Superfluid

A superfluid exhibits superfluidity, with phase stiffness and a phase-gradient current. In an appropriate ordered or quasi-ordered regime it supports low-speed flow without viscous dissipation, and its long-wavelength Goldstone boson has a sound-like dispersion. True thermodynamic phase selection and Bose-Einstein condensation need separate infrared conditions.

The normal component of a superfluid describes the momentum and entropy carried by excitations rather than the phase-gradient condensate current. Its velocity need not equal the superfluid velocity. In phenomenological condensate equations, prescribing its motion supplies a reference flow for dissipation; a full two-component fluid theory would also evolve its momentum and energy.

For a condensate $\psi =\sqrt{n}{e}^{iS}$ with kinetic operator $-{\hslash }^2{\nabla }^2/(2m)$, its particle current is $n\mathbf{u}$ with the displayed velocity. This follows by subtracting the Gross–Pitaevskii equation from its complex conjugate. More generally, in units where the kinetic operator is $-D{\nabla }^2$, the current velocity is $2D\nabla S$. The complex argument gradient and current velocity coincide only when $D=1/2$.

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 $d\le 2$.