The past 50 years have seen much progress in our understanding of the behaviour of complex physical systems, made up of large numbers of strongly interacting particles. This includes a rather detailed, if not complete, understanding of such phenomena as phase transitions of various kinds, macroscopic quantum phenomena such as magnetic order and superconductivity, and the response of such systems to external probes, vital for the interpretation of experimental results. Such unifying concepts as scaling and universality, long-range order (including off-diagonal, long-range order), and spontaneous symmetry breaking have led to a unified understanding of diverse and complex phenomena.

Central to this endeavour has been the detailed and systematic study of lattice models of various genera models which are precisely defined mathematically, which are believed to embody the essential physics of interest, and which are, to a greater or lesser extent, mathematically tractable. Exact analytic treatment of these models is rarely possible. Series expansion techniques, the subject of this book, provide one of the main systematic and powerful approximate methods to treat such lattice models. Our decision to write this book arose from a request from a journal editor to write a review of our group's work over the last decade on series studies of quantum lattice models. On reflection, we came to the view that it would be more useful to write a book covering the entire field, at a level which would be accessible to graduate students and other researchers wishing to learn about these methods.

Disordered systems

The physics of systems with a large degree of random structural disorder is a large and fascinating field (see e.g. Ziman, 1979).We will describe, in this section, three major areas of work where series expansion methods have been successfully used. The discussion will be quite brief but should suffice to give the reader a flavour of the field and a guide to possible future work. We will restrict the discussion, at the outset, in two important ways. Firstly, we will consider only lattice systems, leaving aside the important areas of amorphous and ‘glassy’ systems. Secondly, we will only consider the case of ‘quenched’ disorder, which is frozen into place when the system is created. Strictly speaking, such systems are not in complete thermodynamic equilibrium. However in a large system with short range interactions, all configurations are effectively sampled: the system is said to be ‘self averaging’. In model calculations, such as those discussed here, an average over different disordered configurations has to be taken.

Percolation

Let us consider a square lattice containing two kinds of atoms or sites arranged randomly.We will call them ‘white’ and ‘black’. They might, for example, be nonmagnetic and magnetic atoms in an alloy such as CuMn. Let us denote by p the fraction of black sites. As p is increased larger clusters of connected black sites will be expected and, at a critical probability or ‘percolation threshold’ pc, a black cluster of macroscopic (infinite) size will occur for the first time.

Introduction

In the previous chapters we showed how series expansion methods can be used to study ground state properties and elementary excitations in simple (and not so simple) antiferromagnets. In the present chapter we extend this to a consideration of more complex properties. We begin by discussing various spin-spin correlation functions, or ‘correlators’. These play an important role in characterizing the nature of the ground state and, as we shall show, can be used to locate quantum phase transition points. The correlators can be combined to form the dynamical twospin structure factors S(k, ω), which are measured in inelastic neutron scattering experiments. We will discuss how these can be computed via series methods, and show some results for the integrated (static) forms.

While one-particle excited states are usually the dominant excitations in quantum systems, there are situations in which two-particle excitations, including bound states, and more general multi-particle excitations also play an important role. There are a number of experimental probes which are beginning to show features associated with multi-particle continuum and bound states, including two-magnon Raman spectroscopy, photo-emission and inelastic neutron scattering. For example, Tennant et al. (2003) have measured two-magnon states in copper nitrate, a quasi one-dimensional antiferromagnet (see Section 6.4.4). Thus one can hope to build up a detailed picture of the dynamics of these quasiparticle excitations, and to construct an effective Hamiltonian to describe them. In the last part of the chapter we outline how one may calculate series expansions for the spectrum of multi-particle excitations in a quantum lattice model at T = 0.

This introductory chapter has two goals. The first one is usual for books of this kind and is aimed at providing the reader with a brief outline of the background history of the subject, its main content, form of presentation, and correlation with other related subjects. The second goal is to give this introduction in a way which allows the reader to get a general (although maybe rather rough and superficial) impression of the whole subject and its possibilities, a very brief insight into this trend without reading the corresponding chapters or sections. This is done keeping in mind that quite a number of physicists, chemists, and biologists, who at present are not engaged in the use of the Jahn–Teller effect, may be interested to know in general the status quo in this field and make a fast choice of the parts of it they may be interested in. In other words, the introduction is aimed at giving a very brief qualitative description of the main features of the Jahn–Teller effect theory in a way useful also for the reader who has no intention to read the whole book or its parts. The applications of the theory to chemical problems and molecular systems are given in Chapter 7, while the Jahn–Teller effect in specific solid-state problems is considered in Chapter 8.

This chapter is to introduce the reader to JT problems that may be considered as “less traditional” in the sense that they do not follow exactly from the original (“classical”) JT theorem. Indeed, the PJTE considered below takes place in nondegenerate states, while the RTE is relevant to linear molecules, which are exceptions in the original JT formulation. The product JTE is an approximation to both JT and PJT problems. All these effects are inalienable components of the JT trend as a whole.

