Where two levels must meet

A symmetric operator on a crystal decomposes by wavevector, because the translations are an abelian group and their irreducible representations are the phase factors. That much is bookkeeping. What is left after the decomposition is a finite problem at each wavevector — a small matrix — and the group that acts on it is not the whole plane group but the little group of that wavevector.

The consequence is immediate and is the reason band diagrams are drawn the way they are. At a generic wavevector the little group is trivial, so nothing is forced and every level is alone. At the special points a whole point group survives, and if that group has a two-dimensional irreducible representation, two of the levels there have to be equal.

The claim, stated exactly #

At a wavevector k, the space the operator acts on carries a representation of the little group. That representation decomposes into irreducibles, and an operator commuting with the group takes a single value on each irreducible piece — Schur’s lemma again, arriving in reciprocal space.

So the levels at k come in multiplicities equal to the dimensions of the irreducible representations appearing there, with as many levels of each dimension as that representation’s multiplicity. A two-dimensional representation is two states that no symmetric operator separates.

The claim is therefore a pattern rather than a set of numbers: not where the levels are, which depends on everything, but how many of them there are and how many states each holds, which depends only on the group. That is the thing to compare against a computation, and it is what these figures compare.

How the prediction is made #

Three steps, all exact.

The little group. Which operations fix the wavevector modulo the reciprocal lattice, computed by integer arithmetic, as the previous rung describes.

The Bloch representation. Each of those operations acts on the finite space at k as a matrix, and the matrix is monomial — one non-zero entry per row, and that entry a root of unity — because the operation permutes the sites and multiplies each by the phase of the cell it was carried into. Monomial matrices multiply exactly, so this is arithmetic with exponents rather than numerics.

The decomposition. The character of that representation is the trace, which is the sum of the phases on the sites left where they are; decomposing it against the little group’s character table gives the multiplicities. Every step is an integer computation in the twelfth cyclotomic ring.

How the measurement is made #

The instrument is the least committal operator that commutes with the group, and it is worth being explicit about how uncommitted it is.

The sites are the orbit of a generic point, and the group is rediscovered from the point set before anything is built on it — a motif on a mirror gives an orbit with symmetry nobody asked for, which is the collection’s oldest warning about drawing a comma rather than a dot arriving where it was not expected. The bonds are every pair at the shortest separations, with separations measured exactly in the lattice’s own metric, so the bond set is invariant by construction.

Then the matrix at each wavevector is assembled, its eigenvalues are computed numerically, and levels closer together than a stated gap are counted as one. That last step is the only tolerance in the ladder and the figures carry it, because a multiplicity read off a numerical spectrum is a claim about a gap.

The two routes, and what agreement is worth #

The prediction knows nothing about the operator except which group it commutes with. The measurement knows nothing about representation theory. They agree on fifty-three of the sixty-eight rows of the census, and the seventeen that differ are each explained by one of three named effects.

Agreement of two computations sharing no code is worth more than either alone, and it is the habit this whole collection runs on: a pattern is generated from its group and the group is then rediscovered from the drawing. Here the degeneracy is predicted from the group and then rediscovered from a spectrum.

The disagreements are the interesting part, and each is a rung of its own. Six of them are the sign a glide leaves at the zone edge; five are the operator being real, which pairs a complex representation with its conjugate; one is a coincidence that moves when the weights move; and three are robust doublings this collection’s machinery reports without deciding.

What a doubled level is, concretely #

It is worth resisting the abstraction for a paragraph and saying what the two states of a pair actually are.

At the centre of the zone in a group with a three-fold axis, take the six sites of an orbit and the six numbers giving a wave’s amplitude on them. The combinations that transform as the two-dimensional representation are those in which the amplitudes go round the orbit with a phase advancing by a third of a turn at each step — one combination advancing one way, one the other. A three-fold rotation carries each into a multiple of itself; a mirror carries each into the other. No operator commuting with both can tell them apart, and no basis makes them separate, because the mirror mixes them and the rotation would have to be diagonal in the same basis.

That is the mechanical content of “two-dimensional”: there are two independent ways to be that kind of wave, and the group moves them into each other. A physicist calls them a degenerate doublet, a chemist calls them an E pair, and a crystallographer calls them one representation appearing once.

Where the degeneracies are, group by group #

Ten of the seventeen plane groups have a little group with a two-dimensional representation somewhere in the zone, and the ten are exactly those whose point group is 4mm, 3m or 6mm — which is to say p4m, p4g, p3m1, p31m, p6m, and the ones that inherit such a little group at a special point.

