Module 6 : Superfluidity and the Kosterlitz Thouless transition
Lecture 28 : Low energy rotor description of the superfluid state and transition to                       insulating behaviour
 
This corresponds, in the effective field theory, to a global (space-independent) rotation of the complex field $\psi(x)$ by a constant phase-factor $e^{i\theta}$.
Now, if $\bar{n}$, the average number of particles per site, is quite large, then the dynamical fluctuations in $N_j$ can be treated approximately without worrying about the fact that there is a lower bound
 
$\displaystyle N_j$ $\textstyle \geq$ $\displaystyle 0$
 (3)
that must be satisfied by $N_j$ (here I am intentionally slurring over the distinction between the operator $N_j$ and its eigenvalue to avoid cluttering the notation). Henceforth we assume $\bar{n}$ is an integer. With this assumption, we may define a new operator
 
$\displaystyle n_j = N_j - \bar{n} \; ,$
 (4)
We can approximate the spectrum of this operator by saying $n_j$ can take on all values
$0$, $\pm 1$, $\pm 2$, $\cdots$, $\pm \infty$. In other words, $n_j$ can be thought of as the angular momentum of a planar rotor (or a unit-mass particle on a circle of radius $2\pi$). Let us denote the canonically conjugate variable by $\phi_j$ (which can be thought of as the angular coordinate of the particle on a circle), with $[\phi_j,n_{j'}] = i\delta_{jj'}$.