Which levels join which, on the way out of a point

Where two levels must meet settles what happens at a symmetry point. The operations that leave a wavevector alone form its little group, the states at that wavevector carry a representation of it, and a two-dimensional irreducible representation means two levels that no interaction can separate. The degeneracy is a property of the point.

That leaves a band structure looking like a set of unconnected facts. Here are the levels at Γ, here are the levels at M, here are the levels at X — and nothing whatever about which of the levels at one point becomes which of the levels at another. The lines in every band diagram ever drawn are exactly that missing information, and they are not drawn by interpolation. They are decided by symmetry, and the argument is short.

Move off a symmetry point along a line, and the little group shrinks. Fewer operations fix a general point of a line than fix its end. A representation of the larger group is still a representation of the smaller one — the same matrices, acting on the same space — but with fewer operations available to distinguish vectors, a space that could not be split may now split. It subduces to a sum of the smaller group’s irreducibles, and that sum is the list of levels the degeneracy becomes.

Restriction is the whole of it #

The arithmetic is one line and it is worth writing out because it looks as though something must be missing.

Take an irreducible representation of the point’s little group. Its character is a number for each element of that group. Keep only the values on the elements that survive along the line — that is a character of the line’s little group — and decompose it there by the usual orthogonality sum. The multiplicities that come out are the compatibility row.

Nothing has been done to the representation. The matrices are the same matrices. What changed is the question asked of them: is there a subspace invariant under all of these operations, where “these” is now a shorter list. A subspace can be invariant under a subgroup and not under the group, so a representation irreducible for one can be reducible for the other, and the decomposition is a fact about which operations were dropped.

Two things are checked rather than assumed. A subduction conserves dimension: the parts must sum to what came in, since nothing was added or removed. And the line’s little group must actually be a subgroup of the point’s — restricting a character to a set that is not one evaluates it where the representation is not defined, which is an error that produces numbers rather than a complaint.

The check the model supplies #

A derivation from characters is a derivation, and this collection’s habit is to make claims that can be refused. So the same splitting is measured a second time, by machinery that shares nothing with the first.

The site’s tight-binding model on each plane group produces a Hamiltonian at any wavevector and its eigenvalues at that wavevector. The degeneracies at a symmetry point are counted from those eigenvalues; the degeneracies at a point just off it, along a chosen direction, are counted the same way. The subduction predicts how the first multiset becomes the second, and it has no access to the model — it is character arithmetic over a finite group of matrices. The model has no access to the characters — it is a count of repeated eigenvalues.

They agree everywhere, over thirty-one splittings in six groups.

One detail of how that check is framed is not a technicality. Two of the six groups surveyed — pmm and cmm — have no degeneracy anywhere to split: every irreducible of every little group in them is one-dimensional. A per-group assertion that “something splits” would fail on those two while they are entirely correct, so the assertion is made over the survey as a whole: everything agrees, and something splits, which is what stops the agreement being the agreement of two empty lists.

What the table says that a count of dimensions does not #

The interesting content is not in the sizes. Along a line whose little group is a single mirror there are only two irreducibles, one even and one odd, so a two-dimensional representation splits into one of each and there was never anything else it could do.

The content is in which label each level acquires, and the fact that the answer depends on the direction.

Look at p4m at Γ. The little group is 4mm, of order eight, with four one-dimensional representations and one two-dimensional one. Along the axis [10] the surviving operation is the mirror in the axis; along the diagonal [11] it is the diagonal mirror. Those are different mirrors, and a level can be even under one and odd under the other.

The second row of the table is exactly that: it becomes even along the axis and odd along the diagonal. The fourth row does the reverse.

That is the whole reason a compatibility table is a table rather than a list. Two bands with the same label along a line may not cross — they interact, and the interaction pushes them apart, which is the non-crossing rule applied along a line rather than at a point. Two bands with different labels may cross freely, because there is no matrix element between states of different symmetry. So which crossings are permitted is different in the two directions out of one point, and a band structure drawn without the labels can show a crossing that symmetry forbids and nobody would notice.

A line with no symmetry at all #

One case in the survey is worth pulling out because it looks like a failure and is the argument working.

