theory for this problem[@fradkin04]. We will assign a uniform value to the coarse-grained height field $h=0,1,2,3$ to each reference (columnar) state. Let $n_x(\vec r)$ and $n_y(\vec y)$ represent the coarse grained dimer densities of the horizontal link with endpoints at the pairs of lattice sites $\vec r$ and $\vec r + \vec e_x$, and vertical links with endpoints $\vec r$ and $\vec r + \vec e_y$ respectively. Here $\vec e_x$ and $\vec e_y$ are two lattice unit vectors along the $x$ and $y$ directions respectively, with a lattice spacing of $1$.

We can now define the columnar local order parameter as the two component vector $$\begin{aligned}
\mathcal{O}_x(x,y)&=&n_x(x,y)-n_x(x+1,y)\nonumber \\
\mathcal{O}_y(x,y)&=&n_y(x,y)-n_y(x,y+1)\end{aligned}$$ which clearly correspond to the $(\pi,0)$ and $(0,\pi)$ Fourier components of the dimer densities. This two-component order parameter takes four distinct values for each one of the columnar states, and changes sign under shifts by one lattice spacing in either direction. It also transforms as a vector under $90^{\circ}$ rotations. Thus it is the order parameter for columnar order.

### Effective Field Theory: the non-interacting case {#sec:dimer-qft}

The fluctuations of the free field $h(\vec r)$ are described by a continuum Gaussian (free boson) model. We will find it simpler to work 