Phase rigidity and an energetic velocity threshold answer different questions

For neutral bosons of mass m in a continuum, a smooth phase gradient produces velocity ħ∇θ/m. The energy density of weak flow in the appropriate stationary-normal-component frame is one half the superfluid mass density times velocity squared. Rewriting it as a phase-gradient energy defines the stiffness. In three dimensions its coefficient has units of energy per length. For paired neutral fermions the phase belongs to pairs and the mass in the velocity relation is twice the constituent mass, with the corresponding pair-phase convention.

Superfluid density is a response coefficient, not simply the condensate occupation multiplied by mass. Interactions can strongly deplete condensate occupation while preserving a large superfluid response, and finite-temperature excitations create a normal component. A phase winding around a ring quantizes circulation if the amplitude remains nonzero on the contour. Changing the winding requires a phase slip or vortex process with local amplitude suppression. Such energy barriers help explain long-lived currents, but neither a condensate fraction nor a winding label alone specifies the lifetime of an experimental flow.

For a specified homogeneous Galilean-invariant fluid at zero temperature, creating an excitation of momentum opposite the flow changes energy by ε(p)−vp. It becomes energetically favorable once v exceeds the minimum of ε(p)/p over the relevant positive momenta. This is the Landau energetic threshold for that spectrum. It says when a uniform flow can release energy into those excitations, but it does not compute a nucleation rate or guarantee that a laboratory apparatus reaches that threshold before a vortex enters through a boundary.

In a dilute weakly interacting Bose gas with repulsive contact coupling, the Bogoliubov spectrum becomes linear at small momentum. Its energy-to-momentum ratio approaches the sound speed from above, so the threshold is that sound speed. For ideal free bosons the spectrum is quadratic and the ratio tends to zero. Thus Bose condensation alone does not imply a positive Landau threshold in that model. At finite temperature, excitation scattering and a normal component change the problem; the criterion is neither a universal necessary nor a sufficient test for all superfluid behavior.

An infinite uniform one- or two-dimensional system with short-range interactions and continuous phase symmetry has no true finite-temperature long-range condensate order under the usual equilibrium assumptions. This does not exclude a two-dimensional BKT superfluid with algebraically decaying correlations and a finite phase stiffness below vortex unbinding. Finite size, trapping, and long-range interactions change the relevant assumptions. It is therefore essential to specify which density, correlation, dimension, and thermodynamic limit a claim about a superfluid transition refers to.

The function lab displays the phase-twist energy after assigning x to a normalized phase gradient. Reset and set both controls to a=0.5 and x=1, predicting y=0.5. Reset and set a=0.5 and x=2, predicting y=2. The fourfold increase illustrates the quadratic stiffness cost. It contains no excitation spectrum, vortex nucleation, normal fluid, or thermal phase fluctuations. In particular, its smooth parabola cannot establish a Landau velocity or a BKT transition: those require additional kinematic and statistical information.