The seven with only abelian point groups force no degeneracy anywhere by the unitary argument. Some of them nevertheless show one, and that is the antiunitary effect the third rung of the representations ladder describes: a real operator cannot separate a complex representation from its conjugate.

So a reader looking at a band diagram and seeing two curves touch has three questions to ask and this ladder answers all three: is the little group there large enough to force it, is the touching robust against changing what the symmetry does not fix, and is the operator real.

A path, and why band diagrams look like that #

Drawing levels along a path from one special point to another through the lines between them is not a stylistic choice. The special points are where the little group is large and degeneracies are forced; the lines between them are where a smaller little group survives, so a partial constraint remains; and the interior is where nothing is forced and nothing interesting can be said.

A path visits the constraints in order. Where two curves meet at a labelled point and separate on either side, the meeting is a two-dimensional representation of that point’s little group splitting into the one-dimensional representations of the line’s smaller group. That splitting is called compatibility, and it is what makes a band diagram readable as a sequence of group-theoretic statements rather than as a plot.

What the pattern does not fix #

Two things this rung does not decide, and both matter to anybody reading a real spectrum.

The order of the levels. Which of the two singles lies below the pair is a fact about the operator, not about the group. Changing the weights permutes them freely, and the figures show it happening.

Whether an extra coincidence occurs. Two levels of different symmetry may meet by accident, and the prediction from characters says nothing against it. That is the subject of the accidental-degeneracy essay, whose test — move the weights, watch what survives — is exactly the test this rung’s claim rests on.

The claim is therefore a lower bound with a pattern attached: at least this much degeneracy, arranged in this way. Anything above it is either an accident or a symmetry nobody has written down.

Which wavevectors carry a two-dimensional representation #

The special points of a zone are the wavevectors whose coordinates have small denominators, because a rotation of order n can fix a wavevector only if its coordinates have denominator dividing n. That much is arithmetic. Which of those points has a little group large enough to force a degeneracy is then a short enumeration, and the answer sorts the plane groups into three kinds.

In the hexagonal groups with mirrors — p3m1, p31m, p6m — the centre and the corners carry 3m or larger, so degeneracies are forced at both. In the square groups with mirrors — p4m, p4g — the centre and the corner carry 4mm and the edge centres carry a smaller group, so degeneracies appear at two of the three named points and not the third. In everything else the little groups are abelian everywhere and nothing is forced by the unitary operations at all.

That sorting is a fact about the seventeen and is worth having in the same form as the seventeen themselves: a list produced by enumeration, checked, and not memorised. It is also the reason the same three or four letters label the special points of every crystal in the literature — the letters are attached to wavevectors with denominators two and three, and those are the only denominators a lattice permits a rotation to fix.

The tetragonal case is the one where a reader can watch the sorting happen, because the three named points of a square zone answer the question three different ways. At the centre the little group is the whole of 4mm, order eight, and its two-dimensional representation appears; at the corner the same group survives, for the same reason and with the same consequence; at the edge centre it does not. A wavevector at (½, 0) is fixed by the mirror along that axis and by the mirror across it and by the half-turn, and by nothing else — an abelian group of order four, all of whose representations are one-dimensional. So the edge centre of a square zone forces nothing at all, and every level there is alone.

The three-way sorting is what makes a band diagram’s labelled points worth labelling. A reader who knows the little group at each of them knows, before seeing a single number, where the levels of any crystal with that symmetry may touch and where they may not. Which levels then join which along the lines between those points is the next question and a separate computation — a restriction of characters from the point’s group to the line’s, which the compatibility essay takes as its subject.

A partial constraint, along a line #

Between the special points the little group is usually not trivial and not the full point group: on a mirror line it is the group of order two containing that mirror. Order two is abelian, so no degeneracy is forced — but something else is, and it is worth naming because it is what makes a band diagram legible.

A level on such a line is either even or odd under the mirror, and that label cannot change along the line without the level passing through a place where the mirror stops being a symmetry. So two levels of opposite parity may cross freely on the line, while two of the same parity may not: they repel. That is the standard non-crossing rule, and it is a statement about one-dimensional representations rather than about two-dimensional ones.

The consequence is that the interesting structure of a band diagram lives on the lines as much as at the points. A crossing on a line is a statement that the two levels belong to different representations of the line’s little group; an avoided crossing is a statement that they belong to the same one. Neither is visible in the numbers without the labels, and the labels come from exactly the decomposition this rung performs.

