The cellular automata approach and the related modeling techniques are powerful methods to describe, understand and simulate the behavior of complex systems. The aim of this book is to provide a pedagogical and self-contained introduction to this field and also to introduce recent developments. Our main goal is to present the fundamental theoretical concepts necessary for a researcher to address advanced applications in physics and other scientific areas.

In particular, this book discusses the use of cellular automata in the framework of equilibrium and nonequilibrium statistical physics and in application-oriented problems. The basic ideas and concepts are illustrated on simple examples so as to highlight the method. A selected bibliography is provided in order to guide the reader through this expanding field.

Several relevant domains of application have been mentioned only through references to the bibliography, or are treated superficially. This is not because we feel these topics are less important but, rather, because a somewhat subjective selection was necessary according to the scope of the book. Nevertheless, we think that the topics we have covered are significant enough to give a fair idea of how the cellular automata technique may be applied to other systems.

This book is written for researchers and students working in statistical physics, solid state physics, chemical physics and computer science, and anyone interested in modeling complex systems. A glossary is included to give a definition of several technical terms that are frequently used throughout the text. At the end of the first six chapters, a selection of problems is given.

The recent observation of Bose–Einstein condensation in a polarised alkaline gas is of major significance to the content of this book. Although opinions may differ as to the significance of the observation itself to the physics, everybody must have applauded the impressive success achieved by the novel techniques. Thanks to the discovery which provided us with concrete examples of a dilute Bose gas discussed in Section 1.2, the macroscopic quantum wave function, the central concept in understanding superconductivity and superfluidity, now seems quite familiar to us. Although this subject has been treated in the Appendix of [E-11], we will briefly discuss it here from the viewpoint of the physics of the condensate. We hope that this is read together with the Appendix of [E-11].

The Nobel prize in physics for 1996 was awarded to D.D. Osheroff, R.C. Richardson and D.M. Lee for the discovery of superfluid 3He. The research on superfluidity due to 3P pairing, initiated by this discovery, taught us that the physics of pairing is quite rich. The lessons learned here have been helpful in research on high temperature superconductors, heavy electron systems and, moreover, superconductivity in hadronic matter.

One of the interesting developments in the research on HTSC is an attempt to observe directly the pairing type, now considered likely to be d. This will be discussed in Appendix A2.1. Another unique example of the physics of HTSC is its behaviour in a magnetic field, involving vortex lattice, vortex liquid and glass.

The book consists of two parts. The first part (Chapters 2–4) is purely theoretical; the second part (Chapters 5–7) is mostly experimental. Chapter 2 contains the general theory of instantons and tunneling with dissipation, which is necessary for understanding the rest of the book. Chapters 3 and 4 deal with magnetic tunneling in single-domain particles and bulk materials, respectively. In Chapter 5, the consequences of tunneling for magnetic relaxation are derived and applied to experiments. Non-relaxation experiments are discussed in Chapter 6. Data on resonant spin tunneling in Mn12Ac, and their interpretation, are presented in Chapter 7. In selecting material for this book we were guided by the principle that the theory must be relevant to experiment while experiments must be relevant to magnetic tunneling. Time may prove that some are not. Lengthy theoretical formulas which are difficult to compare with experiments and experimental data the analysis of which is too complicated have been left out. No doubt, there are important results that did not enter the book because we failed to appreciate their significance. Our list of references serves the single purpose of refering the reader to the original papers that we know and understand; in no way does it constitute a complete list of important works on magnetic tunneling.

The work on this book would have been difficult or impossible without support from the National Science Foundation of the USA, the Spanish and Catalan governments, and the Banco Bilbao Vizcaya.

Traditionally, the physics of magnetism has been divided into two almost independent branches. The first branch deals with interactions at the atomic level. When applied to the three-dimensional world, it mostly consists of challenging unsolved problems like the Ising, Habbard, and Heisenberg Hamiltonians, Fermi-liquid models of itinerant magnetism, etc. It is concerned with the ground state and excitations relevant to small scales, typically a few tens of atomic lattice spacings. The Heisenberg ferromagnetic exchange, for example, leads to the formation of a constant local spin density and the quadratic dependence of the energy of short-wavelength spin excitations on the momentum. This has certain consequences for the saturation magnetization, its temperature dependence, the magnetic contribution to the specific heat, neutron diffraction, etc. However, when one turns to the properties of a magnet on a mesoscopic scale of 100 Å and greater, most of what can be derived or predicted from the microscopic theory becomes irrelevant. For instance, the practical question of how a piece of iron magnetizes in an applied magnetic field has nothing to do with the Heisenberg Hamiltonian. This is because processes observed on the mesoscopic and macroscopic scales result from weak interactions unaccounted for in simple quantum-mechanical models. They are the magnetic anisotropy owing to the symmetry of the crystal, the magnetic dipole interaction that breaks the magnet into magnetic domains, interactions of domain walls with defects, impurities, itinerant electrons, etc.

