$r$ is the bare distance from the critical point in the effective LGW theory that takes into account the effect of the vector potential, and $a$, $u$, $d_0$, and $d_2$ are the remaining Landau parameters. As is clear from the above discussion, one expects the bare values of $d_0$ and $d_2$ to be small of order $1/c^2$ compared to the other parameters in natural units. These two parameters are usually set equal to zero in elementary treatments of classical Heisenberg ferromagnets.

Dzyaloshinsky-Moriya interaction, and helimagnets {#subsubsec:II.C}
-------------------------------------------------

The terms in the action so far are all even in the gradient operator, and hence invariant under spatial inversion. The spin-orbit interaction can eliminate this requirement by coupling the electron spins to the underlying lattice, provided the crystal structure is not inversion invariant. Dzyaloshinsky[@Dzyaloshinsky_1958] and Moriya[@Moriya_1960] (DM) showed that to linear order in the spin-orbit interaction $\gso$ the relevant term is \_ = d[x]{} [M]{}([x]{})( ([x]{})), \[eq:2.8\] with $c_{\,\text{DM}} \propto \gso$. At a classical level, this term is purely phenomenological. DM showed how to derive it in the context of quantum mechanics, and in Sec. \[sec:III\] we will see how it arises in the context of field theory.

Quantum Systems {#sec:III}
===============

