The operation that reverses time

Every symmetry on this site so far has been a motion. Something moves the points of a pattern and leaves the pattern looking as it did, and the operation is described by a matrix and a translation.

Magnetism needs one more operation and it moves nothing at all. A magnetic moment is a circulating current, and running time backwards runs the current the other way — so time reversal, written 1′, reverses every magnetic moment and leaves every atom exactly where it was. It is a symmetry operation in the ordinary sense: it maps a magnetic structure onto something, and the question is whether that something is the structure it started from.

Admitting it multiplies the classification. The thirty-two crystal classes become one hundred and twenty-two magnetic point groups, the arithmetic that produces them is the arithmetic of a two-coloured pattern, and one of the steps needs an operation the crystallographic restriction forbids.

The three kinds, and what each describes #

The classification is not a technicality — each of the three kinds is a physically distinct situation, and the distinction can be read off the symbol.

An ordinary group has no 1′ in it and is one of the thirty-two unchanged. Every operation is a plain motion, and a structure with this symmetry is magnetically ordered in a way that happens to need no time reversal to describe.

A grey group contains 1′ by itself, so it has twice as many operations as the class it came from. It is the symmetry of something with no magnetic order: an operation that reverses every moment while moving nothing can only be a symmetry if every moment is zero. Every paramagnet and every diamagnet has a grey group above its ordering temperature, and the word “grey” is Shubnikov’s — a colour that is both black and white at once.

A black-and-white group contains primed operations but not 1′ alone, and so has exactly as many operations as its parent class. Half of its operations preserve the moments and half reverse them, and the split is exactly an index-two subgroup with its complement. This is the interesting case, and it is what an antiferromagnet has.

The count is a subgroup count #

The enumeration is the one this site already had, in a different subject.

A black-and-white group is a class G together with a subgroup H of index two: the elements of H act plainly, and the elements of G outside H act with time reversal attached. So counting black-and-white groups over a class is counting its index-two subgroups, and index-two subgroups are the kernels of homomorphisms onto {±1}.

That is precisely the calculation behind two-colour symmetry, with “moment reversed” in place of “colour swapped”. The two subjects share nothing physically and share their arithmetic entirely — which is the observation this site’s applied field is built on, arriving again in a field where it was not expected.

The subgroups are found by closure inside each class: assign a sign to each element of a generating set, propagate, and keep the assignments that never give one element two signs. Each surviving kernel is checked to be closed and of half the order, and no table of magnetic groups is consulted anywhere.

Sixty-six, and the eight that must be merged #

Running that search over the thirty-two classes and reducing up to conjugacy in integers gives sixty-six black-and-white groups. The literature records fifty-eight.

The difference is eight pairs, and every one of them is a pair of subgroups that differ only in which of two equivalent directions is primed. In the class 422 there are two kinds of two-fold axis in the plane — along ⟨100⟩ and along ⟨110⟩ — and priming one kind or the other gives two descriptions of one magnetic group. The operation that carries one description onto the other is a rotation through forty-five degrees.

Forty-five degrees is an eight-fold operation. No lattice admits one, so no integer matrix in a lattice basis expresses it, and the reduction that merges those two entries cannot be performed in the arithmetic this site works in.

This is the third time this site has met that exact obstruction, and the pattern is worth naming. The ferroic species count came to 247 where the literature says 212, and the difference was the same 45° rotation normalising 4/mmm. The two-colour groups came to 74 where the classical count is 46, and the difference was an equivalence under change of basis.

Every time, the shape is identical: integer arithmetic counts objects, and the literature counts them up to an equivalence whose operations are not integer matrices. Both numbers are right about different questions, and the honest report gives both and names the operation in between — which is what the figure above does, angle by angle.

Reading a magnetic symbol #

The symbols are the ordinary Hermann–Mauguin ones with primes added, and the primes are placed by the same walk that places the letters.

Take each symmetry direction in turn; find the highest-ranking axis on it and the plane across it, exactly as the ordinary derivation does; and mark each with a prime if the operation found is one of the reversing ones. So 2/m has three magnetic descendants — 2′/m, 2/m′ and 2′/m′ — and each is read off its own operation set rather than looked up.