In p31m at Γ the little group is 3m, of order six. Along the diagonal the surviving operation is a mirror, and the two-dimensional representation splits into an even level and an odd one, as expected. Along the axis the little group is trivial — the identity and nothing else — because no mirror of 3m contains that direction.

A trivial group has exactly one irreducible representation, of dimension one, so the subduction says the two-dimensional level becomes two levels labelled the same as everything else — which is to say, labelled nothing. That is the correct answer and it has a consequence: along a line with no symmetry, no crossing is forbidden. Two bands may cross freely, because there is no quantum number to distinguish them and therefore no reason for the matrix element between them to vanish.

The contrast within one group and one point is the clearest statement of what a compatibility relation is for. The same pair of levels, leaving the same point, is constrained along one line and unconstrained along another, and the difference is entirely which operations happen to fix which direction.

How many lines there are #

A point has more directions leaving it than a diagram shows, and the reason the diagram gets away with it is the same reason a band structure is drawn along a path rather than over the whole zone.

The little group at the point acts on the directions leaving it, and two directions in one orbit of that action give the same table: an operation carrying one line to the other carries the whole computation with it. So the number of genuinely distinct lines out of a point is the number of orbits of directions, and it is small — two out of Γ in p4m, and one out of a general point.

That is the same accounting as the star of a wavevector, applied one level down: there, the group acts on wavevectors and the distinct ones are the orbits; here, the little group acts on the directions leaving one wavevector and the distinct lines are the orbits. The path drawn in every band diagram is a set of orbit representatives, and its completeness is a group-theoretic claim rather than a convention.

Why the prediction is a multiset and not an ordering #

The subduction says a two-dimensional level becomes one even level and one odd one. It does not say which goes up.

That is not a gap to be filled by a longer calculation; it is the boundary of what a symmetry argument can reach. Which band rises and which falls is decided by the sizes of the interactions, and those are numbers about the material — hopping integrals, orbital overlaps, whatever the physics of the case supplies. Symmetry decides which kinds of level there are and which may interact; the ordering is a measurement.

This is the same boundary Neumann’s principle draws for properties. A class permits a property or forbids it; the size of a permitted one is not a symmetry question. Here the class permits a splitting into an even and an odd level and forbids anything else; the size and the sign of the splitting are not symmetry questions.

So the check above compares multisets, deliberately. A check that compared ordered lists would be asserting something the argument does not claim, and it would fail on a model where the interactions happened to run the other way — which would be a correct model failing a wrong test.

Where the step has to be small, and what checks it #

Moving off a symmetry point means picking a nearby wavevector, and “nearby” is the one place this could go quietly wrong.

The wavevectors here are exact — a pair of fractions, and a little group decided by equality rather than by nearness — so there is no tolerance. But a step of the wrong size can land on another special point, whose little group is larger than the line’s, and the compatibility table computed from it would be a table about that point instead.

The guard is to take two different steps along the direction and require the little group to be the same at both. A direction along which they differ is a direction along which the step landed somewhere special, and the case is dropped rather than reported. That is cheap, it is exact, and it is the kind of check that only exists because somebody asked what would happen if the step were unlucky.

Where the exactness stops #

Computed here: for six plane groups, every special wavevector with a little group larger than the identity; for each, the directions leaving it whose little group is stable over two steps; the character tables of both little groups, generated rather than tabulated; the subduction of every irreducible; and the degeneracies of a tight-binding model at both wavevectors.

Symmorphic groups only, and the reason is a real one. At a zone boundary of a non-symmorphic group the operators of the little group multiply with a factor system that no rephasing removes — that is what a glide sticks two levels together is about — and the representations are projective rather than ordinary. Subduction still works there; it needs projective character tables, which this site does not build. Every group used here has its factor system checked as trivial before a character is computed, rather than assumed, so a non-symmorphic case is refused instead of being answered wrongly.

And the model is a model. One orbital per site with a chosen set of hoppings is not a material, and its bands are not anybody’s bands. What it is for is to supply degeneracies that the character argument did not know about, and for that it only has to be a legitimate Hamiltonian with the right symmetry — which is a much weaker requirement than being right about a substance.

