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.
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.
We have seen in Chapter 3 , that as temperature is increased quantum statistics, both Bose-Einstein or Fermi-Dirac, tend to classical. High temperatures blur out quantum effects in a system, which can be safely described in classical terms. In this chapter, we will consider some systems which manifest quantum properties at macroscopic level, but at very low temperatures. We recall that helium is the only substance which remains liquid at low temperature, down to absolute zero (see Chapter 6 ), a phenomenon related to the quantum behavior of its constituent atoms. Liquid helium was instrumental to the discovery that some metals are able to sustain an electrical current for long periods of time without any driving force. This phenomenon is named superconductivity, and is also a macroscopic manifestation of the quantum behavior of the electrons in metals. Later on it was discovered that liquid helium itself shows a similar property, named superfluidity, consisting in its ability to flow through a capillary without viscosity. Hence, a persistent flow of liquid can be maintained without any driving force. While the quantum agents of superconductivity are electrons, which are spin- 1/2 fermions, those of superfluidity are 4 He atoms, which are spin- 0 bosons. There are other systems able to display similar behavior as liquid 3 He, atomic nuclei and nuclear matter. In fact, superfluidity and superconductivity are new phases of matter related to the bosonic character of 4 He atoms or of pairs of fermions (either electrons, 3 He atoms or nucleons), the same as Bose-Einstein condensation in ultracold atomic gases. The temperatures where these new phases manifest range from 165 K for superconductivity down to a few nano-Kelvin for Bose-Einstein condensation.
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.
Superfluidity in liquid 3 He, discovered in 1972, is due to pairing, as is the case for superconductivity in ordinary metals. However, the pair is in the 3 P state so that a new aspect of superfluidity appears with internal degrees of freedom. It offers quite an interesting real example of ‘spontaneous symmetry breaking’. This aspect will be emphasised in the following discussion. Fermi liquid 3 He A 3 He atom is a fermion with nuclear spin ½. Although 3 He gas liquefies at low temperatures due to the weak van der Waals force, it does not solidify even at T = 0 unless the pressure exceeds 33.5 atm (3.4 × 10 6 Pa) as shown in the phase diagram (Fig. 6.1 (a)). While Bose liquid 4 He becomes superfluid at about 2K, 3 He is Fermi-degenerate at less than T F ∼ 0.1 K and shows the properties of a normal Fermi gas. For example, the specific heat and the susceptibility are proportional to T at T < T F . It is only at an ultra-low temperature in the mK-region that the system undergoes a transition to the superfluid state. Figure 6.1(b) is the phase diagram showing the superfluid phases with variables T,P and external magnetic field H in the mK-region. It should be noted that there are two thermodynamically distinct phases called the A phase and the B phase even at H = 0, and that for H ≠ 0 the A 1 phase appears.
The goal of this chapter will be to briefly describe the remarkable properties of helium at low temperatures. After stating some of these properties we will see how they can be understood in terms of the phenomenon of Bose–Einstein condensation described in Chapter 7. We will give the main argument in two different formulations, once using the quasi-particle method of Bogoliubov, and then using a Green function approach. We start with some experimental facts. Helium is a remarkable element. It was predicted to exist from observations of the Sun before it was found on Earth. It is the only element which remains a liquid at zero temperature and atmospheric pressure. Experimentally the phase diagram of 4 He is shown in Figure 10.1. Helium I is a normal fluid and has a normal gas-liquid critical point. Helium II is a mixture of a normal fluid and a superfluid. The superfluid is characterized by the vanishing of its viscosity. Helium I and helium II are separated by a line known as the λ -transition line. At T λ = 2.18 K, P λ = 2.29 Pa, helium I, helium II, and helium gas coexist. The specific heat of liquid helium along the vapor transition line forms a logarithmic discontinuity shown in Figure 10.2. The form of this diagram resembles the Greek letter λ and is the reason for calling the transition a λ -transition. The lack of viscosity of helium II leads to some remarkable experimental consequences, one of which we briefly describe. Let two containers A and B be linked by a thin capillary through which only a fluid with zero (or very low) viscosity can pass freely.
In the previous chapter, we discussed both the insulator–superconductor and the insulator–metal transitions. As such transitions are disorder or magnetic-field tuned, thermal fluctuations play no role. Phase transitions of the insulator–superconductor or insulator–metal type are called quantum phase transitions. Such phase transitions are not controlled by changing the temperature, as in the melting of ice or the λ-point of liquid helium, but rather by changing some system parameter, such as the number of defects or the concentration of charge carriers. In all such instances, the tuning parameter transforms the system between quantum mechanical states that either look different topologically (as in the transition between localized and extended electronic states) or have distinctly different magnetic properties. As quantum mechanics underlies such phase transitions, all quantum phase transitions obtain at the absolute zero of temperature and thus are governed by a T = 0 quantum critical point. While initially surprising, this state of affairs is expected, because quantum mechanics is explicitly a zero-temperature theory of matter. Of course, this is of no surprise to chemists who have known for quite some time that numerous materials can exhibit vastly distinct properties simply by changing the chemical composition and, most importantly, that such transformations persist down to zero temperature. Common examples include turning insulators such as the layered cuprates into superconductors simply by chemical doping or semiconductors into metals once again by doping, or ferromagnets such as Li(Ho, Tb) x Y 1- x F 4 into a spin glass (AR1998) by altering x .
Liquid helium is the prototypical example of a superfluid – a liquid that flows without viscosity and transfers heat without a temperature gradient. These properties are intimately related to the Bose condensation that occurs in this strongly interacting liquid. Bose condensation is most directly observed in the single particle atomic momentum distribution, where the Bose condensate appears as a delta function singularity. In this article, we discuss the experimental techniques used to observe the condensate and the current status of measurments of the Bose condensate in liquid helium. Introduction Liquid helium ( 4 He) has fascinated physicists ever since Kammerlingh-Onnes liquified the last of the so called permanent gases in 1908. However, evidence of what is without doubt the most fascinating property of this unique liquid, superfluidity, was not reported until almost 25 years later [1]. The superfluid phase, where heat is transferred without a thermal gradient and mass flows without a driving pressure, is a macroscopic manifestation of microscopic quantum effects governing the behavior of atoms [2]. Bose–Einstein condensation was first proposed by London [3] as the microscopic explanation for these fascinating phenomena. Helium is unique among condensed atomic systems since the bulk properties of the liquid are dominated by quantum effects. The most important of these are the statistical effects that arise for identical particles. For bosons, such as 4 He atoms, there is no limitation on the number of particles that can occupy a single quantum state.