# Spatial symmetry breaking and locality

I consider a system described by a state $\Psi(\mathbf{r})$, where $\mathbf{r}$ are the spatial coordinates. The energy of the system is a functional $E[\Psi]$.

An usual analysis of a phase transition, consists first in finding the group $G$ of the transformations on $\Psi$ that leaves invariant $E[\Psi]$, and secondly in finding the isotropy group $H$ of the ordered phase. Then the order parameter space results to be the broken symmetries $R=G/H$, and interesting information can be learned from the topology of $R$.

The approach is performed for a local order parameter, and this makes sense when the group $G$ is composed of internal symmetries (for example invariance of $E$ under spin rotation).

My question is what is the validity of this description when $G$ contains spatial symmetries. For example, if a system whose energy is invariant under space inversion (i.e. the parity $\mathcal{P}$) exhibits a phase transition spontaneously breaking $\mathcal{P}$. To be more specific, assume the ordered phase is only invariant under a remaining inversion about $x$: $(x,y,z) \rightarrow (-x,y,z)$. In that case it seems difficult to say that the system breaks the symmetry in one way at a given position and in another way somewhere else, since the state at position $(x,y,z)$ must be correlated with the one at $(-x,y,z)$. Therefore, can I still define a local order parameter ? Can I still study the properties of topological defects from the topology of $R$ ?