The halving a lattice will not permit

The operation that reverses time admits one extra operation into a point group and turns thirty-two classes into a hundred and twenty-two. This essay does the same thing to the fourteen lattices, and the answer is thirty-six — but the interesting part is not the number. It is that one of the fourteen contributes nothing to it.

A magnetic moment is a current loop, so running time backwards reverses it and moves nothing else. In a crystal whose atoms carry moments, some translations carry every moment onto a moment pointing the same way, and the rest carry every moment onto one pointing the other way. The first kind form a subgroup; the second kind are what is left over. Two cosets, so the subgroup has index two, and a black-and-white lattice is exactly a pair of lattices, one inside the other, with two cosets between them.

Seven, whatever the lattice is #

The first thing to notice is that the shape of the lattice does not enter the counting of candidates at all.

A subgroup of index two in an abelian group is the kernel of a homomorphism onto the two-element group, and a homomorphism out of a lattice is decided by where it sends a basis. Three basis vectors, two choices each, and the map that sends all three to zero is the whole lattice rather than a proper subgroup. So there are 2³ − 1 = 7 subgroups of index two, in every lattice, in every dimension three. A triclinic lattice with no symmetry beyond the inversion has seven of them and the cubic lattice with forty-eight symmetries has seven of them, and nothing about lengths or angles has been used.

The count is also independent of which basis is chosen, which is not obvious from the way it was just derived. A different basis is a different set of three vectors to send to zero or one, so the seven homomorphisms are relabelled — but they are the same seven maps, because a homomorphism is defined on the lattice and not on a description of it. That matters later, when the seven have to be sorted into types by a group of basis changes: the group is acting on a set that was there before the basis was.

That is worth pausing on, because it inverts the usual expectation. The high-symmetry lattices are not the ones with more ways to be coloured; they are the ones with more ways for a colouring to be wrong. Every operation of the lattice’s own group is another condition the colouring has to satisfy, so a large group is a large obstacle. The face-centred cubic lattice, which has the largest group any lattice in three dimensions can have, turns out to have no admissible colouring at all — and the argument for that is three lines long once the condition is stated properly.

The first filter, and what it is a filter for #

The condition is a statement about what a Bravais lattice is. A Bravais lattice is not merely a lattice; it is a lattice classified by its own symmetry, which is why there are fourteen of them and not infinitely many. Fourteen lattices and no others is the enumeration, and its whole content is that a centred cell describes a lattice of its system only when the centred lattice still has the system’s full point symmetry — why there is no C-centred cubic is the case where it does not.

Colouring the lattice cannot be allowed to break that. A black-and-white Bravais lattice has to keep the whole of its parent’s symmetry with the colours on, or it is a coloured object of some lower type and belongs to a different row of the table. So take an operation U of the lattice’s group and a halving given by the homomorphism φ. The operation sends the kernel of φ to the kernel of φ ∘ U, so it preserves the halving exactly when φ ∘ U = φ, read modulo two. The whole group preserves it when that holds for every U.

This is a linear condition over the two-element field, and it costs nothing to check: reduce each matrix of the group modulo two, act on the seven non-zero row vectors, and keep the ones nothing moves. The cubic primitive lattice is the shortest illustration. Its group has forty-eight elements, and six of the seven halvings are moved by something in it; only the one that sends every basis vector to one survives.

The bars are the point of the figure rather than the verdicts. A halving that fails is not failing narrowly: the halving that colours alternate a layers keeps sixteen of the forty-eight operations, which is to say it demotes a cubic lattice to a tetragonal one. Nothing is being rejected on a technicality. Colouring alternate layers of a cubic crystal really does pick out an axis, and picking out an axis really is the end of cubic symmetry.

The lattice that keeps none #

Run the same test on the face-centred cubic lattice and every one of the seven fails.

The reason is visible in the primitive basis. The face-centred cubic lattice’s primitive vectors are the three face-centring vectors, and the cubic operations permute them and negate them freely; a functional that treats them differently is moved by some operation, and the functional that treats them alike sends all three to one — but the sum of two primitive vectors is a third lattice vector, and the parities do not close. Every candidate is either asymmetric or inconsistent, and the two failures exhaust the seven.

