The exact results presented in the previous chapter allow us to obtain the scaling exponents for d = 1, and reduce the number of independent scaling exponents to one. The same results can be obtained using the dynamic renormalization group method, which we now develop and use to study the scaling properties of the KPZ equation. In particular, we analyze the ‘flow equations’ and extract the exponents describing the KPZ universality class for d = 1. We also discuss numerical results leading to the values of the scaling exponents for higher dimensions.

Basic concepts

So far, we have argued that we can distinguish between various growth models based on the values of the scaling exponents α, β and z. The existence of universal scaling exponents and their calculation for various systems is a central problem of statistical mechanics. A main goal for many years has been to calculate the exponents for the Ising model, a simple spin model that captures the essential features of many magnetic systems. A major breakthrough occurred in 1971, when Wilson introduced the renormalization group (RG) method to permit a systematic calculation of the scaling exponents. Since then the RG has been applied successfully to a large number of interacting systems, by now becoming one of the standard tools of statistical mechanics and condensed matter physics. Depending on the mathematical technique used to obtain the scaling exponents, the RG methods can be partitioned into two main classes: real space and k-space (Fourier space) RG.

It is a truism to remark that no one – not even a theoretical physicist – can predict the future. Nonetheless, after asking the beleaguered reader to indulge in the rather extensive ‘banquet’ of the preceding 27 chapters, it seems only fair to offer a light ‘dessert’ that affords some outlook and perspective on this rapidly-evolving field.

What concepts loom above the details is a question worth addressing at the end of any large meal. Charles Kittel wrote his first edition of Introduction to Solid State Physics almost 50 years ago. He surely realized that solid state physics was a rapidly-evolving field, so his book ran the risk of becoming dated in short order. Therefore the first chapter systematically discusses the various crystal symmetries – and the group theory mathematics that describes these symmetries. The topics comprising solid state physics have changed rather dramatically, and most chapters of Kittel's 7th edition hardly resemble the chapters of the first edition. Nevertheless, the opening chapter of the first edition could serve as well today as an introduction to the essential underpinnings of the subject.

Inspired by Kittel's example, we have attempted in this short book to highlight where possible what seems to us to be the analog for disorderly surface growth of the various symmetries obeyed by crystalline materials. These newer ‘symmetries’, described using terms that may frighten the neophyte – such as scale invariance and self-affinity – are as straightforward to describe as translation, rotation, and inversion.

The increasing interest of researchers in the basic properties of growth processes has provided the initiative for a number of experimental studies designed to check the applicability of various theoretical ideas to experimental systems. While many experimental studies have been inspired by the KPZ theory, most have failed to provide support for the KPZ prediction that α = ½. Instead, most data suggest that α > ½. These experimental results initiated a closer look at the theory, and led to the discovery that quenched noise affects the scaling exponents in unexpected ways. In this chapter, we discuss some of these key experiments, including fluid-flow experiments, paper wetting, propagation of burning fronts, growth of bacterial colonies and paper tearing. Atom deposition in molecular beam epitaxy, which is one important class of experiments on kinetic roughening of interfaces, will be discussed in Chapter 12. The new theoretical ideas needed to understand the effect of atomic diffusion on the roughening process will be developed at that time.

Fluid flow in a porous medium

Two-phase fluid flow experiments have long been used to study various growth phenomena. The Hele–Shaw cell, well-known from studies on growth instabilities in viscous fingering, has proved to be a useful experimental setup for the study of the growth of selfaffine interfaces. A trypical setup used in these experiments (Fig. 11.1) is a thin horizontal Hele–Shaw cell made from two transparent plates.

Most of this book deals with local growth processes, for which the growth rate depends on the local properties of the interface. For example, the interface velocity in the BD model depends only on the height of the interface and its nearest neighbors. However, there are a number of systems for which nonlocal effects contribute to the interface morphology and growth velocity. Such growth processes cannot be described using local growth equations, such as the KPZ equation; if we attempt to do so, we must include nonlocal effects. In this chapter we discuss phenomena that lead to nonlocal effects, and we also discuss models describing nonlocal growth processes.

Diffusion-limited aggregation

Probably the most famous cluster growth model is diffusion-limited aggregation (DLA). The model is illustrated in Fig. 19.1. We fix a seed particle on a central lattice site and release another particle from a random position far from the seed. The released particle moves following a Brownian trajectory, until it reaches one of the four nearest neighbors of the seed, whereupon it sticks, forming a two-particle cluster. Next we release a new particle which can stick to any of the six perimeter sites of this two-particle cluster. This process is then iterated repeatedly. In Fig. 19.2, we show clusters resulting from the deposition of 5 × 105, 5 × 106, and 5 × 107 particles.

