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.

Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived.

Phase transitions and order parameters

It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary.

Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 1023 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.

Partition function for interacting bosons

As a prototypical system with a continuous phase transition we will consider the system of interacting bosons. A well studied physical realization is provided by helium (4He) with the pressure–temperature phase diagram as depicted in Fig. 2.1. Since the atoms of helium are light and interact via weak dipole–dipole interactions, due to quantum zero-point motion helium stays liquid down to the lowest temperatures, at not too high pressures. Instead of solidifying it suffers a continuous normal liquid–superfluid liquid transition at Tc ≈ 2K, also called the λ-transition due to the characteristic form of the specific heat in Fig. 1.6. The λ-transition represents the best quantitatively understood critical point in nature. We have already quoted the specific heat exponent α = -0.0127 ± 0.0003, with the power-law behavior being observed over six decades of the reduced temperature! To achieve this accuracy the experiment had to be performed in the space shuttle so that the small variations in Tc along the height of the sample due to Earth's gravity would be minimized. At higher pressures He eventually solidifies, with the superfluid–solid and the normal liquid–solid phase transitions being discontinuous, the former being so rather weakly.

It has been more than thirty years since the theory of universal behavior of matter near the points of continuous phase transitions was formulated. Since then the principles and the techniques of the theory of such “critical phenomena” have pervaded modern physics. The basic tenets of our understanding of phase transitions, the concepts of scaling and of the renormalization group, have been found to be useful well beyond their original domain, and today constitute some of our basic tools for thinking about systems with many interacting degrees of freedom. When applied to the original problem of continuous phase transitions in liquids, magnets, and superfluids, the theory is in remarkable agreement with measurements, and often even ahead of experiment in precision. For this reason alone the theory of critical phenomena would have to be considered a truly phenomenal physical theory, and ranked as one of the highest achievements of twentieth century physics.

The book before you originated in part from the courses on theory of phase transitions and renormalization group I taught to graduate students at Simon Fraser University. The students typically had a solid prior knowledge of statistical mechanics, and thus had some familiarity with the notions of phase transitions and of the mean-field theory, both being commonly taught nowadays as parts of a graduate course on the subject. In selecting the material and in gauging the technical level of the lectures I had in mind a student who not only wanted to become familiar with the basic concepts of the theory of critical phenomena, but also to learn how to actually use it to explain and compute.