The topic of this chapter is one of the most beautiful and profound concepts in QFT: the concept of criticality, renormalization and scaling. This concept is relevant for systems which have no natural intrinsic scale. In the theory we have been discussing so far, the O(N)-symmetric vector field model, there is such a scale: it is the inverse mass m−1. The correlation functions decay exponentially at distances (or time scales) larger then m−1. Mean while there are extrinsic scales both large and small in the theory: the large scales are 1/T and the size of the system L, and the small scale is a, the lattice size. If we have an intrinsic scale we can apply it to all external agents acting on the system and make qualitative estimates of their strength. However, imagine that m → 0; it seems that the system becomes self-similar on all scales.

Let me give an example of what I mean by self-similar. This example is not taken from physics, but from social institutions. Imagine such a venerable system as an army. An army has a hierarchical structure represented as a collection of units of different size. All this structure is united under the principles of seniority of ranks and subordination of lower ranks to the senior. These principles form a pattern for relations between army personnel inside units of any size. This pattern is very similar for relations between a soldier and a sergeant inside the smallest unit and for relations between a lieutenant of this unit and a captain of the platoon, and between this captain and his colonel, etc.

The objective of this book is to familiarize the reader with the recent achievements of quantum field theory (henceforth abbreviated as QFT). The book is oriented primarily towards condensed matter physicists but, I hope, can be of some interest to physicists in other fields. In the last fifteen years QFT has advanced greatly and changed its language and style. Alas, the fruits of this rapid progress are still unavailable to the vast democratic majority of graduate students, postdoctoral fellows, and even those senior researchers who have not participated directly in this change. This cultural gap is a great obstacle to the communication of ideas in the condensed matter community. The only way to reduce this is to have as many books covering these new achievements as possible. A few good books already exist; these are cited in the select bibliography at the end of the book. Having studied them I found, however, that there was still room for my humble contribution. In the process of writing I have tried to keep things as simple as possible; the amount of formalism is reduced to a minimum. Again, in order to make life easier for the newcomer, I begin the discussion with such traditional subjects as path integrals and Feynman diagrams. It is assumed, however, that the reader is already familiar with these subjects and the corresponding chapters are intended to refresh the memory. I would recommend those who are just starting their research in this area to read the first chapters in parallel with some introductory course in QFT. There are plenty of such courses, including the evergreen book by Abrikosov, Gorkov and Dzyaloshinsky.

It is quite beyond the purpose of this book to give a comprehensive review of the modern theory of metals. I shall touch only on those topics which are related to the mainstream of this course: the problem of strong interactions. As I have noted before, strong interactions can appear as a result of renormalization, and from the aesthetic point of view this is, perhaps, the most interesting case. As we shall see in Part IV, such renormalizations are particularly strong in one-dimensional systems, where practically any interaction leads to quite dramatic effects. Therefore a one-dimensional electron gas neither undergoes a phase transition (this is forbidden due to the low dimensionality), nor behaves like a system of free electrons. On the contrary, in higher dimensions at low temperatures a system of electrons either undergoes a phase transition (superconductivity, magnetic ordering, etc.), or behaves like a free electron gas.

When temperatures as low as several degrees became experimentally available, physicists discovered that an enormous amount of experimental data on normal metals can be described by the model where one neglects electron–electron interactions. This apparent miracle was explained by Landau who demonstrated that the interaction pattern drastically simplifies close to the Fermi surface. Provided the system does not undergo a symmetry breaking phase transition, all interactions except forward scattering effectively vanish on the Fermi surface and low-lying excitations carry quantum numbers of electrons. Therefore they are called quasi-particles. The meaning of the word quasi will become clear in a moment.

In this chapter we consider several typical diagrams in the diagram expansion of the theory (5.1) and discuss certain general problems concerning perturbation expansions in QFT.

The correlation functions are represented as infinite sums and the first thing one should do is to check whether these sums are convergent. But it turns out that even individual members of these sums are often divergent! As we shall learn, there are divergences of two sorts: ultraviolet and infrared. In the first case the integrals diverge at large frequencies and momenta, and in the second case at small. Ultraviolet divergences appear in those field theories where the bare particle spectrum is unbounded. Their removal presents a severe ideological problem in high energy physics, where the unbounded spectrum ∈2 = p2c2 + (mc2)2 follows from the Lorentz invariance. In models of condensed matter physics, ultraviolet divergences present not a problem but a nuisance. Their presence indicates that the continuum description is incomplete, i.e. the behaviour of long-wavelength excitations depends on shorter length scales. It is certainly a nuisance, because the description becomes nonuniversal. Usually it is much easier to write down a continuous field theory; for this purpose one can employ quite general arguments, ones based on symmetry requirements, for example. It is really a disappointment when this beautiful castle built from pure ideas crumbles, and it turns out that a realistic description requires a careful study of processes occurring on the lattice scale.

In this chapter we continue to study the massless scalar field described by the action (22.1). We have seen that this model has a gapless excitation spectrum and the correlation functions of bosonic exponents followpower laws. This behaviour implies that the correlation length is infinite and the system is in its critical phase. It is certainly very important to knowhowstable this critical point is with respect to perturbations. Any model is only an idealization; when one derives it certain interactions are neglected. Howdoes one decide which interactions are important and which are not? Obviously, weak interactions are those whose influence on the correlation functions is small. The trouble is, however, that usually the correlation functions are affected differently on different scales, the long distance asymptotics being affected the most. Therefore it can happen that a certain perturbation causes only tiny changes at short distances, but changes the large distance behaviour profoundly. In the renormalization group picture this is observed as a growth of the coupling constant associated with the perturbing operator. Such growth is a frequent phenomenon in critical theories; a slow decay of their correlation functions gives rise to divergences in their diagram series.

Operators whose influence grows on large scales (small momenta) are called ‘relevant’. The problem of relevancy of perturbations can be formulated and solved in a general form. Suppose we have a system at criticality (i.e. all its correlation functions decay as a power law at large distances).