Superfluid order-parameter effective theory

For a complex superfluid order parameter $\psi =\sqrt{n}{e}^{i\theta }$, the nonrelativistic Landau-Ginzburg kinetic term contains $-n\dot{\theta }$. Thus number density and phase are canonically conjugate, so a phase-selected state necessarily fluctuates in particle number.

Number-phase conjugacy is the canonical relation between a conserved $U(1)$ charge and the phase of its order parameter. A state of sharp number cannot also have a sharply selected broken-symmetry phase.