In this chapter we return to the study of systems that are nearly degenerate. As illustrated in Figure 24.1, a degenerate system of N identical fermions fills the N lowest quantum states, one particle per state. For degenerate bosons, all are in the one single state of lowest energy.

A degenerate system is confined to a small volume in phase space because of either:

• restricted volume in momentum space owing to small masses, or low temperatures, or

• Restricted volume in coordinate space owing to high densities.

Important examples of each case will be studied in this chapter.

The measures of high densities, small masses, and low temperatures are on a relative scale and are interdependent. At earthly densities, many systems become degenerate only if temperatures descend to a few kelvins or lower. Yet these same systems may be degenerate at several million kelvins when squashed together in extremely dense collapsed stars. The conduction electrons in metals have the same density and temperature as the atoms. Yet because of the difference in their masses, the electrons are degenerate and the atoms are not.

At high temperatures and/or low densities, the particles of a system are surrounded by vacant quantum states into which they can move (Figure 24.1). This freedom enables them to give rather smooth and continuous responses to varying environmental conditions. But in degenerate systems, quantum effects are more visible. Imprisoned particles are unable to move into neighboring states, which makes the system unresponsive to external stimuli.

In Chapter 9 we showed that temperature governs thermal interactions, pressure governs mechanical interactions, and chemical potential governs diffusive interactions. They do this in ways that are so familiar to us that we call them “common sense”:

• thermal interaction. Heat flows towards lower temperature.

• mechanical interaction. Boundaries move towards lower pressure.

• diffusive interaction. Particles go towards lower chemical potential.

In this chapter we examine diffusive interactions, working closely with the chemical potential μ and the Gibbs free energy N μ.

The chemical potential

In Chapter 5 we learned that the equilibrium distribution of particles is determined by the fact that particles seek configurations of

• lower potential energy,

• lower particle concentration.

Although the first of these is familiar in our macroscopic world (e.g., balls roll downhill), the second is due to thermal motions, which are significant only in the microscopic world (Figure 14.1).

Both factors trace their influence to the second law. The number of states per particle, and hence the entropy of the system, increases with increased volume in either momentum space or position space. Deeper potential wells release kinetic energy, making available more volume in momentum space, Vp. And lower particle concentrations mean more volume per particle in position space, Vr.

The two factors are interdependent. The preference for regions of lower potential energy affects particle concentrations, and vice versa. There is a trade-off. The reduction in one must more than offset the gain in the other (Figure 14.2).

The Ginzburg–Landau theory describing the Meissner transition in superconductors is introduced, and two types of superconductor are defined. It is shown that fluctuations of the gauge field lead to first-order transition in type-I superconductors. Calculation near four dimensions is performed for type-II superconductors, and the dependence of the flow diagram on the number of components is discussed. Scaling of the correlation length and of the penetration depth near the transition is elaborated.

Meissner effect

Most elemental metals and many alloys go through a sharp phase transition in which the material becomes a perfect diamagnet at low magnetic fields and completely loses its electrical resistance when cooled down to temperatures of several kelvins (Fig. 4.1). Such a “superconducting” transition has now been observed at temperatures as high as ∼150 K, in materials known as high-temperature superconductors. Superconductivity is a closely related phenomenon to superfluidity in He, except that electrons are charged and as such carry electrical current. Even before the advance of the microscopic theory of superconductivity in metals and alloys, V. Ginzburg and L. Landau devised a phenomenological description of the transition and the superconducting state.

Idea

We found that the direct perturbation theory in the Ginzburg–Landau–Wilson theory breaks down below the upper critical dimension because the perturbation parameter grows with the correlation length, and so becomes arbitrarily large as the critical point is approached. If the system were finite, on the other hand, the correlation length would be bound by its size, and perturbation theory could succeed. Singular thermodynamic behavior near the critical point comes from the thermodynamic limit, or, more precisely, from those modes that have arbitrary low energies in an infinitely large system. This is called the infrared singularity. This observation suggests the following strategy to avoid the problem of direct perturbation theory.

First, note that mass m only provides the energy scale in Eq. (2.36). It is practical to rescale it out by absorbing it into the chemical potential, rede- fined as 2mμ/ħ2 → μ, and into the interaction coupling, as 2mλ/ħ2 → λ. Similarly, near the critical point, the temperature T ≈ Tc may be replaced by the critical temperature, and then eliminated by rescaling the action as 2mκBTcS/ħ2 → S.

Dynamical critical exponent

The finite temperature phase transitions studied in previous chapters are the result of the competition between the entropy and the energy terms in the free energy: the weight of entropy increases with temperature, and ultimately destroys the order that may be existing in the system. A sharp phase transition between two phases exhibiting qualitatively different correlations may, however, occur even at zero temperature, by varying a coupling constant in the Hamiltonian. The transition then corresponds to a non-analyticity of the energy of the ground state. A simple example is provided by the interacting bosons, where the superfluid transition may be brought about by tuning the chemical potential at T = 0. Such T = 0 phase transitions are called quantum phase transitions, and will be the subject of the present chapter.

In general, a quantum phase transition may lie at the end of a line of thermal phase transitions, as in the bosonic example mentioned above. It is possible, however, that the system may not even have an ordered state at finite temperatures, but still exhibits a quantum critical point. Two different situations are depicted in Fig. 8.1. Examples of such phase diagrams are provided by the system of interacting bosons which will be discussed in Sections 8.2 and 8.3.