Brillouin zone perpendicular to the glide, every level is doubled, along the whole edge, for reasons that have nothing to do with the little group’s character table. Something in the machinery of the previous rung has broken, and what has broken is worth following exactly.

What the phase is, and where it hides #

An operation acts on a wave by permuting the sites and multiplying by a phase — the phase of the cell each site was carried into. That is all the translation part contributes, since a translation moves no wavevector, and the resulting matrices are monomial: one non-zero entry per row, and that entry a root of unity.

Monomial matrices multiply exactly, so a question about how they compose is a question about exponents modulo twelve. And the composition is where the phase becomes visible. Let D(g) be the operator of the group element g. Then in general

$$D(g_1)\,D(g_2) = \omega(g_1, g_2)\, D(g_1 g_2)$$

with ω a phase rather than one. A representation obeying that weaker rule is called projective, and ω is its factor system.

Why a factor is not automatically a problem #

A factor system can be an artefact of the phases somebody happened to choose. Multiplying each operator by a phase of its own — D'(g) = λ(g) D(g) — changes the factor system by a corresponding amount, and the question is whether some choice of the λ makes every factor one.

That is a finite question here. The λ are roots of unity of order dividing twelve, and the little groups involved have at most eight elements, so the search is at most a few million tries and usually a few thousand. The machinery performs it rather than appealing to a cohomology class, which is the same theorem stated abstractly.

The one that survives every rephasing #

At the edge of the zone in pgg, the search fails: there is no assignment of phases making all sixteen products come out one. The factor system is not an artefact and cannot be removed.

The reason is short enough to state in a line. pgg has two glides, one along each axis, and at the wavevector (½, 0) the Bloch operators of the two anticommute: D(a)D(b) = −D(b)D(a). Scalars commute with everything, so no one-dimensional representation can carry two anticommuting operators. Every irreducible projective representation of that little group with that factor system is therefore at least two-dimensional, and every level at that wavevector is at least a pair.

Nothing about the material enters. The anticommutation is computed from the group’s own translations and the wavevector’s own coordinates, in integers, and it holds for every operator commuting with pgg.

The fact underneath: a glide squares to a translation #

The anticommutation comes from something even shorter, and it is worth isolating because it is the whole of the phenomenon.

A glide applied twice is a lattice translation. That is what a glide is: a reflection composed with half a lattice vector along the mirror, so doing it twice gives the whole vector and nothing else. In reciprocal space a lattice translation acts on a wave by the phase of that translation, which at a general wavevector is some number of modulus one and at the edge of the zone perpendicular to it is exactly −1.

So at the zone edge the glide’s Bloch operator squares to minus the identity. An operator whose square is −1 has eigenvalues ±i; on a one-dimensional space it is one of those two numbers; and combining it with a second such operator forces the two-dimensionality above.

An aside on why the sign cannot be argued away #

A first instinct on meeting a phase that spoils a multiplication rule is that it is a bookkeeping error, or a bad choice of origin, or a convention. All three instincts are correct in most of this subject and wrong here, and it is worth saying why each fails.

It is not a choice of origin. Moving the origin changes the translation parts of every operation together, and the intrinsic part of a glide’s translation is exactly what survives that change. The half-vector along the mirror is intrinsic, so no origin removes it, and the phase it produces at the zone edge is therefore not an origin’s doing.

It is not a choice of phase for the operators. That is precisely what the rephasing search tries, exhaustively, over every assignment of twelfth roots of unity to the elements of the little group. When the search succeeds the machinery says removable and treats the representation as an ordinary one; when it fails, as at pgg’s zone edge, it says so.

It is not a choice of basis inside the space. A change of basis conjugates all the operators at once and leaves every product’s relation to its factor untouched, because the factor is a scalar. Scalars survive conjugation, which is the reason a factor system is a well-defined object at all.

What is left, after those three, is a fact about the group and the wavevector. Two operations of the crystal, at that wavevector, genuinely fail to commute in the way their group elements do, and nothing in the bookkeeping is at fault.

Where it happens, enumerated #

The census over the seventeen finds six wavevectors with a factor system that no rephasing removes. They belong to pmg, pgg and p4g — the non-symmorphic groups with more than one glide — and they are edge centres and corners of the zone.

pg, which is non-symmorphic with a single glide, does not appear on that list, and the reason is instructive. Its little group at the zone edge has order two, and the factor system of a cyclic group of order two is always removable: setting D'(g) = i D(g) makes the square come out +1. So the unitary operations of pg force nothing at all at the zone edge.

