Crystallography & Symmetry
patterns derived from their groups
Concept

Band structure — where it appears

The energies a periodic system allows, plotted as functions of wavevector across one Brillouin zone. Which levels touch, which gaps open and which degeneracies are forced are all decided by the symmetry of the wavevector rather than by the strength of any interaction.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

space-groups · K symmetry 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.

space-groups · K symmetry Four inversion centres, and a band's charge on each. A square crystal with four atoms in each cell, at the points a quarter of the way along both diagonals, and the four points of the cell that an inversion maps onto themselves: the corner, the midpoints of the two edges, and the centre. None of the atoms sits on one. For each of four versions of the crystal, differing only in which bonds are stronger, the ring marks where the lowest band's charge is centred — computed from the band's Berry phase and predicted from four parities. Each version puts it on a different inversion centre, and in every case on a point with no atom.

Four signs say where the charge sits

In a plane crystal whose only symmetry beyond its lattice is a centre, a band's charge must sit on one of the cell's four inversion centres — and which one is decided by four signs: whether the band is even or odd under inversion at the four wavevectors inversion leaves alone. The centre can be a point no atom occupies, and when the four signs multiply to minus one there is no centre at all, only a charge that winds round the cell.

space-groups · K symmetry

Named alongside it

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

Brillouin zoneInversion centreWyckoff positionsBerry phaseCompatibility relationsIrreducible representationLittle groupSubductionTime reversal

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