Physical intuition for spatially constant motion in the XY-model in 2+1D

The XY-model on a 2-torus ($L_1,L_2$) has a lagrangian given by
$$
L_{XY}[theta] = int d^2 x frac{chi}{2}big{(}dot{theta}^2 – (partial_x theta)^2big{)}
$$
Fourier expanding $theta$ as
$$
theta (boldsymbol{x},t) = theta_0(t) + frac{2pi}{L_1}m_1x_1 + frac{2pi}{L_2}m_2x_2 + sum_{boldsymbol{k}}lambda_{boldsymbol{k}}(t) e^{iboldsymbol{k}cdot boldsymbol{x}}
$$
Putting this into our Lagrangian, we get
$$
L=frac{chi}{2}left(L_{1} L_{2} dot{theta}_{0}^{2}-frac{(2 pi)^{2} L_{2}}{L_{1}} m_{1}^{2}-frac{(2 pi)^{2} L_{1}}{L_{2}} m_{2}^{2}+L_{1} L_{2} sum_{k}left(left|dot{lambda}_{k}right|^{2}-k^{2}left|lambda_{k}right|^{2}right)right)
$$
We see that we have a collection of oscillators ($lambda_{boldsymbol{k}}$), a particle on a circle ($theta_0$), and two integers ($m_1,m_2$) which are the winding numbers for our $theta$ field around the torus. The eigenenergies will be
$$
E(m_i,n_kequiv lambda_k, Nequiv p_{theta_0}) =frac{1}{2 chi L_{1} L_{2}} N^{2}+frac{chi(2 pi)^{2} L_{2}}{2 L_{1}} m_{1}^{2}+frac{chi(2 pi)^{2} L_{1}}{2 L_{2}} m_{2}^{2}+sum_{k}|k| n_{k}
$$
where I have used $N in mathbb{Z}$ to label the momentum conjugate to $theta_0$ and $n_boldsymbol{k} in mathbb{N}$ for the occupation number for the $lambda_k$ harmonic oscillator.

My doubt is, in Xiao-gang Wen’s book Quantum field theory of many body systems (Chapter 6, page 263), he mentions that the label $N$ is physically "the total number of bosons minus the number of bosons at equilibrium, namely $N = N_{tot}-N_0$." How do you arrive at that statement?

My understanding is that $N$ just labels the energy from the motion of all the rotors moving uniformly. If I think analogously to the phonon problem, that term is like the energy due to the center of mass motion of the whole crystal. I don’t see how bosons have anything to do with it.