Magnetic molecules represent another interesting area for MQT studies. In this chapter we will describe experiments on spin tunneling in macroscopic crystals of Mn12Ac molecules. A Mn12Ac molecule has spin 10 and is equivalent to a very small single-domain magnetic particle. A crystal consists of a macroscopic number of molecules arranged in a tetragonal lattice. Spins of the molecules interact weakly with each other and are subject to a very strong uniaxial anisotropy. A Mn12Ac crystal is, therefore, equivalent to a large system of identical uniformly oriented magnetic particles, which has been wanted so much by experimentalists. Experiments on Mn12Ac [16] are the first experiments on spin tunneling which have been understood quantitatively without any free parameters. Whether the moment of the molecule, 20μB, is macroscopic enough to allow one to talk about MQT is left to the judgement of the reader.

The Mn-12 acetate complex

The chemical formula of the manganese acetate complex discussed below is [Mn12O12(CH3COO)16(H2O)4]·2CH3COOH·4H2O. We will denote it by Mn12Ac. This compound was first synthesized by Liz and characterized by Xray diffraction methods [128]. It forms a molecular crystal of tetragonal symmetry with the lattice parameters a = 1.732 nm and b = 1.239 nm. The unit cell contains two Mn12O12 molecules surrounded by four water molecules and two acetic acid molecules. The structure of the Mn12O12 molecule is shown in Fig. 7.1. The molecule possesses S4 symmetry.

An overview of the problem of spin tunneling

In this chapter we shall apply conceptions from the previous chapter to tunneling of the magnetic moment. We will begin with the problem of spin tunneling in a nanometer-size single-domain ferromagnetic particle. The particle will be considered small enough that one can treat its moment as a fixed-length vector that can only rotate as a whole. This approximation, which should be good in some but not all cases, is discussed in some detail below.

The absolute value of the magnetization in ferromagnets, M0, defined as the magnetic moment of 1 cm3 of the material, is formed by the strong exchange interaction between individual spins. This interaction has its origin in the nondiagonal (in spin) matrix elements of the Coulomb interaction between electrons of neighboring atoms. For that reason the energy, ∈ex, required to rotate one atomic spin with respect to its neighbors is of the order of 1 eV times a factor proportional to the overlap of the corresponding wave functions. A good estimate of that energy is provided by the Curie temperature. If one takes a material that is ferromagnetic at room temperature and cools it down to a few kelvins, the probability that any individual atomic spin flips with respect to other spins is proportional to exp (–∈ex/T), that is, exponentially small. For that reason, at low temperature, treating the magnetization of a bulk ferromagnet as a fixed-length vector must always be a good approximation.

The physics of slow relaxation in solids

At a given temperature any magnetic system has a certain state that corresponds to the absolute minimum of its free energy. For a bulk ferromagnetic crystal it is a certain configuration of domains. For a system of interacting monodomain particles, it is a certain orientation of individual magnetic moments. For a superconductor it is a certain structure of the flux-line lattice, etc. In practice, however, the minimum-energy state is difficult to achieve because there are various metastable states. In spin glasses this situation is generic: the complexity of the potential makes the energy minimum physically unattainable even in a microscopic volume. In contrast, in conventional magnetic crystals the spin structure on the scale of 100–1000 Å is usually the one that minimizes local interactions. On bigger scales, however, even well-ordered systems begin to exhibit metastability. This is because grain boundaries, dislocations, impurities, etc., do not allow domain walls to move freely in order to establish the configuration of domains that corresponds to the true energy minimum. Pinning of flux lines produces a similar effect in superconductors. A separate case is a system of monodomain particles embedded within a non-magnetic solid matrix. Here each particle can be in a metastable state due to energy barriers produced by the magnetic anisotropy.

The presence of metastable states results in magnetic hysteresis. All magnetic systems that exhibit hysteresis are expected to relax slowly toward the minimum of the free energy. Note that metastability is common among non-magnetic solids as well.