The one-line version of a long history #

The statement that a two-dimensional representation forces a pair of levels is older than the crystallography it is used in. It is Wigner’s, from the late nineteen-twenties, and the argument he gave for atomic spectra is the argument used here for a periodic solid: an operator commuting with a group cannot distinguish states the group treats as one object.

What crystallography adds is that the relevant group is different at every wavevector, and that the wavevector’s own group is computable from the crystal’s group by integer arithmetic. The two-dimensional representations are then not exotic objects but the ordinary furniture of the hexagonal and tetragonal classes, and their consequences are visible at the labelled points of any band diagram anybody has ever drawn.

The arithmetic behind it is the arithmetic of this whole collection. A lattice permits rotations of order 1, 2, 3, 4 and 6; those orders make point groups of orders up to 48; those orders make representations of dimensions at most 2 in the plane and 3 in space. A degeneracy of five is impossible in a crystal for the same reason a five-fold axis is.

The list of dimensions available is short enough to state in a sentence, and stating it is the sharpest form of the whole rung. Of the ten plane crystal classes, seven have nothing but one-dimensional representations and three have a two-dimensional one; none has anything larger. So a level in a plane crystal is single or double and never triple, whatever the operator, whatever the material and whatever the wavevector. A spectrum showing three states at one value in a plane-periodic structure has either an accident in it, or a symmetry nobody has written down, or an error — and the three possibilities are distinguishable by moving the weights, which is the test the next section is about.

In three dimensions the same argument gives dimensions up to three, and the reason is the same arithmetic one rung further out: the point groups are larger, the largest is of order forty-eight, and its representations reach three. Nothing reaches four. That a degeneracy of five is impossible in a crystal and a five-fold axis is impossible in a crystal are not two facts; they are the crystallographic restriction, read once about a rotation and once about a representation.

What connects a point to the line beside it #

The observation that a level’s label cannot change along a line has an exact form, it is the piece of machinery a band diagram is actually drawn with, and it deserves stating rather than gesturing at.

Move off a special wavevector along a line. The little group shrinks — the operations fixing the line are a subgroup of those fixing the point — so a representation that was irreducible at the point need not stay irreducible when restricted to the smaller group. Where it does not, it decomposes, and the decomposition says which levels the point’s pair separates into.

Those decompositions are the compatibility relations, and computing one is a restriction of a character: take the representation’s character, evaluate it only on the operations that survive, and decompose the result in the smaller group’s table. It is the same arithmetic as everywhere else on this page, run between two groups rather than inside one.

The relations do two things. They say how a pair splits, which is what a band diagram shows as two curves leaving one point. And they constrain how the levels join up, because a level arriving at a special point from one side must land on a representation the line’s label is compatible with — so a diagram in which a level of one label runs into a point that has no such representation is wrong, and can be seen to be wrong without any computation of eigenvalues.

That is why band diagrams are drawn along paths rather than as surfaces. A path through the special points and the lines between them visits every group in the chain, and the compatibility relations are the rules for what may connect to what.

The sum that has to come out the same everywhere #

There is a check available at every wavevector at once, it is trivial, and it is the one that catches an error in the whole apparatus rather than in one entry.

The space the operator acts on is the same space at every wavevector: one amplitude per site of the orbit. So the total number of levels is the orbit’s size, and it is the same number at the centre of the zone, at the corners, and at every general point in between. The multiplicities predicted at each wavevector must therefore sum to the orbit size, and so must the multiplicities measured.

That is a weak check on any individual entry and a strong one on the procedure. A little group computed with an operation missing produces a decomposition that is short by a level, and the sum fails. A representation counted with the wrong multiplicity fails it too. And a measured spectrum whose degeneracies were merged too aggressively — a gap threshold set too wide — reports fewer levels than the orbit has, which is the same failure arriving from the instrument rather than from the theory.

The census tests both columns against it, which is what makes the agreement between prediction and measurement worth something: two computations agreeing on a number they are both required to produce is weaker evidence than two computations agreeing on a pattern, and the sum is what separates the two claims.

Where this ladder goes #

The next rung is the case this one has been carefully not handling: the little groups whose operators do not multiply the way the group does. There the character table cannot be used as it stands, the levels double along a whole line of the zone rather than at isolated points, and the reason is a sign that no choice of convention removes.

And the corner of the honeycomb is this rung’s argument on the smallest interesting case, where the two-dimensional representation of 3m forces a crossing that turns out to be an identity between three cube roots of unity — exact, in the ring these phases live in, rather than a number that came out small.