As discussed in the previous chapters, we can distinguish the various growth processes based on the concept of universality. Interfaces that belong to the same universality class are described by the same scaling exponents, and they are also described by the same continuum equation.

The universality class is determined by the physical processes taking place at the surface. There are three basic microscopic processes that can play a major role in this respect: deposition, desorption, and surface diffusion. In addition to these, nonlocal effects such as shadowing may play a decisive role in shaping the interface morphology. While deposition must occur, the other microscopic processes may be irrelevant or even absent altogether. For example, in many systems desorption is negligible, while at low temperatures surface diffusion may be negligible.

A number of recent experiments support the existence of kinetic roughening in various deposition processes. It is possible to measure both the roughness exponent α characterizing the interface morphology, and the exponent β quantifying the dynamics of the roughening process. However, the emerging picture is far from complete, and there is no unambiguous support for the various universality classes.

There are a number of reasons for this situation. First, it is only recently that experimental groups have initiated systematic investigations of the various roughening processes. While the results are quite encouraging, more work is needed to obtain a coherent picture. Second, due to the complicated nature of the competing effects discussed in the previous chapters, the interpretation of the data is often not straightforward.

Introduction

The discovery that scaling laws and continuum theories are applicable to molecular beam epitaxy (MBE) has generated increasing interest among both experimentalists and theorists. The closer study of these deposition processes reveals the decisive role played by surface diffusion of the deposited particles. From the experimental point of view, these studies re-focus attention on a neglected aspect of MBE growth processes: roughening of a growing interface.

There are two complementary approaches to crystal growth:

(a) Atomistic approaches, in which the position of every atom is well defined. Our knowledge of the behavior of individual atoms has increased due to the high resolution of scanning tunneling microscopy (STM). STM is capable of identifying not only the structure of the lattice, but the positions of the individual atoms as well. First principles calculations provide insight into the energetics of atomic motion on solid surfaces. Based on this detailed information, modeling of different growth processes on the atomic level is becoming a widely used tool to gain deeper insight on the collective nature of atomic motion and deposition processes.

(b) Continuum approaches view the interface on a coarse-grained scale, in which every property is averaged over a small volume containing many atoms. Neglecting the discrete nature of the growth process, continuum theories attempt to capture the essential mechanisms determining the growth morphology. Their predictive power is limited to length scales larger than the typical interatom distance, providing information on the collective nature of the growth process, such as the variation in the interface roughness or correlation functions.

Abstract

We review generally accepted definitions of Bose–Einstein condensation and superfluidity, emphasizing that they are independent concepts. These ideas are illustrated in a dilute hard-sphere Bose gas, which is relevant to experiments on excitons and spin-aligned atomic hydrogen. We then discuss superfluid He in porous media, as simulated by different models in different regimes. At low coverage, we model it by a dilute hard-sphere Bose gas in random potentials, and show that superfluidity is destroyed through the pinning of the Bose condensate by the external potentials. At full coverage, we model the random medium by an ohmic network of random resistors, and argue that the superfluid transition is a percolation transition in d = 3, with critical exponent 1.7.

This book is devoted to the phenomenon of Bose–Einstein condensation [1, 2] and inevitably, its relevance to superfluidity [3]. To provide some background for other articles in this volume, I would like to summarize some commonly accepted views on these phenomena, and illustrate them in the context of a dilute hard-sphere Bose gas, a model in which we have some control over the approximations made. I will also describe some recent work on the effect of randomness on the Bose condensate, which shows that Bose–Einstein condensation does not automatically give rise to superfluidity.

Abstract

We present a simple review of the basic physical processes allowing one to control, with laser beams, the velocity and the position of neutral atoms. The control of the velocity corresponds to a cooling of atoms, that is, a reduction of the atomic velocity spread around a given value. The control of the position means a trapping of atoms in real space. The best present performances will be given, in terms of the lowest temperatures and the highest densities. The corresponding highest quantum degeneracy will also be estimated. It is imposed by fundamental limits, which will be described briefly. We also give the general trends in this field of research and outline the new directions which look promising for observing quantum statistical effects in laser cooled atomic samples, but which are for the moment restricted by unsolved problems.

