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.

A thermal phase fluctuation is a finite-temperature fluctuation of the phase of a complex order parameter. For a linearly dispersing phase mode, its infrared variance is proportional to $T\int d^{d}k/k^2$, which prevents true finite-temperature long-range order when $d\le 2$.