Winston Smith, the main character in George Orwell's apocalyptic novel 1984, perceived while reading the secret Brotherhood's book that the best books are those that tell you what you know already. If you are encountering the field of molecular organic materials for the first time and do not want to contradict Orwell, you are invited to read the book you are holding in your hands more than once. If you are an expert in the field I hope you will discover new and perhaps unexpected issues.

This book, my opera prima, was conceived for both beginner and experienced chemists, physicists and material scientists interested in the amazing field of molecular organic materials. Some basic notions of solid-state physics and chemistry and of quantum mechanics are required, but the book is written trying to reach a broad multidisciplinary audience.

When entering this field for the first time one is faced with myriad names and materials and it becomes an arduous task to find one's way. This volume is based on my own experience, aware of the difficulty that non-specialists encounter.

Introduction: simple antiferromagnets

In the previous chapter we showed how to derive perturbation expansions for quantum spin models at T = 0, for both ground state bulk properties and for excitations. We used, as an example, the transverse field Ising model. This is arguably the simplest model to show a quantum phase transition. Moreover the model has a natural perturbation parameter, Γ/J or J/Γ, and the series are relatively straightforward to derive and interpret.

In the present chapter we turn to quantum antiferromagnets. These are systems in which the exchange interaction favours anti-alignment of neighbouring spins. The physics will turn out to be much richer, and we will use series methods to explore not only simple antiferromagnets on bipartite lattices but also the effects of frustration, competing interactions, destruction of antiferromagnetism due to singlet formation on dimers or plaquettes, and other related topics. With some minor modifications the techniques are the same as in Chapter 4. In this chapter, however, we will provide fewer details and concentrate more on results.

The Ising expansion

Let us start by considering a spin-½ system on a bipartite lattice, with nearest-neighbour antiferromagnetic exchange. A bipartite lattice is one which consists of two interpenetrating sub-lattices A, B such that the nearest neighbours of A spins all lie on sub-lattice B and vice versa. Examples are the linear chain, the square and honeycomb lattices in two dimensions, the simple cubic and body-centred cubic lattices in three dimensions.

This appendix provides a brief introduction to topics in the mathematical field of Graph Theory, which are pertinent to the subject of series expansions. For further details the reader is referred to Domb (1974) and to Chartrand (1977).

We start with some definitions.

1. (i) A graph is a collection of points (vertices) and lines (bonds) (see Figure A1.1).

2. (ii) A connected graph is one in which there is a path between any pair of points. A disconnected graph is one which is not connected. The number of components of a disconnected graph can be 2, 3, …

3. (iii) An articulation point (articulation vertex) is a vertex, the removal of which, with all of its incident lines, breaks the connectivity of the graph.

4. (iV) The order (degree) of a vertex is the number of lines incident on the vertex. Note that if a vertex is of order 1, then the vertex to which it is joined is an articulation vertex.

5. (v) A star graph is a connected graph with no articulation points.

6. (vi) A tree graph is a connected graph with at least one vertex of order 1 (Note that this differs from more usual definitions, but is most convenient for our purposes).

7. (vii) Asimple graph is one in which there is at most one line joining any pair of vertices. A multi-graph is one in which there is more than one line between at least one pair of vertices.

8. […]

Various computer programs have been referred to in the text and used to obtain some of the results presented in the preceding chapters. These programs can be accessed at www.cambridge.org/978052184242. This Appendix lists the various programs, together with a paragraph or two about each one.

The programs are written in fairly old-fashioned FORTRAN, and have been tested and used in our work over the years. They are relatively efficient and, as a result, not always as transparent as they might be. We attempt to point out particularly subtle sections when these occur. However we do not claim the ultimate in efficiency or complete freedom from ‘bugs’. Readers are invited to let us know of any.

A few comments on programming style are worth making. Some may prefer a single program of tens of thousands of lines which essentially does everything and covers all possible cases. In our view this is generally both inefficient and is difficult to adapt to particular problems. Our preference is for maximum modularity where a problem is broken down into different parts which are performed sequentially. This does lead to greater overheads, requiring storage of intermediate results, but has major advantages in efficiency, flexibility, and transparency.

