Superfluidity

Core Idea

Superfluidity is a macroscopic quantum state in which a fluid flows without viscosity. Helium-4 becomes superfluid below T_lambda = 2.17 K, exhibiting zero viscosity, quantized vortices (circulation = nh/m_4), a two-fluid behavior (superfluid and normal components), and a linear phonon-roton excitation spectrum. The Landau criterion states that superfluidity persists as long as the flow velocity is below a critical velocity v_c = min(epsilon(p)/p) set by the excitation spectrum. Superfluidity is intimately connected to Bose-Einstein condensation, though not identical: in liquid helium-4, only ~8% of atoms are in the condensate at T = 0 due to strong interactions, yet the superfluid fraction is 100%.

Explainer

Superfluidity — the flow of a liquid without any viscosity — was discovered in helium-4 by Kapitza and by Allen and Misener in 1938, shortly after the theoretical prediction of Bose-Einstein condensation. Below the lambda temperature T_lambda = 2.17 K (named for the lambda-shaped specific heat anomaly), liquid helium-4 enters a state with astonishing properties: it flows through capillaries with zero viscous resistance, it creeps up and over the walls of containers as a thin film, and it supports quantized vortices where the circulation is restricted to integer multiples of h/m_4.

The theoretical framework begins with Landau's excitation spectrum for the interacting Bose liquid. At low momenta, the excitations are phonons (ε = cp, with c the speed of sound). At higher momenta, a local minimum in the spectrum — the roton minimum — represents a remnant of the tendency toward short-range solidlike order. Landau showed that a superfluid can only lose energy to its surroundings by creating excitations, and this is kinematically possible only if the flow velocity exceeds v_c = min(ε(p)/p). For helium-4, v_c is set by the roton minimum at about 58 m/s — below this velocity, the superfluid cannot dissipate energy and flows without resistance.

The two-fluid model describes the phenomenology below T_lambda. The liquid is treated as two interpenetrating components: a superfluid fraction rho_s (zero viscosity, zero entropy, irrotational flow) and a normal fraction rho_n (ordinary viscous fluid carrying all the entropy, consisting of thermally excited phonons and rotons). At T = 0, rho_s = rho and rho_n = 0; at T_lambda, rho_s = 0. This picture explains second sound — a propagating temperature wave unique to superfluids, where the normal and superfluid components oscillate out of phase. It also explains the fountain effect: a temperature difference drives a superfluid flow (because superfluid carries no entropy, it flows to equalize the free energy, not the pressure).

The connection between superfluidity and BEC is deep but not simple. In an ideal Bose gas, BEC occurs but the critical velocity is zero (quadratic spectrum). In liquid helium-4, strong interactions deplete the condensate to only ~8% of atoms at T = 0, yet 100% of the liquid is superfluid. Interactions convert the spectrum from quadratic to linear, enabling the Landau criterion to be satisfied. The relationship is that superfluidity requires the phase coherence associated with a condensate, but the superfluid fraction is determined by the response of the whole system to a velocity field, not by the condensate fraction alone. The discovery of superfluidity in fermionic helium-3 (at 2.5 mK, through Cooper-like pairing) and in ultracold atomic gases has extended these ideas to entirely new physical regimes.