Its levels double there anyway, robustly, at every weight scheme tried. The doubling is real and is not the group’s doing, and this is the honest place to say what the machinery here does and does not settle.

The claim that there are six of them and no more is worth putting on the same footing as the rest of the arithmetic, which means asking the question of all seventeen groups at once rather than of the three that were expected to answer. Every special wavevector of every plane group, on a grid of sixths, has a little group; each little group has a factor system; and each factor system either is trivial, or yields to a rephasing, or does not. That is a hundred and twenty-three separate exhaustive searches, and the count that comes back is six.

One more thing follows from the census being complete rather than illustrative, and it is a negative worth stating. Fourteen of the seventeen plane groups have no wavevector anywhere with an irremovable factor system, and thirteen of those have no factor system at all — every product of Bloch operators comes back with a plus sign at every special wavevector. The fourteenth is pg, which carries one at ten of its wavevectors and has it removed by a rephasing at every one of them, which is exactly the case the search exists to distinguish. In the thirteen a degeneracy at the zone boundary is either forced by the little group’s ordinary characters, or forced by the operator being real, or it is a coincidence that moves when the operator does. The phenomenon this essay is about simply does not arise there, and a reader looking for it in a symmorphic crystal’s bands will not find it however carefully they look. That is the useful shape of an exhaustive answer: it says where to look and, just as usefully, where not to.

The part this collection does not decide #

At pg’s zone edge the factor system is removable, so what is left is an ordinary representation of a group of order two, whose two representations are one-dimensional and whose characters are real. Nothing unitary forces a doubling. The doubling is nevertheless there and survives every perturbation.

What accounts for it is an antiunitary symmetry: the operator is real, so complex conjugation commutes with it, and conjugation is not a motion of the plane. In the gauge that removes the factor system the two levels carry +i and −i, which conjugation exchanges, so a real operator cannot separate them. That is Herring’s case, and deciding it in general requires a sum over the antiunitary coset that this collection has not built.

So the census reports three rows as unaccounted: robust, not explained by the unitary operations, with the antiunitary candidate named and not computed. Naming a gap is not filling it, and the difference between the two is worth keeping visible — the same position the decidability essays take about an answer a bounded search did not reach.

Why the doubling is along a line rather than at a point #

This is the difference worth carrying away from the rung, because it is visible in any band diagram of a non-symmorphic crystal.

A degeneracy forced by a two-dimensional representation happens at the special wavevectors where the little group is large enough to have one, and those are isolated points. Move off the point and the little group shrinks, the representation is no longer available, and the pair separates.

A degeneracy forced by a factor system happens wherever the factor system survives, and the factor system is decided by the phase of a lattice translation — which depends on only one component of the wavevector. Every wavevector on the zone edge perpendicular to the glide has the same phase of −1, so the doubling holds along the whole edge and not at points of it.

That is why the band structure of a non-symmorphic crystal has whole surfaces of the zone boundary where its levels stick together in pairs, and why the effect disappears entirely when the same point group is realised symmorphically. The bands of pgg and pmm differ in a way that has nothing to do with their point groups, which are the same, and everything to do with a half-translation neither of them can see in reciprocal space directly.

The same phenomenon, three fields apart #

It is worth noticing how many of this collection’s arguments are the same half-translation seen from different sides, because the non-symmorphic groups are where crystallography’s arithmetic is least visible in the geometry.

In direct space the half-translation is what makes a glide a glide, and it is intrinsic: a quantity that survives every change of origin, which is what that phrase is defined to mean.

In a diffraction pattern it is a systematic absence — a whole row of reflections extinguished, because the half-translation makes the contributions of two halves of the cell cancel exactly for those indices.

In reciprocal space, on the levels, it is this: a factor system at the boundary, forcing levels into pairs along a whole edge.

Three quite different-looking phenomena, all of them the same ½ in the same place. Whether that counts as one fact or three is a matter of taste; what is not a matter of taste is that a reader who has met all three can predict any of them from any other, which is the kind of unification worth building a collection around.

The bookkeeping this replaces #

A reader who has met this material elsewhere will have met it as loaded representations or ray representations of the space group, or as the representations of a central extension of the little co-group. Those are three names for the same repair: a projective representation of a group is an ordinary representation of a larger group built from it and the factor system.

The construction is standard and this collection does not build it, for the reason it does not build many standard things — the question here was whether a level is forced to be a pair, and the anticommutation answers it in a line. What the larger machinery buys is the classification of the projective representations, which matters when there is more than one and the levels have labels that need distinguishing.