Two conventions are worth stating because they are not obvious. A grey group is written with 1′ appended, since 1′ is exactly what it has that the ordinary group does not. And an axis is called primed when the operations of its highest type are primed: a four-fold whose quarter-turns reverse the moments is 4′, and its square is then an ordinary two-fold, which is why 4′ is a sensible symbol rather than a contradiction.

Why grey groups are the ones with no magnetism #

The grey case deserves its own paragraph because it is the clearest thing time reversal does.

In a grey group every operation occurs twice: once plain, once primed. So for any quantity that reverses under time reversal — a magnetic moment, most obviously — the two occurrences contribute with opposite signs and cancel exactly. Averaged over the group, such a quantity is zero, and it is zero for every grey group, whatever the class.

That is the arithmetic form of a physical statement: a material whose symmetry includes time reversal on its own has no magnetic order. Above its ordering temperature every magnetic material is in that state, so the grey groups are the symmetry of ordinary crystallography — which never had to mention any of this, because a property that does not reverse under time reversal cannot tell a grey group from an ordinary one.

The consequence for measurement is direct. X-rays scatter from charge, which is time-even, so an X-ray experiment sees the grey group whatever the moments are doing; neutrons scatter from moments as well as from nuclei, so a neutron experiment sees the black-and-white group. The magnetic structure of a solid was therefore unavailable until neutron scattering existed, which is 1949 and after.

Shubnikov, Heesch, and the ornament that got there first #

The mathematics is older than the physics that needed it, which is the usual order on this site.

Heinrich Heesch introduced antisymmetry in 1929 — a group with an extra operation of order two attached — as an abstract extension of the classification, with no application in mind. Alexei Shubnikov developed it through the 1940s and 1950s and gave it a name and a book, and the groups have been called Shubnikov groups ever since. Both were thinking about coloured ornament: two-coloured patterns, in which the extra operation swaps black for white.

The magnetic reading came from Landau and Lifshitz and from Tavger and Zaitsev around 1956, once neutron diffraction had made magnetic structures measurable. What they contributed was the identification: the abstract operation of order two that Heesch attached to a group is time reversal, and the two-coloured groups are the magnetic ones.

So a classification built for wallpaper turned out to be the classification of antiferromagnets, unchanged, twenty-five years later. The counts had been published; the symbols had been printed; nobody had to derive anything.

That is the fourth instance of this site’s standing observation and the one with the shortest gap between the two subjects. A crystal form is an orbit, a twin law is a coset, a domain state is a coset, a layer’s two faces are a homomorphism onto ±1 — and now a reversal of time is the same homomorphism again, with the sign meaning something entirely different and the arithmetic meaning exactly the same thing.

What a magnetic group is a statement about #

It is worth being precise about the object, because “the symmetry of a magnetic crystal” is ambiguous in a way that matters.

A magnetic point group describes the symmetry of the arrangement of moments, taken as axial vectors sitting at the atoms. It is not a statement about the atoms alone — that is the ordinary point group, and it is generally larger. And it is not a statement about the electronic structure in any deeper sense; the moments are treated as classical arrows, which is exactly enough to decide which macroscopic properties are permitted and not enough to compute any of them.

The relation between the two groups is the useful one. The magnetic group is a subgroup of the grey group of the ordinary structure — the moments can only break symmetry, never create it — and the index says how many magnetic domains the ordering produces. That is the coset count again, in a field where the parent group has time reversal in it and the child does not.

So an antiferromagnet with magnetic group m′m′m in a crystal whose grey group is mmm1′ has index two, two domains, and a domain wall across which every moment reverses. Nothing in that sentence is a measurement, and all of it follows from the two groups.

Why the answer has exactly three kinds in it #

The three kinds are usually presented as a taxonomy — here are ordinary groups, here are grey ones, here are black-and-white ones — and they are better derived, because the derivation says why there is no fourth.

