Crystallography & Symmetry
patterns derived from their groups
Series

K symmetry — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. p4: (1/2, 0) has a star of 2. The first Brillouin zone of the square lattice, with the reciprocal lattice points at its corners and centre, and the whole star of the wavevector (1/2, 0) under p4. The star has 2 members and the little group — the operations that leave the wavevector where it is, modulo the reciprocal lattice — has order 2. The two multiply to the order of the point group, which is the orbit–stabiliser theorem and is checked rather than displayed. Each member is drawn at whichever of its equivalent copies lies nearest the origin, because that is where a reader expects a wavevector to be.

    The star of a wavevector

    A plane group acts on the plane, and this collection has spent two hundred essays watching it. It also acts on the reciprocal lattice, where the action is different in one decisive way: the translations move no wavevector at all, and come back instead as a phase.

    part 1 · space-groups
  2. 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.

    part 2 · space-groups
  3. 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.

    part 3 · space-groups
  4. 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.

    part 4 · space-groups
  5. Compatibility at (0, 0) in p4m. Every representation of the little group at (0, 0) in p4m, and what it becomes along two lines out of that point. A one-dimensional representation stays one level and acquires a label; a two-dimensional one splits into two levels of opposite label. The rows where the two columns differ are the point: the same level is even under the mirror that survives along one line and odd under the mirror that survives along the other, so which bands may cross and which must repel is different in the two directions out of one point. Labels are the characters on the classes, computed rather than named.

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

    A degeneracy at a symmetry point is forced by the little group there. Move off the point and the little group shrinks, the degeneracy is free to split, and which pieces it splits into is decided by restricting a character. That restriction is what joins a table of isolated points into a band structure.

    part 5 · space-groups
  6. 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.

    part 6 · space-groups
  7. Ten chains, two phases. The Zak phase of the lower band of a two-site chain, as the ratio of the two hoppings is swept. Every value is exactly zero or exactly π and nothing lies between them, because the chain has an inversion centre and inversion maps the zone loop to itself reversed — which forces the phase to equal its own negative modulo a full turn. The switch happens where the two hoppings are equal, which is the one place the band gap closes and the phase belongs to no band.

    The phase a symmetry turns into a number

    Carry a band's state once across the Brillouin zone and it returns with a phase. In a chain with an inversion centre that phase is exactly zero or exactly π and never anything else — and the two values turn out to be the two positions in the cell that an inversion centre fixes. Remove the centre and the phase moves continuously, which is what a quantisation claim has to be able to lose.

    part 7 · space-groups

All series