Superfluidity and superconductivity are striking examples of how quantum mechanics can affect the collective behaviour of large numbers of identical particles. When the temperature of a many-particle fluid is lowered below a critical threshold, thermal fluctuations are no longer able to prevent it from collapsing into a purely quantum state. If the particles are electrically neutral, like helium–3 and helium–4 atoms, such a 'coherent' state can flow without friction and is known as a superfluid. If, on the other hand, the particles are charged, the coherent state loses its resistance to electrical current and becomes a superconductor.
Superfluidity and superconductivity are striking examples of how quantum mechanics can affect the collective behaviour of large numbers of identical particles. When the temperature of a many-particle fluid is lowered below a critical threshold, thermal fluctuations are no longer able to prevent it from collapsing into a purely quantum state. If the particles are electrically neutral, like helium–3 and helium–4 atoms, such a 'coherent' state can flow without friction and is known as a superfluid. If, on the other hand, the particles are charged, the coherent state loses its resistance to electrical current and becomes a superconductor.
What is a superfluid? The word “superfluid” was coined to describe a qualitatively different state of a fluid that can occur at low temperatures, in which the resistance to flow is identically zero, so that flow round a closed path lasts for ever – a persistent current. Superfluidity can occur either for uncharged particles such as helium atoms or for charged particles such as the electrons in a metal. In the latter case the flow constitutes an electric current and we have a superconductor. Since an electric current is accompanied by a magnetic field, it is much easier to demonstrate the presence of a persistent current in a superconductor than in a neutral superfluid. In this chapter I shall describe the properties of superfluids, starting with the simplest and working up to more complicated examples. But in this introductory section I shall depart from the historical order even further by turning to the last page of the detective story so as to catch a glimpse of the conclusion from almost a century of experimental and theoretical research; we shall then have an idea of where we are heading. The conclusion is that superfluidity is more than just flow without resistance. It is an example of a transition to a more ordered state as the temperature is lowered, like the transition to ferromagnetism. At such a transition new macroscopically measurable quantities appear: in the case of a ferromagnet, the spontaneous magnetization. Such new measurable quantities are known as the order parameter of the low-temperature phase. The new quantities in the case of a superfluid are more subtle and more surprising: the amplitude and phase of the de Broglie wave associated with the motion of the superfluid particles. A superfluid can therefore exhibit quantummechanical effects on a macroscopic scale.
A system composed of particles, whose quantum mechanical zero point energy is large compared to their interaction energy, does not solidify even at absolute zero temperature and remains a so-called quantum liquid. Typical examples of such systems are conduction electrons in metals and liquid helium. Many conductors undergo a phase transition at their critical temperature T c and become superconducting below it. Similarly, liquid 4 He under its vapour pressure becomes a superfluid at T c = 2.17 K ( T λ is often used instead of T c ), and liquid 3 He at T c = 0.9 mK. Although the phenomena are called superconductivity for a charged system like conduction electrons, and superfluidity for a neutral system like liquid helium, they are characterised by the same basic property that the wave nature of the particles manifests itself on a macroscopic scale, as we shall see below. Phenomena The most important factor in determining the properties of a quantum liquid is the statistics of the constituent particles. Liquid 3 He becomes a typical Fermi liquid below 0.1 K and a superfluid below a few mK, in contrast with the bosonic liquid 4 He mentioned above. The two superfluids are quite different both phenomenologically and microscopically. Table 1.1 shows systems which exhibit, or are predicted to exhibit (in brackets), superconductivity or superfluidity. In the case of fermion systems pair formation is responsible for the phenomena so that their properties differ according to the pairing type.
Superfluidity is one of the most striking manifestations of quantum mechanics at macroscopic scales, characterized by frictionless flow and the emergence of collective quantum phenomena. The theoretical foundation of superfluidity in fermionic systems was revolutionized by the Bardeen-Cooper-Schrieffer (BCS) theory, developed in 1957 to explain superconductivity in metals. This groundbreaking work provided a microscopic description of how attractive interactions between electrons near the Fermi surface can lead to the formation of Cooper pairs-correlated pairs of fermions that condense into a coherent quantum state. The BCS theory not only explained superconductivity but also laid the groundwork for understanding superfluidity in systems like liquid helium-3 and ultracold fermionic gases, becoming a cornerstone of condensed matter physics.
Quantum matter is everywhere, from the interiors of neutron stars to the electrons in everyday metals. Like ordinary, classical matter, it is made up of many interacting particles. In classical matter, however, it is possible to think of each particle as an individual entity, whereas in quantum matter Heisenberg's uncertainty principle prevents us from telling individual particles apart: their behaviour can only be described collectively. In spite of this, many types of quantum matter are well understood from a theoretical point of view. For example, the “electron liquid” that is responsible for the flow of electricity through ordinary metals, the magnetic properties of many insulating materials and the normal and superfluid phases of helium at very low temperatures have all succumbed to the probing of theorists.
In the previous chapter we described quantized vortex lines, which are one of the characteristic features of superfluids. In a classical fluid, the circulation of vortex lines is not quantized and, in addition, vortex lines decay because of viscous processes. Another feature of a superfluid, the lack of response to rotation for a small enough angular velocity, was also demonstrated. This is analogous to the Meissner effect for superconductors. One characteristic common to superfluids and superconductors is the ability to carry currents without dissipation. Such current-carrying states are not the lowest-energy state of the system. They are metastable states, the existence of which is intimately connected to the nature of the low-lying elementary excitations. The word ‘superfluidity’ does not refer to a single property of the system, but it is used to describe a variety of different phenomena (see Ref. [1]). Historically, the connection between superfluidity and the existence of a condensate, a macroscopically occupied quantum state, dates back to Fritz London's suggestion in 1938, as we have described in Chapter 1. However, the connection between Bose–Einstein condensation and superfluidity is a subtle one. A Bose–Einstein condensed system does not necessarily exhibit superfluidity, an example being the ideal Bose gas for which the critical velocity vanishes, as demonstrated in Sec. 10.1 below. Also lower-dimensional systems may exhibit superfluid behaviour in the absence of a true condensate, as we shall see in Chapter 15.
This book is about a certain class of superfluids, namely the high temperature superconductors. Perhaps the simplest of the superfluids is liquid He4. Since the very early days of quantum mechanics it has been known that the helium atom He4 is a boson; it is built of an even number of particles (two electrons, two neutrons and two protons), so that a wave function ^(ri,F2) must be symmetrical in FI — r? where FI, r-z are the electron's coordinates. This has the result that half the rotational states of the molecule H62 are absent and that the scattering of alpha particles by helium for low energies deviates from the Rutherford scattering law. It also has the remarkable result that this is not so for the light isotope He3, which shows that the number of neutrons in the nucleus affects properties depending otherwise only on the electrons and nuclear charge.