The speed below which nothing can be made

A marble dropped into honey slows because it drags honey along with it, and the dragging happens molecule by molecule: the marble knocks molecules sideways, gives them momentum and energy, and slows by exactly as much as it has given away. The liquid’s ability to take momentum and energy from a moving body in small amounts, at any speed, is what drag is. Below 2.17 kelvin, liquid helium flows through gaps no ordinary liquid could pass and does not slow down. The obvious question is what has happened to its ability to take energy from things moving through it, and Lev Landau’s answer in 1941 was an argument of a single page. A body moving through a liquid can give it energy only by creating something in it — an excitation, a quantum of motion — and it can create an excitation only if doing so conserves energy and momentum together. Whether it can is a question about the shape of one curve.

The argument #

Consider a heavy body moving at velocity $v$ through a liquid at rest, at zero temperature, so that the liquid is in its ground state with no excitations in it. Suppose the body creates one excitation with momentum $p$ and energy $\varepsilon(p)$. The body loses the momentum $p$, and its kinetic energy changes by $-v\cdot p$ to first order, because a heavy body’s velocity barely changes. Energy conservation then requires that the energy the body loses equal the energy of the excitation:

$$\varepsilon(p) = v\,p\cos\theta \le v\,p,$$

where $\theta$ is the angle between the excitation’s momentum and the body’s velocity. An excitation can be made only if $\varepsilon(p) \le vp$ — only if the body moves faster than $\varepsilon(p)/p$. Landau’s critical velocity is the smallest value of that ratio over the whole spectrum:

$$v_L = \min_p \frac{\varepsilon(p)}{p}.$$

Below $v_L$ there is no excitation the body can make, no way for it to lose energy to the liquid, and so no drag. The same result can be reached the other way round, by keeping the body still and letting the liquid flow past it: energy depends on the frame it is measured in, and an excitation of energy $\varepsilon$ in the liquid’s own frame has energy $\varepsilon - v\cdot p$ in the frame of the wall. If that can be negative, the flowing liquid lowers its energy by making excitations and slows down; if it cannot, the flow is stable. The criterion is a statement about energy and momentum conserved together in a collision, and it needs nothing about what the excitations are except the curve $\varepsilon(p)$.

Geometrically, $\varepsilon(p)/p$ is the slope of the line from the origin to the point on the curve. The smallest such slope belongs to the line from the origin that just touches the curve — a tangent, if the curve bends upward anywhere. For helium that tangent touches just past the roton minimum, and its slope is 57 metres per second.

What the shape of the spectrum has to do with it #

The criterion makes superfluidity a property of a spectrum, and comparing three spectra shows what property.

An ideal gas of bosons has the spectrum of free particles, $\varepsilon = p^2/2m$, and $\varepsilon/p = p/2m$, which goes to zero as $p$ does. The tangent from the origin is horizontal: a body moving at any speed at all can make excitations of low enough momentum. An ideal Bose gas can condense — every particle can pile into the lowest state — and it is still not a superfluid, because nothing protects the condensate from being disturbed by the slowest obstacle. Condensation and superfluidity are different things, and the difference is the shape of the spectrum at low momentum.

A weakly interacting Bose gas, such as the dilute clouds of atoms cooled in laboratories since 1995, has a spectrum Bogoliubov worked out in 1947: $\varepsilon = \sqrt{(cp)^2 + (p^2/2m)^2}$. The interactions change the bottom of the spectrum from a parabola into a straight line — sound, with speed $c$ set by the interactions and the density — and above it the free-particle parabola takes over. The ratio $\varepsilon/p$ never falls below $c$, so the critical velocity is the speed of sound, and the gas is a superfluid. Experiments stirring such clouds with a laser beam have found the onset of dissipation at a fraction of the sound speed, which, as in helium, is set by the formation of vortices before Landau’s limit is reached.

Helium is a dense liquid, not a dilute gas, and its spectrum has structure a dilute gas does not. The phonons at low momentum are the sound that carries heat in the superfluid as a wave as well as the ordinary sound; above them the curve rises to a maximum and then dips to the roton minimum, at a momentum corresponding to a wavelength comparable to the spacing between atoms. The dip is the liquid’s short-range order showing through: an excitation with a wavelength matching the interatomic spacing costs less energy than one slightly longer or shorter. And because the dip sits at large momentum, it brings the ratio $\varepsilon/p$ down to a quarter of the sound speed. Helium’s critical velocity is set by its rotons, not by its phonons.

