The Dance of Liquid Helium

Why Superfluid Atoms Refuse to Stir Below Two Kelvin

Cool liquid Helium-4 below its lambda transition temperature of 2.17 Kelvin at atmospheric pressure, and the liquid loses every trace of mechanical viscosity. In laboratory cryostats in Moscow and Cambridge during the late 1930s, experimenters Pyotr Kapitsa, John Allen, and Don Misener watched superfluid helium perform feats that defied classical hydrodynamics. The cold liquid crept up the vertical glass walls of a beaker against gravity. It squeezed through ceramic micropores narrower than $10^{-5}$ centimeters without any pressure drop, and it conducted heat thousands of times faster than solid copper. Classical fluid mechanics could not explain why dense atoms crowded less than four angstroms apart suddenly refused to experience friction.

Soviet theorist Lev Landau had explained the thermodynamic data in 1941 by proposing a phenomenological curve of allowed quantum excitations. At low momentum $p$, the liquid supported longitudinal sound waves called phonons with linear energy $E = cp$. At higher momentum near inverse atomic spacing, the curve dipped into a local minimum called the roton gap $\Delta$. Because a moving wall or paddle cannot transfer energy to the fluid below a critical velocity $v_c = \min(E(p)/p)$, slow flow cannot dissipate heat and viscosity vanishes. Yet Landau had guessed that excitation curve from macroscopic fluid equations without deriving it from the Schrödinger wave function of individual helium atoms.

Between 1953 and 1955 at the California Institute of Technology, Feynman set out to construct the microscopic many-body wave function $\Psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N)$ from first principles. Every Helium-4 nucleus contains two protons and two neutrons orbited by two electrons. That even particle count makes each atom a composite spin-zero boson. Quantum mechanics requires the total wave function of identical bosons to remain completely unchanged when any two atoms swap positions.

Feynman realized that hard-core electrostatic repulsion between helium electron shells keeps the atoms from overlapping. At the same time, Bose symmetry prevents low-energy stirring motions from existing as distinct quantum states. Imagine a closed ring of ten helium atoms inside the liquid rotating by one atomic spacing around a loop. They move in unison like children on a merry-go-round stepping to the next horse. In a classical fluid of distinguishable billiard balls, that shifted ring is a new physical configuration. It can therefore absorb low-energy thermal shear.

In a Bose liquid, however, shifting every identical atom by one slot produces the exact same set of occupied spatial coordinates. Because a wave function orthogonal to the ground state must change sign somewhere in configuration space, a simple ring permutation cannot form an excited state. Because one helium atom is quantum-mechanically indistinguishable from another, a ring of atoms sliding past its neighbors leaves the collective quantum state completely unchanged.

With all low-energy stirring modes erased by Bose permutation symmetry, the only possible elementary excitations are density waves of the variational form $\Psi = \sum_i \exp(i\mathbf{k}\cdot\mathbf{r}_i)\Phi_0$. Feynman then used the liquid's static structure factor $S(k)$, which was measured directly by X-ray and neutron scattering. From it he derived the microscopic energy relation $E(k) = \frac{\hbar^2 k^2}{2m S(k)}$. Because atomic crowding creates a sharp diffraction peak in $S(k)$ at $k \approx 1.9\text{ \AA}^{-1}$, the denominator pulls $E(k)$ down into Landau's exact roton minimum.

In 1955, Feynman pushed the model one step further to explain how an irrotational superfluid ($\nabla \times \mathbf{v}_s = 0$) could rotate inside a spinning bucket. He showed that the liquid must perforate itself with microscopic hollow vortex lines whose cores are just 1.2 angstroms wide. Around each core, the quantum phase winds by integer multiples of $2\pi$ and circulation locks to multiples of $h/m$. Two years after Feynman predicted that rotating superfluid helium must break into a regular lattice of quantized whirlpools, W. F. Vinen measured the exact single-quantum circulation steps in a vibrating wire experiment at Cambridge.