It is worth saying what the levels above are levels of, because the argument would be empty if the operator had been chosen to produce the answer. The model is the least committal one available: an orbit of four sites under pgg, generated from a single point in a general position, with a bond between each pair of nearest neighbours and one orbital on each site. Nothing about it was tuned. The group is then recovered from the point set before anything is built on it — a motif that landed on a special position would give an orbit with more symmetry than the group it was asked for, and a model built on that orbit would exhibit degeneracies belonging to a group nobody named, which is a way of proving a theorem about pgg using pmm by accident.

The two glides are what relate the four sites in pairs, and everything above is about what those two operations do to a wave rather than to the sites. The sites are permuted; the wave picks up a phase for each cell a site is carried into; and the whole of the argument lives in those phases. So the four-site orbit is not an example the conclusion depends on. It is the smallest thing the conclusion can be tested on, and the test is that an operator with no reason to cooperate produces exactly the pairing the exponent arithmetic predicts.

What is exact and what is measured #

Everything on the left-hand side of this argument is exact. The wavevector is a pair of rationals; the little group is found by integer arithmetic; the Bloch operators are permutations with exponents attached; their products are exponent sums modulo twelve; the rephasing search is a finite enumeration over roots of unity.

Everything on the right-hand side is a measurement. The levels are eigenvalues of a matrix, computed numerically, and a pair is called a pair when two of them agree to within a stated gap.

The two sides are compared as patterns, and that comparison is what a phase-system argument is worth: an exact statement that certain levels cannot be separated, tested against an operator that had no reason to cooperate. When the two agree, as they do at every one of the six wavevectors above, the exactness has been earned rather than assumed.

Where the factor systems live #

The rephasing search asks, for one wavevector in one group, whether a particular factor system can be argued away. The answer belongs to a classification, and naming it says why the search is finite and what it is searching.

Two factor systems that differ by a rephasing are equivalent, and the classes form a group under multiplication — the Schur multiplier of the little co-group, which is its second cohomology with coefficients in the circle. So the question is this factor system removable is the question is its class the identity, and the number of genuinely different answers available is the order of that group.

That is the same machinery as the extension arithmetic that produces the seventeen, one level along: there the cocycles took values in the lattice and the classes were plane groups; here they take values in phases and the classes are the ways a set of operators can fail to represent a group. Both are second cohomology, both are annihilated by the order of the group, and that annihilation is exactly why the rephasing search over twelfth roots of unity is exhaustive rather than a sample — a class killed by the group’s order needs no finer phase than that.

The finiteness is therefore a theorem rather than a bound chosen for the computation. A search over a continuum of phases would prove nothing; a search over the roots of unity the cohomology can hold proves that no phase whatever removes the factor, which is the statement the essay needs.

What a whole surface of stuck levels forbids #

The last section’s observation — that the doubling runs along a line rather than sitting at a point — has a consequence for counting that is worth extracting, because it is what makes non-symmorphic groups matter outside the band diagram.

If the levels stick in pairs across an entire face of the zone boundary, then no gap can open between the two members of a pair anywhere on that face. A gap in the spectrum has to separate whole levels from whole levels at every wavevector, so a gap can only fall where the pairing permits — after an even number of levels, never after an odd one.

That is a statement about how many states a system may hold before a gap is available, and it is fixed by the group before any model is written down. A symmorphic group imposes no such condition: its degeneracies sit at isolated points, and a gap can open everywhere else and simply avoid them. A non-symmorphic one has a surface of degeneracy that cannot be avoided, and the counting constraint follows.

It is the same half-translation once more. In direct space it makes a glide intrinsic; in a diffraction pattern it extinguishes a row of reflections; on the levels it sticks pairs together along a surface; and now it forbids a gap at an odd count. Four consequences, one ½, and none of them derived from the others.

Where this ladder goes #

The last rung is the smallest case in the subject and the one with the shortest proof: the honeycomb’s two levels at the corner of its zone, where the degeneracy is an identity between three cube roots of unity and the gap that opens when the two sites differ is exactly their difference.

And the machinery carries on into the applied field, where the same wavevector-by-wavevector analysis is applied not to levels but to the rank of a rigidity matrix, and a mechanism turns out to be a wave that a finite cell can only see if the cell happens to be the right size.

The two additions above are the same observation from opposite ends. The cohomological framing says the obstruction is a class rather than an accident of how the operators were written, which is what makes the search’s negative answer trustworthy. The counting constraint says what that class costs a material, which is what makes it worth searching for. Neither is a statement about any particular crystal, and both are decided by the group before a model exists.