Why there is a roton at all #

The dip that sets helium’s critical velocity has an explanation that ties the dynamics of the liquid to a snapshot of its structure, and it is one of the more surprising results in the physics of liquids.

Feynman argued in 1954 that the lowest excitation of a Bose liquid at wavenumber $k$ is, to a good approximation, a density ripple of that wavelength imposed on the ground state, and that its energy is then fixed by two things: the kinetic energy a free atom would have at that wavenumber, $\hbar^2k^2/2m$, and how strongly the liquid’s atoms are already correlated at that wavelength. The correlation is measured by the static structure factor $S(k)$, the quantity an X-ray or neutron diffraction pattern records, and Feynman’s relation is

$$\varepsilon(k) = \frac{\hbar^2 k^2}{2m\,S(k)}.$$

At small $k$ the structure factor rises linearly with $k$, because long-wavelength density fluctuations in a liquid are suppressed by its compressibility, and the relation gives a linear spectrum — sound. At the wavenumber matching the spacing between neighbouring atoms, $S(k)$ has a strong peak, because the atoms sit at nearly that spacing from each other; dividing by a large $S(k)$ pulls the energy down, and the result is a minimum. The roton is the liquid’s short-range order, read off a diffraction pattern and turned into an energy.

Feynman’s relation overestimates the roton energy by about a factor of two — the ripple is not the whole excitation, and later refinements that let the atoms flow around each other as the ripple passes bring the number close to the measured 8.6 K — but it gets the shape right, and the shape is what makes the critical velocity what it is. A liquid with a sharper diffraction peak would have a deeper roton and a lower critical velocity. As helium is compressed its atoms pack more tightly, the peak in $S(k)$ grows and moves to larger $k$, and the roton deepens and moves outward, which is exactly the trend in the pressure figure. Pushed further, as the liquid approaches solidification, the roton energy would fall towards zero at the wavenumber of the peak, and a spectrum whose minimum touches zero at finite $k$ is a liquid about to freeze into a crystal with that spacing.

What a moving body is allowed to make #

The criterion does more than give a threshold. For any speed above it, it says which excitations the body can create.

Just above the threshold a body can make rotons, and only rotons with momenta in a narrow band near the minimum; as it speeds up the band widens. Phonons — ordinary sound — cannot be emitted until the body moves faster than sound, and at that speed the emission is the familiar one: a source outrunning its own waves leaves them in a cone, and the condition for the cone to exist, speed greater than wave speed, is exactly Landau’s criterion applied to a linear spectrum. Cherenkov radiation from a charge moving faster than light in a medium is the same criterion applied to photons, whose $\varepsilon/p$ in the medium is the phase velocity of light. Landau’s argument is the general statement of which a sonic boom and a Cherenkov glow are two special cases: a body emits the excitations whose $\varepsilon/p$ it outruns.

Squeezing the liquid lowers the limit #

The roton’s energy and momentum change as the liquid is compressed, and the critical velocity changes with them.

Near the roton minimum the spectrum is nearly $\Delta + (p - p_0)^2/2\mu$, and the tangent from the origin touches close to $p_0$, so Landau’s velocity is close to $\Delta/p_0$ — a little above the exact tangent, which touches slightly past the minimum. Squeezing the liquid packs its atoms closer, which moves the roton to larger momentum, and lowers its energy, and both reduce the ratio: from about 59 m/s at the liquid’s vapour pressure to about 46 m/s near 25 bar, just below the pressure at which helium solidifies.

The limit has been reached, but only by an unusual probe. Negative ions in liquid helium are electrons that have pushed the liquid away to form a bubble a couple of nanometres across, and an electric field drags them through the liquid. In the superfluid at low temperature they accelerate with almost no drag until they approach a limiting speed, where they begin to shed rotons — exactly the process the emission figure shows switching on — and experiments with ions in pressurised helium found that limit close to Landau’s velocity for the pressure. The ions are the one kind of body small and smooth enough to reach the roton threshold before something else intervenes.

What intervenes everywhere else #

For helium flowing through a tube, or a wire moving through the liquid, the flow begins to dissipate at speeds far below 57 m/s, and the reason is an excitation the spectrum does not contain.

The spectrum describes excitations that are small — a phonon or a roton is a ripple a few atoms across. A superfluid can also contain quantised vortices, lines around which the liquid circulates with exactly one quantum of circulation, and a vortex is a large object whose energy grows with its length and whose momentum grows with the area it encloses. A vortex ring or a vortex line spanning a channel has a small ratio of energy to momentum when it is large, and Feynman estimated in 1955 that making one becomes possible at a speed of order $(\hbar/md)\ln(d/a_0)$ in a channel of width $d$ — centimetres per second in a micrometre-wide channel, millimetres per second in a millimetre one. Measured critical velocities in channels follow that scale, not Landau’s. The flow breaks down by nucleating vortices at the walls, and only in channels a few nanometres wide, where no vortex fits, does the vortex limit approach the roton limit.

