In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators âk. There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation.

Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general.

In Chapter 5, ø4-theory was introduced as an archetypal model of interacting continuum theories. Motivated by the existence of nonlinearities inherent in the model, a full perturbative scheme was developed, namely Wick contractions and their diagrammatic implementation. However, from a critical perspective, one may say that such perturbative approaches present only a limited understanding. Firstly, the validity of the ø4-action as a useful model theory was left unjustified, i.e. the ø4-continuum description was obtained as a gradient expansion of, in that case, a d-dimensional Ising model. But what controls the validity of the low-order expansion? Indeed, the same question could be applied to any one of the many continuum approximations we have performed throughout the first chapters of the text.

In previous chapters we have emphasized repeatedly that the majority of many-particle problems cannot be solved in closed form. Therefore, in general, one is compelled to think about approximation strategies. One promising ansatz leans on the fact that, when approaching the low-temperature physics of a many-particle system, we often have some idea, however vague, of its preferred states and/or its low-energy excitations. One may then set out to explore the system by using these prospective ground state configurations as a working platform. For example, one might expand the Hamiltonian in the vicinity of the reference state and check that, indeed, the residual “perturbations” acting in the low-energy sector of the Hilbert space are weak and can be dealt with by some kind of approximate expansion. Consider, for example, the quantum Heisenberg magnet. In dimensions higher than one, an exact solution of this system is out of the question. However, we know (or, more conservatively, “expect”) that, at zero temperature, the spins will be frozen into configurations aligned along some (domain-wise) constant magnetisation axes.

In the past few decades, the field of quantum condensed matter physics has seen rapid and, at times, almost revolutionary development. Undoubtedly, the success of the field owes much to ground-breaking advances in experiment: already the controlled fabrication of phase coherent electron devices on the nanoscale is commonplace (if not yet routine), while the realization of ultra–cold atomic gases presents a new arena in which to explore strong interaction and condensation phenomena in Fermi and Bose systems. These, along with many other examples, have opened entirely new perspectives on the quantum physics of many-particle systems. Yet, important as it is, experimental progress alone does not, perhaps, fully explain the appeal of modern condensed matter physics. Indeed, in concert with these experimental developments, there has been a “quiet revolution” in condensed matter theory, which has seen phenomena in seemingly quite different systems united by common physical mechanisms. This relentless “unification” of condensed matter theory, which has drawn increasingly on the language of low-energy quantum field theory, betrays the astonishing degree of universality, not fully appreciated in the early literature.

On a truly microscopic level, all forms of quantum matter can be formulated as a many body Hamiltonian encoding the fundamental interactions of the constituent particles. However, in contrast with many other areas of physics, in practically all cases of interest in condensed matter the structure of this operator conveys as much information about the properties of the system as, say, the knowledge of the basic chemical constituents tells us about the behavior of a living organism! Rather, in the condensed matter environment, it has been a long-standing tenet that the degrees of freedom relevant to the low-energy properties of a system are very often not the microscopic.

In the previous chapters we have introduced important elements of the theory of quantum many-body systems. Perhaps most importantly, we have learned how to map the basic microscopic representations of many-body systems onto effective low-energy models. However, to actually test the power of these theories, we need to understand how they can be related to experiment. This will be the principal subject of the present chapter.

Modern condensed matter physics benefits from a plethora of sophisticated and highly refined techniques of experimental analysis including the following: electric and thermal transport; neutron, electron, Raman, and X-ray scattering; calorimetric measurements; induction experiments; and many more (for a short glossary of prominent experimental techniques, see Section 7.1.2 below). While a comprehensive discussion of modern experimental condensed matter would reach well beyond the scope of the present text, it is certainly profitable to attempt an identification of some structures common to most experimental work in many-body physics.

In the preceding chapters we encountered a wide range of long-range orders, or, to put it more technically, different types of mean–fields. Reflecting the feature of (average) translational invariance, the large majority of these mean-fields turned out to be spatially homogeneous. However, there have also been a number of exceptions: under certain conditions, mean-field configurations with long-range spatial textures are likely to form. One mechanism driving the formation of inhomogeneities is the perpetual competition of energy and entropy: being in conflict with the (average) translational invariance of extended systems, a spatially non-uniform mean-field is energetically costly. On the other hand, this very “disadvantage” implies a state of lowered degree of order, or higher entropy. (Remember, for example, instanton formation in the quantum double well: although energetically unfavorable, once it has been created it can occur at any “time,” which brings about a huge entropic factor.) It then depends on the spatio-temporal extension of the system whether or not an entropic proliferation of mean-field textures is favorable.

Even minor departures from equilibrium call for a theoretical description entirely different from the ‘Matsubara formalism’ developed in earlier chapters of this book. The applicability of the latter is rigidly tied to the existence of a quantum grand canonical density operator, the hallmark of a many particle equilibrium system. But how then, can an functional theory of nonequilibrium quantum systems be developed? An obvious idea would be to once more go through the different elements of classical nonequilibrium theory developed in the previous chapter, and subject every one of them to an individual quantization scheme. Luckily, there are more efficient ways to achieve our goal: some decades ago, a many particle formalism suitable to describe nonequilibrium systems under the most general conditions was introduced by Keldysh.

As mentioned in Chapter 5, the perturbative machinery is but one part of a larger framework. In fact, the diagrammatic series already contained hints indicating that a straightforward expansion of a theory in the interaction operator might not always be an optimal strategy: all previous examples that contained a large parameter “N” – and usually it is only problems of this type that are amenable to controlled analytical analysis – shared the property that the diagrammatic analysis bore structures of “higher complexity.” (For example, series of polarization operators appeared rather than series of the elementary Green functions, etc.) This phenomenon suggests that large-N problems should qualify for a more efficient and, indeed, a more physical formulation.

The world is a place full of nonequilibrium phenomena: an avalanche sliding down a sandpile, a traffic jam forming at rush hour, the dynamics of electrons inside a strongly voltage-biased electronic device, the turmoil of markets following an economic instability, the diffusion-limited reaction of chemical constituents and many others are examples of situations where a large number of correlated constituents evolves under “out-of-equilibrium” conditions. Statistical nonequilibrium physics is concerned with the identification and understanding of universal structures that characterise these phenomena. Notwithstanding the existence of powerful principles of unification, it is clear that a theory addressing the dazzling multitude of nonequilibrium phenomena must be multi-faceted. And indeed, nonequilibrium statistical physics is a strongly diversified field comprising many independent sub-disciplines. But this is not the only remarkable feature of this branch of physics.