Introducing the Simple Schemes

The radiative forces acting on atoms in a light field can be split into two parts, a reactive one and a dissipative one. The dissipative force (radiation pressure), which basically involves scattering processes, is velocity dependent. We will see that this dependence leads to the Doppler cooling scheme and to the concept of optical molasses, and we will give the corresponding minimal achievable temperature. The dissipative force can be made position dependent, through a gradient of the magnetic field, so that the atoms are also trapped in the so-called magneto-optical trap.

Abstract

We predict and analyze non-trivial relaxational behavior of magnetically trapped gases near the Bose condensation temperature Tc. Due to strong compression of the condensate by the inhomogeneous trapping field, particularly at low densities, the relaxation rate shows a strong, almost jump wise, increase below Tc. As a consequence the maximum fraction of condensate particles is limited to a few percent. This phenomenon can be called a “relaxation explosion”. We discuss its implications for the detectability of BEC in atomic hydrogen.

Magnetostatic traps offer the possibility to study gases of Bose particles in the truly dilute limit, and have proved particularly fruitful [1, 2, 3, 4, 5] in the study of atomic hydrogen (H). In these traps, proposed for H by Hess [6], the effective elimination of physical boundaries is accomplished by creating a magnetic field minimum in free space. This minimum forms a potential well for electron spin-up polarized atoms (H↑), called low-field seekers. The occurrence of Bose–Einstein condensation (BEC) in such systems introduces qualitatively different behavior from the case of a homogeneous Bose gas. This is related to the explosive increase of the dipolar relaxation rate associated with the strong compression of the condensate in an external potential.

Abstract

Initially put forward by Moskalenko and Blatt et al., the idea of a possible Bose–Einstein condensation (BEC) of excitons in semiconductors has attracted the attention of both experimentalists and theoreticians for more than three decades. At different stages of this long history, the results of their efforts have been described and discussed in review articles. A brief introduction and summary of the main qualitative conclusions of this older work is presented here (Sections 1 and 2), followed by a more detailed discussion of some more recent developments (Sections 3 and 4).

Electronic Excitations in Semiconductors

Schematically presented in Fig. 1 is the typical electronic spectrum of a semiconductor: two bands (or two groups of bands) of continuous spectrum – conduction (c) and valence (v) – separated by the energy gap Eg = Ec,min – EV,max. In the ground state, all of the states in the valence band(s) are occupied by valence electrons of the semiconductor, and all states in the conduction band are empty.

The lowest single-particle electronic excitations are an additional electron (e) in the conduction band or a single empty state – a hole (h) – in the valence band. Both of these excitation types are mobile fermions (spin = 1/2) characterized by effective masses, me and mh, and effective charges ee = e and eh = −e, respectively. Here e is the usual (negative) elementary charge.

Abstract

The bosonization method developed in nuclear physics in the last 20 years is briefly reviewed.

Introduction

In the last 20 years, a considerable amount of work has gone into the development of bosonization methods for highly correlated finite fermion systems. This work has been stimulated by the phenomenological successes of the interacting boson model introduced in 1974 in which atomic nuclei composed of nucleons (fermions) are treated in terms of an interacting system of bosons (fermion pairs). In this article the logic scheme of the bosonization method in nuclear physics will be briefly reviewed. The purpose here is to provide the basic references upon which the review is built and sources where further references can be found. Bosonization methods were introduced much earlier than 1974 in connection with infinite Fermi systems. The method developed in nuclear physics has some similarities with those developed in other areas of physics, but it also has major differences. In nuclear physics one maps not only operators but also states and seeks a description not only of the ground state but also of the entire excitation spectrum. Furthermore, number projection plays a very important role contrary to the case of infinite systems where number projection is not relevant. These differences, and others, will be briefly remarked upon in this article.

Strongly Interacting Finite Fermion Systems

Fermions and the Shell Model

Quite often in physics one has to deal with a system of interacting fermions.

Abstract

Spin-polarized atomic hydrogen continues to be one of the most promising candidates for Bose condensation of an atomic system. In contrast to liquid helium, hydrogen is gaseous and therefore its density can be changed to vary its behavior from the weakly to the strongly interacting Bose gas. Until now, efforts to Bose condense hydrogen have been thwarted by recombination and relaxation phenomena. After a long introduction to the subject, two promising approaches to observe quantum degenerate behavior are discussed: the microwave trap and a two-dimensional gas of hydrogen.