So Landau’s criterion is a bound, not a prediction, and a rigorous one: below $v_L$ no small excitation can be made, whatever else happens, and the superfluid is stable against being broken up into rotons and phonons. What it cannot rule out is dissipation through objects its spectrum omits. A proof that something cannot happen by one route is not a proof that it cannot happen, and the gap between 57 m/s and the millimetres per second of real channels is the width of that distinction.

The same criterion in a superconductor #

A superconductor carries current without resistance for the same reason, and the parallel is exact. Its electrons are bound into pairs, and the pairs flow together as one quantum state; breaking a pair costs at least the energy gap $2\Delta$. Moving the whole electron fluid at velocity $v$ lets a pair be broken with momentum near the Fermi momentum $p_F$, and Landau’s argument gives a critical velocity of about $\Delta/p_F$, beyond which the current can dissipate by breaking pairs. The corresponding current density — the depairing current — is the theoretical maximum a superconductor can carry. As in helium, real superconductors usually fail earlier, by letting magnetic vortices in and pushing them around, and which of the two lengths of a superconductor is longer decides how easily those vortices enter. The analogy is not loose: both are condensates whose excitation spectrum has a floor on $\varepsilon/p$, and both lose that protection to vortices first.

Where the argument’s assumptions sit #

Zero temperature. At any finite temperature the liquid already contains thermal excitations — the normal fluid of the two-fluid picture — and a moving body can scatter off them, exchanging energy without creating anything new. Landau’s criterion concerns the superfluid component; the normal component always has viscosity.

A heavy body. The argument treats the body’s velocity as unchanged by one emission, which requires it to be much heavier than the momentum it gives away. For a light body — a single atom moving through the liquid — the exact kinematics shifts the threshold.

One excitation at a time. The body might create two excitations whose momenta nearly cancel while their energies add, which is allowed by energy conservation at lower speeds for some spectra. For helium’s spectrum the single-roton process has the lowest threshold, but the general criterion includes multi-excitation processes.

The liquid fills all space. Near a wall or a free surface the liquid supports excitations the bulk spectrum does not list: ripples on the surface, whose restoring force is surface tension and gravity, and waves in thin films, whose speed depends on the film’s thickness. Each has its own ratio of energy to momentum, and a body moving close to a surface can reach the threshold for making them well below the bulk limit.

The spectrum is drawn as a fit. The curve in the figures is a smooth interpolation through measured values, not a theory, and the critical velocity drawn from it inherits a few per cent of uncertainty from the fit’s shape near the roton.

Where the spectrum stops being one curve #

The spectrum is a single curve because the excitations are treated as independent, each with a definite energy and momentum. At higher momenta the real spectrum ends — beyond about 2.5 Å⁻¹ a roton can decay into two lower-energy excitations and ceases to be a sharp quasiparticle — and the drawing simply stops. What happens to a body moving fast enough to make excitations there is not in the picture.

Nor can a curve of $\varepsilon(p)$ show vortices, which are the real limit in almost every flow. They are not points on the spectrum but objects with their own energy and momentum depending on their size and shape, and a figure of the spectrum is a figure of what the liquid can do in small pieces, silent about what it does in large ones.

Still open: how a vortex is born #

Feynman’s estimate says when a vortex becomes energetically possible; it does not say how one forms, and the energy barrier to forming a vortex from nothing in a clean channel is large. Critical velocities measured in different experiments vary by factors of ten, depend on the history of the sample, and in many cases appear to be set by vortices already present — pinned to roughness on the walls since the liquid was cooled — rather than by new ones being made. Whether an ideal, vortex-free superfluid in a perfectly smooth channel would nucleate vortices by thermal activation, by quantum tunnelling at the lowest temperatures, or only at speeds approaching Landau’s is a question several experiments have addressed with different answers, and it is not settled.

Past it lies the turbulent superfluid, a tangle of quantised vortices sustained by flow, whose behaviour at large scales resembles ordinary turbulence and at small scales does not. The habit worth carrying from here is to read a spectrum as a list of what can be emitted. Anything moving through a medium can give it only the excitations whose ratio of energy to momentum it outruns — which makes a sonic boom, a Cherenkov glow and a superfluid’s frictionless flow three readings of one line drawn from the origin.