The programs dealing with graph generation and counting are largely based on programs written by Dr C. J. (Chuck) Elliott at the University of Alberta in the early 1970s and we gratefully acknowledge this.

Lattice models in theoretical physics

A major part of theoretical physics involves the construction and systematic analysis of mathematical models as a description of the physical world. For microscopic phenomena, in particular, the reference to models becomes quite explicit. The phenomena are often complex, and any theory that attempts to include every detail soon becomes intractable. It is more fruitful to develop models which ignore irrelevant details but, hopefully, capture the essential physics of the phenomena of interest. Thus we have the Heisenberg model of magnetic order, the Bardeen–Cooper– Schrieffer (BCS) model of superconductivity, and so on. It is important to note, at the outset, that the models we shall be discussing describe strongly interacting, and therefore highly correlated, systems of particles. These are difficult and interesting problems. Where interactions are absent, or weak, elementary treatments are possible and the resulting phenomena are generally unspectacular.

Why lattice models? In solid-state phenomena there is usually an underlying lattice structure, and the symmetry properties of this lattice play an important role in the analysis. Even the process of electrical conduction in metals or semiconductors can be equally well described in terms of localized quantum states or in terms of the more usual continuum picture. In quantum field theory, which is formulated in a space–time continuum, a lattice is often introduced for computational purposes. One can think of this in two ways: as an approximation to the continuum, which is then recovered as a limit at the end of the calculations, or as a necessary means of regulating the theory (i.e. controlling divergences) in calculating the Feynman path integral.

Introduction

In the limit of zero temperature, the partition function is dominated by the ground state of the system, the free energy becomes equivalent to the ground state energy, and the central object of interest is the Hamiltonian of the system H(λ), dependent in general on one or more parameters λ. A ‘quantum’ Hamiltonian, as noted previously, is one which contains non-commuting operators. Here we concentrate on quantum spin Hamiltonians, of which the prime example is the Heisenberg model. Of particular interest is the possibility of a quantum phase transition, i.e. nonanalytic behaviour as a function of coupling λ. A quantum phase transition may exert a dominating influence on the system over a range of low temperatures above T = 0. Such phenomena have figured prominently in theoretical discussions of experiments on the cuprate superconductors, the heavy fermion materials, organic conductors, and related materials (e.g. Sachdev, 1999).

In some cases, a correspondence can be found between a quantum system at zero temperature in 1 time and (d – 1) space dimensions, and an equivalent classical system at finite temperature in d space dimensions, based on the Feynman path integral formalism – see Chapter 9 and Appendix 8 for further discussions. Under this mapping, the temperature kT in the classical system corresponds to a coupling λ in the equivalent quantum system. If a phase transition occurs, the same universal critical exponents are expected to apply in both cases.

Introduction

In the preceding chapters we have developed a number of series expansion methods, and shown how to use these to study lattice spin models of various kinds. While this encompasses a great deal of interesting physics it does not include perhaps the richest, most interesting, and most challenging area of all: strongly correlated electron systems. Of particular interest here are the cuprate high Tc superconductors and the manganite ‘colossal magnetoresistance’ (CMR) materials. These systems are far from fully understood but it seems clear that a complex and subtle interplay between charge, spin, and perhaps other degrees of freedom plays a vital role.

A number of generic lattice models for strongly correlated electron systems exist. Besides the Hubbard model, which was briefly introduced in Chapter 1, there is its derivative the ‘t–J model’, the Anderson model, the Kondo lattice model, the Falicov–Kimball model, and others. These all present difficult problems, with little in the way of exact or rigorous results. Many approximate approaches, both analytical and numerical, have been used. In this chapter we will discuss the application of series methods, both at T = 0 and at finite temperatures, to some of these models. Some successes have been achieved but, generally speaking, the series are shorter and the analysis more problematic than for spin models. A particular difficulty, for ground-state calculations, is that series are not well suited to studying continuously variable electron density, although we will illustrate one attempt to overcome this.

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.