Crystallography & Symmetry
patterns derived from their groups
Concept

Degeneracy — where it appears

Two or more states of an operator sharing one value. Symmetry forces a degeneracy equal to the dimension of an irreducible representation, which in the plane is at most two and in space at most three.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

The ten plane classes, and the dimensions they permit. Every crystallographic point group of the plane, with one block per irreducible representation and each block as wide as its dimension. Nine of the ten have only one-dimensional representations; 4mm, 3m and 6mm carry a two-dimensional one, drawn in the measured colour. Nothing is wider than two, and the sum of the squares of the widths in each row is the order of that group — the identity that says the row is complete.

How large a degeneracy may be

Symmetry can force two things to have the same value, and in a crystal it can force three. It can never force five, and the reason is a sum of squares — the same kind of arithmetic that forbids a five-fold axis, arriving at a question about levels rather than about rotations.

point-groups · Representations p3m1 at (0, 0): the levels the little group requires, and the ones measured. The levels of the p3m1 model at (0, 0), with degenerate ones drawn thick. The little group there has order 6, and its characters predict levels of dimensions 1, 1, 2, 2. The measurement is 1, 2, 2, 1, and the account is "unitary". The values are numerical and the multiplicities are read at a stated gap; the prediction they are compared against is exact.

Where two levels must meet

At most wavevectors nothing of a crystal's symmetry survives, and its levels are as unconstrained as any operator's. At a handful of them a whole point group survives, and where that group has a two-dimensional representation, two levels are obliged to coincide — before anything about the material is known.

space-groups · K symmetry 4mm: a degeneracy tuned into existence, and gone at the next weight. Four invariant operators on one orbit of 8 points under 4mm, differing only in the weight given to a single class of pairs. The first column is not a choice: it is the value that weight has to take for two levels of different symmetry to arrive at the same number, found by sweeping the weight and closing on the crossing, and the two levels there agree to 1.0e-9. The character table predicts levels of sizes 1, 1, 1, 1, 2, 2; the tuned column shows 1, 1, 1, 2, 3 and every other column shows the predicted pattern again. That is the whole of what an accidental degeneracy is — a property of one choice of weights, not of the group — and it is why the weights have to be moved before a degeneracy is called forced. A degeneracy the group requires would be in all four columns, because nothing respecting the symmetry can lift it.

A coincidence the group did not ask for

Two levels sitting at the same value look identical whether symmetry required it or not. The difference is testable: move the numbers the symmetry does not decide and watch what survives, because a degeneracy the group forces cannot be shifted by anything the group leaves alone.

point-groups · Representations pgg at (1/2, 0): the operators multiply up to a sign. Every product of two Bloch operators of the little group of (1/2, 0) in pgg, against the operator of the product. They agree up to a scalar, and the scalar is +1 or −1: 4 of the 16 products come back with a minus sign. No rephasing removes them, and the search that says so tries every assignment of twelfth roots of unity to the operators. A representation that multiplies only up to this sign cannot be one-dimensional, because scalars commute and these operators do not. Each entry is an exponent modulo twelve, so the table is exact.

A glide sticks two levels together

The translation attached to a glide moves no wavevector at all. It comes back as a phase factor, and at the edge of the zone the factor is minus one — after which the operators of the little group no longer multiply the way the group does, and no rephasing repairs it.

space-groups · K symmetry The honeycomb's two levels meet at K, exactly. The two levels of the honeycomb net along a line from the centre of the zone to its corner. The off-diagonal entry of its two-by-two matrix is the sum of the phases of three bonds, and at the corner those phases are the three cube roots of unity, whose sum is zero — exactly, as an identity in the ring the phases live in rather than as a number that came out small. So the matrix there is the zero matrix and both levels are zero. It is the shortest exact statement of a crossing in this collection.

The crossing at the corner

The honeycomb's two levels meet at the corner of its zone, and the meeting is not approximate. Three phases sum to zero there — an identity between cube roots of unity — so the matrix is the zero matrix, and making the two sites differ opens a gap of exactly that difference.

space-groups · K symmetry 4mm: 17 of 25 transitions forbidden. Every pair of irreducible representations of 4mm, with the number of times the identity occurs in the product of the two with the vector operator. A zero is a prohibition: the integral that would give the transition rate vanishes for every choice of functions carrying those representations, whatever the material is made of. A positive number is a permission and nothing more. Rows are final states and columns initial ones; the labels are the dimensions of the representations, so the twos are the degenerate levels. Every one of the 17 prohibitions here was checked again against explicit polynomials.

What a group forbids to happen

Two levels and a thing that might carry a crystal from one to the other. Whether it can is one sum over the group — and a zero there is a prohibition that no material, no temperature and no intensity of light gets round.

point-groups · Representations Where time reversal does something, and what. Every wavevector of every plane group at which time reversal changes the answer, with the square of each antiunitary operator, the unitary prediction, Herring's corrected prediction and the measured degeneracies. Case (b) is Kramers' theorem in a crystal with no spin, and it happens exactly where a glide's operator squares to −1. Case (c) is a representation being carried to a different one by the antiunitary operator, so the two become one level. Nine of these rows were open before the criterion was built — three the earlier census called unaccounted and six it could not reach at all.

The degeneracy time reversal forces

A crystal with a glide has levels that stick together at the edge of its zone for a reason no character table contains. The operation responsible is antiunitary, it squares to minus one, and Kramers' theorem then applies to a model with no spin anywhere in it — which closes nine rows an earlier census in this collection had to leave open.

space-groups · K symmetry A spin needs two full turns to come back. The number a rotation about a fixed axis multiplies a state by, against the angle turned through. A vector — anything of integer spin — is back where it started after one full turn; a spin-one-half state is multiplied by minus one and needs a second turn. So the operators acting on such a state do not form the rotation group: a full turn is an operation distinct from doing nothing, and the group is twice as large.

Two turns to come back

A rotation through a full turn does nothing to a crystal and multiplies a spin-one-half state by minus one, so the group acting on such a state is not the point group but a group twice its size. Building those eleven double groups from quaternions and averaging a random operator over each gives the degeneracies a spin may have — and shows that the doubling everybody calls Kramers' is time reversal's doing and not the double group's.

point-groups · Representations

Named alongside it

The objects these essays reach for when they reach for this one.

Irreducible representationLittle groupCharacter tableTime reversalBloch stateCharacterFactor systemFrobenius schur indicatorProjective representationRepresentationWavevectorAbelian group

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