# Landau theory; irrelevence of the lattice strcture?

In Ginzburg-Landau theory the derivative terms in the free energy depend on the structure of the lattice1. That said when looking at e.g. the O(3)-model the only derivative term kept is $$(\vec \nabla \vec m)^2$$ however if the lattice is not completely isotropic other terms may exist e.g. $\partial_x \vec m \cdot \partial_y \vec m$. Is it possible to show that all such terms which are present for some lattice structures and not for others are irrelevant (in the renormalization group sense)?

1 This is mentioned in a number of sources e.g. (Giuseppe, 2013; pg224) and (Kardar, 2007; pg23)

Let us take a lattice with spatial inversion symmetry. The lowest order term is then1: $$A_x \partial_x \ldots \partial_x \ldots+A_y \partial_y \ldots \partial_y \ldots+A_z \partial_z\ldots \partial_z \ldots$$ We can then scale $x$, $y$ and $z$ so all the $A$'s are equal thus getting2: $$A(\nabla \ldots)^2$$ higher order terms will (in general) be irrelevant and can be ignored. Now the fact that critical exponents are independent of lattice structure follows from the fact that all lattices have spatial inversion symmetry and thus the argument given here holds.