It is worth saying what this does not claim. It does not claim that a face-centred cubic crystal cannot be antiferromagnetic; a great many are, and their magnetic structures are described by magnetic space groups whose translation lattices are of some other type. What it claims is narrower and sharper: the lattice itself has no two-colour form of its own type. A face-centred cubic array of moments in which the reversed and unreversed sites form a lattice of index two is necessarily an array whose symmetry is no longer cubic.

That is the same kind of statement as the holohedry is the ceiling makes about point groups, arriving from the other side. There, the lattice’s own symmetry bounds what a crystal on it can have. Here, the lattice’s own symmetry bounds what a colouring of it can be, and for one of the fourteen the bound is empty.

The second filter: the axes have no names #

Surviving the holohedry is necessary and it is not sufficient for two halvings to be different lattices, because two halvings can be carried onto each other by a change of basis that leaves the lattice type alone.

Orthorhombic P is the clean case. Its group is mmm, which is eight diagonal matrices, and modulo two every one of them is the identity — so all seven halvings survive the first filter. But three of the seven are the colourings that alternate along a, along b and along c, and nothing distinguishes those three, because nothing distinguishes the axes of an orthorhombic lattice beyond which letter somebody wrote on them. Relabelling is a change of basis, it is not an operation of the lattice’s group, and it must identify the three.

The group doing the identifying is the normaliser of the lattice type: every integer matrix that carries the type’s whole set of Gram matrices back to itself. It is strictly larger than the lattice’s own group for every type below cubic — an orthorhombic lattice’s group has eight elements and its normaliser contains the six axis permutations as well — and it is infinite for the low-symmetry types, since a monoclinic lattice can be sheared along its unique axis by any integer at all.

Two things about the normaliser are worth stating, because both look like shortcuts and neither is. The first is that it has to be the normaliser of the type and not of the particular lattice: a monoclinic lattice with some specific angle has a normaliser that preserves that angle, and what is wanted is every matrix that carries any monoclinic Gram matrix to another one, which is a condition on the whole cone rather than on one point of it. That is why the search tests a basis of the cone instead of testing the lattice in hand.

The second is that the cone itself is computed rather than read off the parameter list. The set of Gram matrices a lattice type admits is the set the type’s group fixes, so it is the solution of a linear system in six unknowns, and its dimension comes out as six for triclinic, four for monoclinic, three for orthorhombic, two for tetragonal, hexagonal and rhombohedral, and one for cubic. Those are the parameter counts of the seven systems, arriving as the rank of a matrix rather than as a convention about which cell parameters are free.

Infinite is not an obstacle here, because everything happens modulo two, where the image of the normaliser is a subgroup of a group of order 168. Searching integer matrices with entries between minus one and one for the ones that preserve the type’s Gram-matrix cone, and then closing that set modulo two, gives the image; the triclinic lattice’s image is the whole of GL(3, 𝔽₂), which acts transitively on the seven, so the triclinic lattice has exactly one black-and-white form however it is coloured.

This is the same equivalence question that seventy-four colourings, forty-six groups settles in the plane, and it has the same shape: the number of homomorphisms is not the number of designs, and the gap between them is a group acting. It is also the reason the ten ways of being mm2 exists — a normaliser is what decides when two descriptions of the same object have been counted twice, and computing it is the content of a classification rather than a step in applying one.

The count #

Apply both filters to the fourteen and the total is twenty-two.

By system the twenty-two are one triclinic, five monoclinic, eight orthorhombic, four tetragonal, one rhombohedral, one hexagonal and two cubic, and the shape of that distribution is the argument of this essay written as a table. The systems with the least symmetry contribute the most, not because they have more halvings — everything has seven — but because they impose the fewest conditions on one and identify the fewest with each other. Symmetry is subtractive here in both filters at once, which is unusual enough to be worth naming: it is rare for a group to reduce a count by making candidates fail and by making survivors coincide.

The fourteen grey lattices deserve a sentence of their own, because they are counted and it is easy to think they are a formality. A grey lattice is one in which time reversal is a symmetry on its own, without being combined with any translation — every moment reverses and the structure is unchanged, which for a magnetic structure means there is no moment to reverse. They are the paramagnetic lattices, and they are genuinely fourteen distinct objects rather than fourteen copies of one, because the underlying lattice is still what it was. Adding them to the twenty-two is not padding; it is the statement that a lattice with no colouring at all is also a magnetic lattice.