Time reversal acts on a degree of freedom no spatial operation touches. It reverses a moment and moves nothing, so it commutes with every rotation and every mirror, and the group generated by a crystal class G together with 1′ is the direct product G × {1, 1′}. Every magnetic point group is a subgroup of that product, and the physically meaningful ones are the subgroups that use every spatial operation of G exactly once — anything less is a description of a smaller class.

There are exactly three ways to do that. Take no primed operation at all, and the result is G: an ordinary group. Take every operation both ways, and the result is the whole product: a grey group. Or prime some of the operations and not others, which is possible precisely when there is a rule for deciding which — and the rule has to be a homomorphism from G onto the two-element group, because priming twice must unprime.

That last sentence is the whole enumeration. A black-and-white group is a surjective homomorphism G → {±1}, its kernel is the unprimed half, and the number of them for a given class is 2ʳ − 1, where r counts the independent such homomorphisms the class admits. Summed over the thirty-two classes it gives sixty-six, and the merges above reduce it to fifty-eight.

It also says which classes have no black-and-white descendant at all. A class of odd order admits no homomorphism onto a group of order two, since the image would be a quotient of odd order containing an element of order two. The classes 1 and 3 are the only ones of odd order among the thirty-two, so those two have an ordinary group and a grey group and nothing else — and the reason is a divisibility argument rather than anything about magnetism.

What the primes forbid, quantity by quantity #

The classification earns its keep by deciding, for each measurable property, whether a crystal in a given group may have it. The rule is one line and the interesting cases follow from it.

Every physical quantity is either even or odd under time reversal. Electric polarisation is even — it is a separation of charge, and running time backwards does not move charge. Magnetisation is odd — it is a current loop, and reversing time reverses the current. A quantity may be non-zero only if it is invariant under every operation of the group, primes included, so an odd quantity is killed by any operation whose prime it does not survive.

That immediately gives the grey-group result: a grey group contains 1′, which negates every odd quantity while fixing every point, so no magnetisation of any kind is permitted. It also gives the converse — a group with no 1′ may permit magnetisation, and whether it does depends on the ordinary invariance of an axial vector under the unprimed part.

The case worth naming is the one that needs both halves. A magnetoelectric crystal is one in which a magnetic field induces an electric polarisation and an electric field induces a magnetisation, and the coefficient relating them is a tensor that is odd under time reversal and odd under inversion. So it is forbidden in any grey group, and forbidden in any group containing an unprimed inversion — and permitted in a group whose inversion is primed. Chromium sesquioxide, in the group 3̅′m′, is where the effect was first predicted and then found, and the prediction was made by exactly this argument before any measurement existed.

Where the exactness stops #

Three limits.

These are point groups, not space groups. The magnetic space groups number 1,651 in the standard enumeration — 230 ordinary, 230 grey, 674 black-and-white with the same point group and 517 with an enlarged cell — and this site does not derive them, for the same reason it does not derive the 230.

Time reversal is being used as a symmetry of a static arrangement. The operation as physics is deeper than that, and its full statement involves the antiunitary structure of quantum mechanics rather than a sign on a moment. What is used here is the consequence for a magnetic structure, which is the sign, and the essays that need more say so.

A group says what is permitted, never what occurs. The hundred and twenty-two are the symmetries a magnetic arrangement may have; which one a given material takes is decided by exchange interactions and anisotropies that no symmetry argument supplies, and this site computes none of them. That is the standing line the point-groups field draws under every one of its counts, and it applies here word for word.

And the merge count is a claim about an equivalence rather than about groups. Sixty-six and fifty-eight are both correct answers to questions that differ by whether a 45° rotation is allowed to relabel a class. The literature’s question is the standard one and this site’s arithmetic cannot ask it directly, which is stated here rather than papered over — and is why the merge figure names the angle for every one of the eight.

Where the ladder goes next #

A magnetic group, like an ordinary one, decides which physical properties a crystal may have — and the character sum that counts them needs only one change: a primed operation’s contribution is multiplied by the sign the property takes under time reversal. Doing that gives the classes that may be ferromagnetic and the classes that may show the magnetoelectric effect, and both counts come out of the same average that counted elastic constants. That is the next rung.