Liquid helium-4 flows through a narrow capillary at very low velocity with no measurable pressure drop. The correct explanation is:
AAt low temperatures, helium atoms move slowly enough that they cannot transfer momentum to the capillary walls
BThe zero-point motion of helium atoms prevents them from forming the clusters needed for viscous drag
CThe flow velocity is below the Landau critical velocity, so the superfluid cannot shed energy into phonon excitations — dissipation is energetically forbidden
DThe condensate fraction is so large that inter-atomic collisions are suppressed, eliminating the source of viscosity
The Landau criterion is the correct physical explanation. Frictionless flow is not merely due to slow atomic motion or suppressed collisions — it follows from the excitation spectrum. For the moving superfluid to lose energy (to dissipate), it must create excitations. Energy-momentum conservation requires the flow speed to exceed ε(p)/p for some excitation. If the minimum of ε(p)/p (the Landau critical velocity v_c) is nonzero, no excitations can be created below that speed and the flow is frictionless. The linear (phonon) excitation spectrum from interactions gives v_c > 0, which is what makes ⁴He superfluid.
Question 2 Multiple Choice
An ideal (non-interacting) Bose gas undergoes Bose-Einstein condensation at low temperature, with a macroscopic fraction of atoms in the ground state. Yet it is not technically a superfluid. Why?
AThe condensate fraction in an ideal gas is too small to support coherent flow
BAn ideal Bose gas lacks quantized vortices, which are required for superfluidity by definition
CWithout interactions, the excitation spectrum is parabolic (ε ∝ p²), giving a Landau critical velocity of zero — the gas can shed energy into excitations at any flow speed, so flow is never truly frictionless
DBEC requires a periodic lattice, which blocks the long-range phase coherence needed for superfluidity
This is a subtle and important distinction. BEC (macroscopic ground-state occupation) is necessary but not sufficient for superfluidity. The Landau critical velocity v_c = min[ε(p)/p] for a free Bose gas is zero because the parabolic spectrum ε = p²/2m gives ε(p)/p = p/2m → 0 as p → 0. This means the gas can always create excitations at arbitrarily small flow speeds — there is no protection against dissipation. Interactions are essential: they convert the parabolic spectrum into a linear (phonon-like) spectrum at small momenta, raising v_c above zero and enabling genuine superfluidity.
Question 3 True / False
A rotating superfluid accommodates angular momentum through an array of quantized vortices rather than through uniform rotation, because the phase coherence of the condensate constrains the allowed velocity fields.
TTrue
FFalse
Answer: True
The superfluid velocity is vs = (ℏ/m)∇φ, which is the gradient of a scalar (the phase). This means ∇ × vs = 0 everywhere except at singularities — the superfluid cannot have uniform rigid-body rotation (which requires ∇ × v ≠ 0). The solution is vortices: topological defects where the phase winds by 2π around a core, the condensate density vanishes at the core, and the circulation around each vortex is exactly h/m. Under rotation, many such vortices form an array that mimics rigid-body rotation on average. Quantization of circulation is a direct consequence of the phase structure of the order parameter.
Question 4 True / False
The superfluid order parameter is simply a number proportional to the condensate density — it carries no phase information relevant to the flow properties of the superfluid.
TTrue
FFalse
Answer: False
The order parameter is a complex field Ψ(r) = √(n₀) e^{iφ(r)}, and the phase φ(r) is the physically crucial part for flow. The superfluid velocity is entirely determined by the phase gradient: vs = (ℏ/m)∇φ. The amplitude √(n₀) sets the condensate density, but all flow, quantized vortices, and the Josephson effect arise from the phase. Phase rigidity — the tendency of the phase to be spatially uniform in the ground state — is what makes the superfluid resistant to creating the phase gradients associated with excitations, which is the microscopic origin of frictionless flow.
Question 5 Short Answer
What is the role of the order parameter's phase in superfluidity, and why does phase rigidity lead to frictionless flow?
Think about your answer, then reveal below.
Model answer: The superfluid order parameter is a complex field Ψ(r) = √(n₀) e^{iφ(r)}. The superfluid velocity is vs = (ℏ/m)∇φ — it is entirely determined by the spatial gradient of the phase. Phase rigidity means that in the ground state, the phase is spatially uniform, and deforming it costs energy. Frictionless flow follows from the Landau criterion: for flow to dissipate, the superfluid must create phonon or other excitations by transferring energy and momentum to the fluid. The linear (phonon) excitation spectrum produced by interactions sets a minimum threshold for this process — the Landau critical velocity. Below this speed, energy-momentum conservation forbids excitation creation, so the phase gradient (and hence the flow) remains unchanged. Flow is frictionless because phase rigidity and the excitation spectrum together make dissipation energetically impossible at low velocities.
This connects BEC to superfluidity precisely: BEC creates the order parameter (and hence the phase), but interactions are what shape the excitation spectrum into the linear form that guarantees a nonzero Landau critical velocity. Without interactions, the phase exists but v_c = 0, and no frictionless flow results.