Crystallography & Symmetry
patterns derived from their groups
Concept

Double group — where it appears

The group of unit quaternions lying over a rotation group, of twice its order, in which a full turn is a distinct element rather than the identity. It is what acts on a half-integer spin, and it is also the fundamental group of the space an orientationally ordered medium takes its values in.

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

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 The kinds of line defect each breaking allows. The eleven proper crystal classes, each with the order of the binary group that covers it, the number of conjugacy classes of that group other than the identity — which is the number of kinds of line defect — and whether the group commutes. 6 of the eleven do not, and in those media two defect lines cannot pass through each other without leaving a third line behind.

The defect that needs two laps

Which defects a medium can have is not a fact about the medium. It is a fact about the space its order parameter lives in, and for a rotational symmetry broken down to a point group that space has a fundamental group twice the size of the point group. The kinds of line defect are its conjugacy classes — and in six of the eleven cases they do not commute, which means two defect lines cannot pass through each other.

applied · Defects

Named alongside it

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

CharacterConjugacy classCrystal fieldDefectDegeneracyDisclinationHomotopyLocal symmetryOrder parameterPoint groupQuaternionRepresentation

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