Introduction

Since the stabilization of atomic hydrogen as a spin-polarized gas (H↓), reported in 1980 [1], there has been a continuous effort to observe Bose–Einstein condensation (BEC) or other effects of quantum degeneracy in this Bose gas. This challenge has not yet been realized due to the instability of H↓ towards recombination to H2 or relaxation among hyperfine states as the conditions for BEC are approached. Although this difficulty is formidable, hydrogen presents a unique opportunity to study BEC and the related superfluidity, as it remains a gas to T = 0 [2]. By comparison, 4He, the only experimentally observed boson fluid, is a strongly interacting superfluid Bose liquid, with little flexibility for varying its density. As a result of the gaseous nature of hydrogen, its density, n, can be varied over several orders of magnitude with the possibility of studying BEC and its relationship to superfluidity, ranging from the weakly to the strongly interacting boson gas.

Abstract

Recent experiments on the near-infrared absorption, thermal conductivity and the critical field Hc2 in several high-Tc oxides are interpreted as a manifestation of the Bose–Einstein condensation of small bipolarons.

Basic Model for High-TcOxides

To describe low-energy spin and charge excitations of metal oxides and doped fullerenes with bipolarons, Alexandrov and Mott [1, 2, 3, 4] have suggested that bipolarons are intersite in two possible spin states (S = 0 or 1), and a proportion of bipolarons are in Anderson localised states.

Our assumption is that all electrons are bound in small singlet or triplet bipolarons and they are responsible for the spin excitations. Hole pairs, which appear with doping, are responsible for the low-energy charge excitations of the CuO2 plane. Above Tc, a material such as YBCO contains a non-degenerate gas of these hole bipolarons in a singlet or in a triplet state, with a slightly lower mass due to the lower binding energy.

The low-energy band-structure includes two bosonic bands (singlets and triplets), separated by the singlet–triplet exchange energy J, estimated to be of the order of a few hundred meV. The half-bandwidth w is of the same order. The bipolaron binding energy is assumed to be large (Δ >> T), and therefore single polarons are irrelevant in the temperature region under consideration.

We argue that many features of spin and charge excitations in metal oxides can be described within our simple model.

Abstract

The short coherence lengths and the low carrier concentrations in high temperature superconductors (HTcSC) quite naturally favor a scenario where preformed bound electron pairs undergo a Bose–Einstein condensation (BEC). There are numerous experimental indications in the normal state properties (anomalous diamagnetic susceptibility, NMR spin-lattice relaxation, Knight shift and crystal electric field transitions, as well as entropy non-linear in T and strong local lattice deformations seen by EXAFS), all of which indicate a characteristic temperature Tpb associated with the breaking of such preformed electron pairs and the existence of a pseudogap above Tc. As the number of charge carriers is increased by chemical doping, Tpb decreases and the critical temperature Tc, where superconductivity sets in, in this so-called “underdoped regime”, increases. The maximum value for Tc is reached when Tc ≲ TPB. Upon further doping, going into the so-called “overdoped regime”, Tc decreases. We take this as an indication that the superconducting state depends on the existence of preformed electron pairs. The normal state properties of the “underdoped regime” seem to show significant differences from Fermi liquid behavior, while in the “overdoped regime” they seem to be typical of ordinary metals.

In order to capture this empirically established scenario in HTCSC, we discuss a phenomenological model based on a mixture of localized bosons (bound electron pairs such as bipolarons) and itinerant fermions (electrons), with a local exchange between bosons and fermion pairs.

Abstract

We review the proposal for Bose–Einstein condensation of positronium atoms. All of the ingredients necessary to achieve BEC of Ps atoms are currently available.

Introduction

In this volume several authors have discussed and described a variety of weakly interacting systems which might display BEC. In this short contribution we suggest that a dense gas of positronium (Ps) atoms in vacuum is a rather ideal but somewhat more exotic system that might be a very good candidate for observing a weakly interacting BEC.

Recent investigations of the interactions of positrons (e+) and (Ps) with solids have led to extraordinary improvements in the kinds of low energy experiments we can do with the positron [1, 2]. All of the ingredients necessary to achieve BEC of Ps atoms are currently available. We envision a scenario where roughly N ≅ 105 Ps atoms are trapped in a volume v ≃ 10−13 cm3 and allowed to cool through the Bose transition temperature of 20 – 30K in a time of the order of nanoseconds. In the following we discuss the relevant interactions and describe how Ps BEC can be achieved.

Single Positronium Physics

Ps is comprised of an e+ − e− bound in a hydrogenic orbit. Its mass, 2me, is extremely light compared to H, an important ingredient for achieving reasonable Bose condensation temperatures. Its binding energy (6.8 eV) is half that of H.