# Section 3: Emergent Behavior in the Helium Liquids

The property that physicists call spin plays an essential role in the nature and emergent behavior of particles, atoms, and other units of matter. As we have noted previously, fermions have an intrinsic half-integer spin; no two fermions can occupy the same quantum state. And as we learned in Unit 6, because bosons have integral spin, any number of bosons can occupy the same quantum state. Those differences play out in the behavior at very low temperatures of the two isotopes of He—the fermion ^{3}He with spin of 1/2 owing to its single unpaired neutron and the boson ^{4}He with no net spin because it has two neutrons whose antiparallel spins sum to zero, as do the spins of the two protons in the He nucleus.

**Figure 9:** Temperature-pressure phase diagrams of the two quantum materials, ^{3}He and ^{4}He, that remain liquid down to the lowest temperatures in the absence of pressure compared to a typical liquid-solid phase diagram.

**Source: **Recreated from graphics by the Low Temperature Laboratory, Helsinki University of Technology. More info

The liquid forms of the two isotopes of He are the only two quantum liquids found in nature. Unlike all other atomic materials, because they are exceptionally light they do not freeze upon cooling; their zero point energy prevents them from freezing. As may be seen in Figure 9 (phase transitions) at low temperatures, the two isotopes of He exhibit remarkably different emergent behavior. Below 2.18 Kelvin (K), liquid ^{4}He becomes a superfluid that flows without appreciable resistance. Liquid ^{3}He behaves quite differently, flowing like a normal liquid down to temperatures in the millikelvin regime, some three orders of magnitude cooler, before it exhibits a transition to the superfluid state.

The reason is simple. Atoms of ^{4}He obey Bose-Einstein statistics. Below 2.18 K, a single quantum state, the Bose condensate, becomes macroscopically occupied; its coherent motion is responsible for its superfluid behavior. On the other hand, ^{3}He obeys Fermi-Dirac statistics, which specify that no two particles can occupy the same quantum state. While, as we shall see, its superfluidity also represents condensate motion, the condensate forms only as a result of a weak effective attraction between its quasiparticles—a bare particle plus its associated exchange and correlation cloud—rather than as an elementary consequence of its statistics.

Although physicists understood the properties of ^{3}He much later than those of ^{4}He, we shall begin by considering Landau's Fermi liquid theory that describes the emergent behavior displayed by the quasiparticles found in the normal state of liquid ^{3}He. We shall put off a consideration of their superfluid behavior until after we have discussed Bose liquid theory and its application to liquid ^{4}He, and explained, with the aid of the BCS theory that we will also meet later in this unit, how a net attractive interaction can bring about superconductivity in electronic matter and superfluidity in ^{3}He and other Fermi liquids, such as neutron matter.

## Landau Fermi liquid theory

There are three gateways to the protected emergent behavior in the "Landau Fermi liquids" that include liquid ^{3}He and some simple metals: 1) adiabaticity; 2) effective fields to represent the influence of particle interactions; 3) a focus on long-wavelength, low-frequency, and low-temperature behavior. By incorporating these in his theory, Lev Landau was able to determine the compressibility, spin susceptibility, specific heat, and some transport properties of liquid ^{3}He at low temperatures.

Adiabaticity means that one can imagine turning on the interaction between particles gradually, in such a way that one can establish a one-to-one correspondence between the particle states of the noninteracting system and the quasiparticle states of the actual material. The principal effective fields introduced by Landau were scalar internal long-wavelength effective density fields, which determine the compressibility and spin susceptibility and can give rise to zero sound, and a vector effective field describing backflow that produces an increased quasiparticle mass. The focus on low-energy behavior then enabled him to determine the quasiparticle scattering amplitudes that specify its viscosity, thermal conductivity, and spin diffusion.

**Figure 10:** Landau impacted theoretical physics over much of the 20th century.

**Source: **© AIP Emilio Segrè Visual Archives, Physics Today Collection. More info

The restoring force for zero sound, a collective mode found in neutral Fermi liquids, is an internal density fluctuation field that is a generalization of that found in the random phase approximation (RPA). Its phenomenological strength is determined by the spin-symmetric spatial average of the effective interactions between parallel spin and antiparallel spin He atoms; the "Fermi liquid" correction to the Pauli spin susceptibility is determined by the corresponding spin antisymmetric average of the interactions between He atoms, i.e., the difference between the spatially averaged effective interactions between atoms of parallel and antiparallel spin. The vector field representing the strength of the effective current induced by a bare particle is just the backflow field familiar from a study of the motion of a sphere in an incompressible fluid. The collisions between quasiparticles produce an inverse quasiparticle lifetime that varies as the square of the temperature, and when modified by suitable geometric factors, give rise to the viscosity and thermal conductivity found experimentally in the normal state of liquid ^{3}He.

As we noted above, Landau's theory also works for electrons in comparatively simple metals, for which the adiabatic assumption is applicable. For these materials, Landau's quasiparticle interaction is the sum of the bare electrostatic interaction and a phenomenological interaction; in other words, it contains an add-on to the screening fields familiar to us from the RPA.

It is in the nonsimple metals capable of exhibiting the localized behavior predicted by Nevill Mott that is brought on by very strong electrostatic repulsion or magnetic coupling between their spins that one sees a breakdown of Landau's adiabatic assumption. This is accompanied by fluctuating fields and electronic scattering mechanisms that are much stronger than those considered by Landau. For these "non-Landau" Fermi liquids, the inverse quasiparticle lifetime may not vary as T^{2}, and the electron-electron interaction contribution to the resistivity will no longer vary as T^{2}.