The two cubic entries and the one empty row are the whole of the cubic system’s contribution, and they are the part of the table a reader is most likely to disbelieve. Three cubic lattices, two black-and-white forms, and the missing one is the lattice most crystals actually adopt.

What the surviving translations are #

The census says how many, and it does not say what. For that the halved lattice has to be identified, and identification is harder than it looks, because the three cubic lattices share a group of order forty-eight and share a one-parameter family of Gram matrices. Nothing computed from the group alone can tell them apart.

What tells them apart is how many vectors of each length they hold. Counting lattice vectors by norm gives 6, 12, 8 for the primitive cubic lattice, 8, 6, 12 for the body-centred one and 12, 6, 24 for the face-centred one, and those three sequences are different in the first term. So the halved lattice can be named by counting its own shells.

The first of those is the classical picture and worth stating in words. Colour a primitive cubic lattice point by the parity of x + y + z. The even points are the face-centred cubic lattice — that is what “all faces centred” means, read as a parity condition rather than as a picture of a cell — and the odd points are the coset. So the black-and-white primitive cubic lattice is a face-centred cubic lattice of unreversed moments interleaved with a face-centred cubic lattice of reversed ones, which is the structure of a great many simple antiferromagnets and arrives here as arithmetic rather than as a model.

There is a reading of that result which makes the whole classification feel inevitable, and it is worth having. The primitive cubic lattice is the odd-and-even points of itself; the face-centred lattice is the even half; and halving is therefore a map that takes a cubic lattice to the sublattice of even points, which is a different cubic lattice. Iterating it would give a chain, and the chain is short because the third cubic lattice is not the even half of anything cubic. The classification is not a list of cases so much as a very short orbit.

The second is the same statement one step down. The body-centred lattice’s parity condition leaves the primitive lattice, so its black-and-white form is a primitive cubic array of one sign interleaved with the body centres of the other. And then the chain stops, because the face-centred lattice’s own halvings all break the symmetry. Each of these is a sublattice of index two in the sense that essay makes precise, and how many sublattices counts them at every index; what is new here is the demand that the parent’s symmetry survive, which no count of sublattices imposes.

One of them, drawn #

The tables are the argument and a picture is still worth having, so here is one type drawn as points rather than as counts.

What the picture makes obvious, and the tables do not, is how little of a lattice a colouring is. The two colours alternate along one direction and are constant in the plane perpendicular to it, and the whole classification above is about which directions may be chosen and which choices are the same choice. It is also, deliberately, a picture that would look identical for a different lattice with different lengths — which is the point of doing the classification on Gram matrices rather than on drawings, and the reason the same pattern described twice is a question with an answer.

What the machinery has to refuse #

An assertion that has never rejected anything proves nothing, so the claims this census would have to make if it were wrong are made deliberately and refused.

The second is the one that matters most, because it is the boundary between a subgroup and the whole group. The zero functional has the whole lattice as its kernel, which is index one and not two, and a construction that quietly admitted it would report eight halvings everywhere and a count of thirty-six that came out right for the wrong reason. The determinant of the kernel basis is checked to be two on every one of the seven, in every one of the fourteen, and that check is the only thing standing between this argument and a plausible wrong one.

Where this stops #

Twenty-two black-and-white lattices and fourteen grey ones is the lattice half of the magnetic classification, and it is not the magnetic space groups. Those are 1,651 of them, built from these thirty-six lattices and the hundred and twenty-two magnetic point groups by the same extension arithmetic that builds two hundred and thirty from thirty-two — and that count is not computed in this collection. It needs the cocycle machinery applied to each of the four types of magnetic group separately, and the four types differ in whether time reversal is combined with a translation, a point operation or nothing at all.

The boundary is worth drawing precisely, because “magnetic lattice” and “magnetic space group” are close enough in ordinary use to be run together. A magnetic Bravais lattice answers a question about translations alone: which pairs of a lattice and an index-two sublattice keep the parent’s full point symmetry. A magnetic space group answers a question about the whole of an operation, translation and point part together, and the four types in its classification are distinguished by where time reversal is attached. The first question has thirty-six answers and is settled here; the second has 1,651 and is not.

What is computed here is the piece the rest stands on, and the piece with the sharpest single result in it. Which magnetism a class permits asks what a magnetic point group allows a crystal to do; this asks what a lattice allows a magnetic structure to be, and for one of the fourteen the answer is nothing.