# 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 $$L_{XY}[\theta] = \int d^2 x \frac{\chi}{2}\big{(}\dot{\theta}^2 - (\partial_x \theta)^2\big{)}$$ Fourier expanding $$\theta$$ as $$\theta (\boldsymbol{x},t) = \theta_0(t) + \frac{2\pi}{L_1}m_1x_1 + \frac{2\pi}{L_2}m_2x_2 + \sum_{\boldsymbol{k}}\lambda_{\boldsymbol{k}}(t) e^{i\boldsymbol{k}\cdot \boldsymbol{x}}$$ 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 $$E(m_i,n_k\equiv \lambda_k, N\equiv p_{\theta_0}) =\frac{1}{2 \chi L_{1} L_{2}} N^{2}+\frac{\chi(2 \pi)^{2} L_{2}}{2 L_{1}} m_{1}^{2}+\frac{\chi(2 \pi)^{2} L_{1}}{2 L_{2}} m_{2}^{2}+\sum_{k}|k| n_{k}$$ 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.

• Any suggestions for editing the question such that it becomes more understandable is also appreciated. Jul 27, 2020 at 14:15
• I think this might be more suitable for math.stackexchange.
– Wyuw
Jul 29, 2020 at 20:53
• Thanks for that suggestion. I was looking to understand the physical picture he had so I thought this might be the better place to do it. Jul 31, 2020 at 7:01