We will consider Mott localization and some of the quantum states of matter that contain such non-Landau Fermi liquids in Section 7.

It is straightforward, but no longer exact, to extend Landau's picture of interacting quasiparticles to short-range behavior, and thereby obtain a physical picture of a quasiparticle in liquid ^{3}He. The theory that achieves this turns out to be equally applicable to ^{3}He and ^{4}He and provides insight into the relative importance of quantum statistics and the strong repulsive interaction between ^{3}He atoms.

## The superfluid Bose liquid

While Heike Kamerlingh Onnes liquefied He in its natural ^{4}He form and then studied its properties in the 1920s, its superfluidity remained elusive until 1938. Jack Allen of Cambridge and Piotr Kapitsa in Moscow almost simultaneously found that, as the flowing liquid was cooled below 2.18 K, its viscosity suddenly dropped to an almost immeasurably low value. German-American physicist Fritz London quickly understood the gateway to the emergence of this remarkable new state of quantum matter. It was Bose condensation, the condensation of the ^{4}He atoms into a single quantum state that began at 2.18 K.

Superfluidity in liquid ^{4}He and other Bose liquids, such as those produced in the atomic condensates, is a simple consequence of statistics. Nothing prevents the particles from occupying the same momentum state. In fact, they prefer to do this, thereby creating a macroscopically occupied single quantum state, the condensate, that can move without friction at low velocities. On the other hand, the elementary excitations of the condensate—phonons and rotons in the case of ^{4}He—can and do scatter against each other and against walls or obstacles, such as paddles, inserted in the liquid. In doing so, they resemble a normal fluid.

This is the microscopic basis for the two-fluid model of ^{4}He developed by MIT's Laszlo Tisza in 1940. This posits that liquid He consists of a superfluid and a normal fluid, whose ratio changes as the temperature falls through the transition point of 2.18 K. Tisza, who died in 2009 at the age of 101, showed that the model had a striking consequence; it predicted the existence of a temperature wave, which he called second sound, and which Kapitsa's student, Vasilii Peshkov subsequently found experimentally.

**Figure 11:** Geometry of a straight vortex line in a superfluid, showing how the superfluid velocity rotates about a normal core (shown in blue) whose size is the superfluid coherence length, .

The superfluid flow of a condensate can also involve rotation. Norwegian-American scientist Lars Onsager and Richard Feynman independently realized that the rotational flow of the condensate of ^{4}He would be characterized by the presence of quantized vortex lines—singularities around which the liquid is rotating, whose motion describes the rotational flow.
See the math

W.F. Vinen was subsequently able to detect a single vortex line, while Feynman showed that their production through friction between the superfluid flow and the pipe walls could be responsible for the existence of a critical velocity for superfluid flow in a pipe.

## The introduction of rotons

It was Landau who proposed that the long wavelength elementary excitations in superfluid liquid He would be phonons; he initially expected additional phenomena at long length scales connected with vorticity, which he called rotons. He subsequently realized that to explain the experiment, his rotons must be part of the general density fluctuation spectrum, and would be located at a wave vector of the order of the inverse interatomic distance. Feynman then developed a microscopic theory of the phonon-roton spectrum, which was subsequently measured in detail by inelastic neutron scattering experiments. A key component of his work was the development with his student Michael Cohen of a ground state wavefunction that incorporated backflow, the current induced in the background liquid by the moving atom, thereby obtaining a spectrum closer to the experimental findings.

**Figure 12:** Lars Onsager, a physical chemist and theoretical physicist who possessed extraordinary mathematical talent and physical insight.

**Source: **© AIP Emilio Segrè Visual Archives, Segrè Collection. More info

By introducing response functions and making use of sum rules and simple physical arguments, it is possible to show that the long-wavelength behavior of a Bose liquid is protected, obtain simple quantitative expressions for the elementary excitation spectrum, and, since the superfluid cannot respond to a slowly rotating external probe, obtain an exact expression for the normal fluid density.

An elementary calculation shows that above about 1 K, the dominant excitations in liquid ^{4}He are rotons. Suggestions about their physical nature have ranged from Feynman's poetic tribute to Landau—"a roton is the ghost of a vanishing vortex ring"—to the more prosaic arguments by Allen Miller, Nozières, and myself that we can best imagine a roton as a quasiparticle—a He atom plus its polarization and backflow cloud. The interaction between rotons can be described through roton liquid theory, a generalization of Fermi liquid theory. K.S. Bedell, A. Zawadowski, and I subsequently made a strong argument in favor of their quasiparticle-like nature. We described their effective interaction in terms of an effective quasiparticle interaction potential modeled after that used to obtain the phonon-roton spectrum. By doing so, we explained a number of remarkable effects associated with two-roton bound state effects found in Raman scattering experiments.

In conclusion we note that the extension of Landau's theory to finite wave vectors enables one to explain in detail the similarities and the differences between the excitation spectra of liquid ^{3}He and liquid ^{4}He in terms of modest changes in the pseudopotentials used to obtain the effective fields responsible for the zero sound spectrum found in both liquids. Thus, like zero sound, the phonon-roton spectrum represents a collisionless sound wave and the finding of well-defined phonons in the normal state of liquid ^{4}He in neutron scattering experiments confirms this perspective.

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