Two-level and multilevel pseudo JT (PJT) problems. Uniqueness of the PJT origin of configuration instability and its bonding nature

Among the JT vibronic coupling effects the pseudo JT (PJT) effect occupies an outstanding place. Indeed, while the occurrence of the JT and RT effects is limited to polyatomic systems in degenerate electronic states, the PJT effect may take place, in principle, in any system without a priori limitations. This circumstance enlarges essentially the possible subjects of the JT vibronic coupling effects, transforming the JTE theory into a general tool for better understanding and solving problems of structure and properties of molecules and crystals. The waiver of the degeneracy restrictions together with the proof of the uniqueness of the PJT origin of structural instabilities of polyatomic high-symmetry configurations elevates the JT effect theory to a general approach to molecular and crystal problems.

The PJT effect emerges directly from the basic equations (2.6) for the vibronically coupled electronic states.

With this chapter we begin the applications of the general theory of the JTE to all-range spectroscopy, molecular structure, and solid-state physics (Chapters 6–8). Among them the JTE in spectral properties is the most significant due to the sensitivity of spectra to changes in electronic structure and vibronic coupling. On the other hand, the influence of vibronic coupling on spectra is very specific; it depends on both the system parameters and the spectral range under consideration. Still there are some general features common to all systems with the same JT problem.

This chapter is devoted to such a general theory of the JTE in spectroscopy. More particular problems are considered in Chapters 7 and 8, together with the corresponding specific systems, but the separation of the general theory from more system-oriented questions is to a large extent conventional. Obviously, there is also substantial overlap with calculations of vibronic states presented in Chapter 5: spectroscopy is inalienable from energies and wavefunctions of the system.

Electronic transitions

Optical band shapes

Optical band shapes with the JTE were subjected to multiple investigations by many authors (see the reviews [6.1–6.10] and references therein).

The term band shape means an envelope of elementary transitions between vibronic states, each of which has a specific width so that the vibronic lines merge into a continuous band.

The JTE in local properties of solids

Local properties in crystals are strongly affected by JT vibronic coupling phenomena, quite similar to molecular systems discussed in the previous chapter. There is no sharp border between clusters and coordination systems, considered in Sections 7.6.1 and 7.6.2, and local formations in crystals (which are often modeled by clusters). Differences occur for impurity centers for which the JT problem is in fact a multimode one (Sections 3.5 and 5.5) with a very large (infinite) number of JT active modes that are significantly involved in the vibronic coupling. Another important crystal problem emerges when there are multiple JT impurity centers at sufficiently small intercenter distances with sufficiently strong interaction between them. In the limit of an infinite number of such JT centers that occupy regular positions in the lattice we come to the cooperative JTE (CJTE) considered in Sections 8.2 and 8.3. The JTE in superconductivity and colossal magnetoresistance is discussed in Section 8.4.

Impurity centers in crystals

We start with a single JT center in a regular crystal in the form of an impurity center or a lattice defect that has localized electronic states. The latter may be either orbitally degenerate and subject to the JTE (when the crystal field of the environment has at least one axis of symmetry of third or higher order), or pseudodegenerate with a PJTE (for which there are no symmetry restrictions, but the ground state should be force-equilibrated in the reference configuration, Section 4.1).

The Jahn–Teller effect (JTE) is one of the most fascinating phenomena in modern physics and chemistry. It emerged in 1934 in a discussion between two famous physicists, L. Landau and E. Teller, and grew into a general tool for understanding and an approach to solving molecular and crystal problems, which is applicable to any polyatomic system. The first formulation of this effect as instability of molecular configurations in electronically degenerate states proved to be the beginning of a whole trend which rationalizes the origin of all possible instabilities of high-symmetry configurations, and the peculiar nuclear dynamics resulting from these instabilities as well as the origins of all structural symmetry breakings in molecular systems and condensed matter.

Intensive development of the JTE theory began in the late 1950s together with a wave of main applications to spectroscopy, stereochemistry, and structural phase transitions, which lasted a couple of decades. The next significant resurgence of interest in the Jahn–Teller effect is related to the late 1980s and is still continuing. It was triggered by one of the most important Nobel Prize discoveries in physics of our times inspired by the Jahn–Teller effect: the high-temperature superconductivity. As explained by the authors of this discovery, “the guiding idea in developing this concept was influenced by the Jahn–Teller polaron model” (J. G. Bednorz and K. A. Müller, in Nobel Lectures: Physics, Ed. G. Ekspong, World Scientific, Singapore, 1993, p. 424).

The main effects of JT vibronic couplings are due to the special dynamics of the nuclear configuration that follows from the JT instability. The energy levels and wavefunctions describing these effects are solutions of the system of coupled equations (2.6) in Section 2.1. They were obtained for the most important JT problems formulated in Chapters 3 and 4, and are discussed in this chapter.

Weak vibronic coupling, perturbation theory

Calculation of the energy spectrum and wavefunctions of a JT or PJT molecule as solutions of the coupled equations (2.6) is a very complicated problem which cannot be solved in a general form, without simplifications, for arbitrary systems. However, as in similar quantum-mechanical situations, analytical solutions for some limiting cases in combination with exact numerical solutions of some particular cases yield the general trends and provide understanding of the origin and mechanism of the phenomenon as a whole.

For vibronic problems the limiting cases of weak and strong vibronic coupling with relatively small and large vibronic coupling constants, respectively, can be solved analytically. A quantitative criterion of weak and strong coupling can be defined by comparing the JT stabilization energy with the zero-point energy of -fold degenerate vibrations. Denote. Then, if, the vibronic coupling will be regarded as weak, and if λΓ ≫ 1, the coupling is strong; λΓ is the dimensionless vibronic coupling constant.