Two dimensions, again. Everything is a plane group, so the little groups are plane point groups, of which the largest here is of order twelve. In three dimensions the same argument runs with the same words and the little groups reach order forty-eight, the special points multiply, and the tables in the standard references run to pages. Nothing in the method changes and every count does.

Who put the lines in #

The compatibility relations are Bouckaert, Smoluchowski and Wigner’s, from 1936, in the paper that gave the special points of the cubic zone the letters Γ, X, L, W that everyone still uses. Their table for the cubic groups is the ancestor of every band diagram since, and it is exactly the subduction above, computed by hand from character tables computed by hand.

The reason it mattered so quickly is that band structures in the 1930s were computed at a handful of points and interpolated. Knowing which levels may connect turned a scatter of numbers into a set of curves, and knowing which crossings were forbidden turned some apparent crossings into avoided ones — which is the difference between a metal and an insulator in more than one famous case.

The modern form of the same table is a database, and the modern use of it is inverted: instead of drawing bands from computed points, one reads the symmetry labels off a computed band structure and asks whether the set of labels is one that any atomic arrangement could produce. The answer is sometimes no, and that is what a topological insulator is — a band structure whose compatibility relations are satisfied and whose labels are not the labels of any local orbital picture.

Where the ladder goes next #

Back, to what happens at the point rather than on the way out of it: where two levels must meet, and the two-dimensional representation that forces the degeneracy this essay splits.

Sideways, to the case this method refuses: a glide sticks two levels together, where the factor system survives every rephasing and the representations stop being ordinary ones.

And back further, to the object every one of these tables is about: the star of a wavevector, which is where the little group comes from in the first place, and what a group does to a function, which is where a character comes from.

Two levels with the same label #

The table says which labels appear along each line and it does not always say which level goes where, and the gap is worth naming because it is where symmetry hands over to the material.

Suppose two levels along a line carry the same label. They transform identically under every operation that survives, so nothing in the group distinguishes them, and no character computation can say which of them connects to which level at the far end of the line.

What the group does say is that they may not cross. Two levels of the same label are coupled by the Hamiltonian — the matrix element between them is not forbidden, because it is an invariant — so as a parameter is varied the two eigenvalues repel rather than meet. Bringing them together would require tuning that matrix element to zero, and there is only one parameter along the line, so it does not happen.

Two levels of different labels cross freely. Their coupling is zero by symmetry at every point of the line, so the two eigenvalues pass through each other with nothing to stop them, and the crossing is a genuine degeneracy at a wavevector that is not a symmetry point.

So the table decides which crossings are permitted and the material decides whether they occur. A pair of bands with different labels along a line may cross; whether they do depends on the ordering of the levels at the two ends, which is a question about hopping integrals and orbital energies. The compatibility relation is a permission, in exactly the sense a selection rule is one.

Solving the relations for a whole band structure #

The relations here are computed one point and one direction at a time, and there is a use for them that only appears when all of them are imposed at once.

Take every symmetry point, every line joining them, and every relation. The result is a system of constraints: at each end of each line, the labels of the levels are fixed by the little group, and along the line they must match up according to the subduction. Ask which assignments of bands to labels satisfy all of the constraints simultaneously and the question becomes combinatorial rather than representation-theoretic.

The system has finitely many solutions, and each is a way a set of bands could be connected across the whole zone. Some solutions have the set joined into one connected group; others have it splitting into two groups separated by a gap everywhere.

That distinction has become the tool it sounds like. A set of bands generated by placing orbitals on the atoms of a crystal — the bands an ordinary chemical picture predicts — corresponds to a particular solution of the constraints. If the only solutions available to that set are connected ones, then any material realising it has no gap there. If the set can also be realised in a disconnected way, then a gapped material exists whose occupied bands are not the ones the atomic picture supplies, and such a material has electronic properties an atomic picture does not describe.

Which is a great deal to get from a character table. The whole calculation uses only the little groups, their representations and the subductions of this essay — no Hamiltonian, no material, no numbers — and it returns a statement about which band structures a given space group can and cannot support. The compatibility relations stopped being bookkeeping between two rows of a table and became a constraint satisfaction problem whose solutions are classified.