The extension of these ideas to the vortex lattice and Wigner crystal is more complicated. In this case, we can write the density as
where the order parameters 
satisfy
and 
forms the set of smallest basis vectors in the reciprocal
lattice. For the common case of a triangular lattice, there are three
complex order parameters . Under the translation 
, these transform as
The business of writing a Landau theory in this case is somewhat more complicated, and I will not discuss it here. We can, however, still easily make the connection to the elastic description,
Lattice systems are also characterized by another order parameter, describing broken rotational invariance. This orientational order parameter is defined as
where 
is the bond angle between the 
and
points on the lattice. The integer p characterizes the
degree of broken rotational symmetry. In the usual case of a
triagular lattice, the appropriate choice is the hexatic order
parameter, with p=6. Long-range orientational order is
automatically present in any lattice phase, since
has finite fluctuations provided 
is single-valued and
well-defined. Distinct p-atic phases may also exist, however, in
which 
, while 
for all 
.