Crystallography & Symmetry
patterns derived from their groups
Concept

Cubic invariant — where it appears

A third-degree polynomial in an order parameter's components that the parent group leaves alone. Its existence forbids a continuous transition, because a free energy with an odd term has its minimum away from zero the moment the quadratic coefficient does.

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

4: three generators in two variables, and the one relation between them. The invariant ring of 4 needs 3 generators, of degrees 2, 4, 4, and three functions of two variables cannot be algebraically independent. The relation between them is found rather than quoted: every monomial in the generators of the degree at which they can first be dependent is written out, the map back to polynomials in x and y is formed, and its kernel is the relation. It is then evaluated at points of the lattice, where all three generators take integer values and the combination comes to exactly zero. A group with a reflection has no such relation, which is the same statement as its ring being free.

Three invariants and one relation

Four of the ten plane classes need three invariants where two variables can only support two, so exactly one polynomial identity ties them together. The identity is not recognised or recalled: it is the kernel of a linear map, computed and then checked at lattice points where every term is an integer.

point-groups · Invariants Which order parameters carry a cubic invariant, and therefore cannot grow from zero. Every order parameter of every plane class, with the number of independent cubic invariants it admits. The count is the degree-three coefficient of the Molien series of the representation's image — the same computation the invariant-ring figures make for a different reason — and Landau's condition is that it be zero. Where it is not, a free energy in that order parameter has a term of odd degree, which puts its minimum away from zero the moment the quadratic coefficient does anything at all, so the parameter jumps rather than growing. In the plane exactly two order parameters carry one, and both are the two-dimensional representation of a class with a threefold axis and no sixfold.

The cubic term that forbids a continuous change

A crystal may lose a symmetry gradually only if the quantity measuring the loss admits no cubic invariant. Whether it does is the third coefficient of a Molien series — so a question about how a material changes is answered by counting polynomials.

point-groups · Invariants

Named alongside it

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

Invariant polynomialMolien seriesAlgebraic independenceChevalley theoremFirst order transitionHypersurfaceInvariant ringLandau conditionLifshitz invariantOrder parameterPhase transitionReflection group

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