This chapter addresses the stochastic dynamics of interacting particle systems, specifically reaction-diffusion models that, for example, capture chemical reactions in a gel such that convective transport is inhibited. Generic reaction-diffusion models are in fact utilized to describe a multitude of phenomena in various disciplines, ranging from population dynamics in ecology, competition of bacterial colonies in microbiology, dynamics of magnetic monopoles in the early Universe in cosmology, equity trading on the stock market in economy, opinion exchange in sociology, etc. More concrete physical applications systems encompass excitons kinetics in organic semconductors, domain wall interactions in magnets, and interface dynamics in growth models. Yet most of our current knowledge in this area stems from extensive computer simulations, and actual experimental realizations allowing accurate quantitative analysis are still deplorably rare. We begin with a brief review of mean-field and scaling arguments including Smoluchowski's self-consistent treatment of diffusion-limited binary annihilation. The main goal of this chapter is to demonstrate how one may systematically proceed from a microscopic master equation for interacting particles, which perhaps represents the most straightforward description of a system far from equilibrium, to a non-Hermitean bosonic ‘quantum’ many-body Hamiltonian, and thence to a continuum field theory representation that permits subsequent perturbative expansions and renormalization group treatment. The ensuing rich physics is illustrated with simple examples that include the annihilation reactions k A → l A (l < k) and A + B ∅, their generalization to multiple particle species, as well as reversible recombination A + A ⇌ B.

This chapter addresses phase transitions and dynamic scaling occurring in systems comprised of interacting indistinguishable quantum particles, for which entanglement correlations are crucial. It first describes how the dynamics (in real time) and thermodynamics (in imaginary time) of quantum many-particle Hamiltonians can be mapped onto field theories based on coherent-state path integrals. While bosons are described by complex-valued fields, fermions are represented by anticommuting Grassmann variables. Since quantum-mechanical systems are of inherently dynamical nature, the corresponding field theory action entails d + 1 dimensions, with time playing a special role. For Hamiltonians that incorporate only two-particle interactions, we can make contact with the previously studied Langevin equations, yet with effectively multiplicative rather than additive noise. As an illustration, this formalism is applied to deduce fundamental properties of weakly interacting boson superfluids. Whereas Landau–Ginzburg theory already provides a basic hydrodynamic description, the Gaussian approximation allows the computation of density correlations, the Bose condensate fraction, and the normal- and superfluid densities from the particle current correlations. We next establish that quantum fluctuations are typically irrelevant for thermodynamic critical phenomena, provided that Tc > 0, and readily extend finite-size scaling theory to the imaginary time axis to arrive at general scaling forms for the free energy. Intriguing novel phenomena emerge in the realm of genuine quantum phase transitions at zero temperature, governed by other control parameters such as particle density, interaction or disorder strengths.

Continuous phase transitions from active to inactive, absorbing states represent prime examples of genuine non-equilibrium processes whose properties are strongly influenced by fluctuations. They arise in a broad variety of macroscopic phenomena, ranging from extinction thresholds in population dynamics and epidemic spreading models to certain diffusion-limited chemical reactions, and even turbulent kinetics in magnetic fluids. Intriguingly, the generic universality class for such active to absorbing phase transitions is intimately related to the scaling properties of critical directed percolation clusters. After elucidating this remarkable connection of stochastic kinetics with an originally geometric problem through mappings of both a specific microscopic interacting particle model and a more general mesoscopic Langevin description onto the corresponding Reggeon field theory action, we exploit the mathematical and conceptual techniques developed in previous chapters to compute the associated critical exponents to lowest non-trivial order in a dimensional ∊ expansion about the upper critical dimension dc = 4. We then set out to explore generalizations to systems with multiple particle species, and to investigate the dynamic percolation model variant that generates isotropic critical percolation clusters in the quasi-static limit. Particle spreading via long-range Lévy flights rather than nearest-neighbor hopping and coupling to an additional conserved field that may cause a fluctuation-induced first-order transition are also discussed. Motivated by the domain wall kinetics in non-equilibrium Ising systems, we address more general stochastic reaction systems of branching and annihilating random walks, and study the ensuing non-equilibrium phase diagrams and continuous transitions, including the parity-conserving universality class.

To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify dc = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.

In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.

The goal of this advanced graduate-level textbook is to provide a description of the field-theoretic renormalization group approach for the study of time-dependent phenomena in systems either close to a critical point, or displaying generic scale invariance. Its general aim is a unifying treatment of classical near-equilibrium, as well as quantum and non-equilibrium systems, providing the reader with a thorough grasp of the fundamental principles and physical ideas underlying the subject.

Scaling ideas and the renormalization group philosophy and its various mathematical formulations were developed in the 1960s and early 1970s. In the realm of statistical physics, they led to a profound understanding of critical singularities near continuous phase transitions in thermal equilibrium. Beginning in the late 1960s, these concepts were subsequently generalized and applied to dynamic critical phenomena. By the mid-1980s, when I began my research career, critical dynamics had become a mature but still exciting field with many novel applications. Specifically, extensions to quantum critical points and to systems either driven or initialized far away from thermal equilibrium opened fertile new areas for in-depth analytical and numerical investigations.

By now there exists a fair sample of excellent textbooks that provide profound expositions of the renormalization group method for static critical phenomena, adequately introducing statistical field theory as the basic tool, and properly connecting it with its parent, quantum field theory. However, novice researchers who wish to familiarize themselves with the basic techniques and results in the study of dynamic critical phenomena still must resort largely to the original literature, supplemented with a number of very good review articles.

Originally, the term ‘dynamic critical phenomena’ was coined for time-dependent properties near second-order phase transitions in thermal equilibrium. The kinetics of phase transitions in magnets, at the gas–liquid transition, and at the normal- to superfluid phase transition in helium 4 were among the prominent examples investigated already in the 1960s. The dynamic scaling hypothesis, generalizing the scaling ansatz for the static correlation function and introducing an additional dynamic critical exponent, successfully described a variety of these experiments. Yet only the development of the systematic renormalization group (RG) approach for critical phenomena in the subsequent decade provided a solid conceptual foundation for phenomenological scaling theories. Supplemented with exact solutions for certain idealized model systems, and guided by invaluable input from computer simulations in addition to experimental data, the renormalization group now provides a general framework to explore not only the static and dynamic properties near a critical point, but also the large-scale and low-frequency response in stable thermodynamic phases. Scaling concepts and the renormalization group have also been successfully applied to phase transitions at zero temperature driven by quantum rather than thermal fluctuations. It is to be hoped that RG methods may help to classify the strikingly rich phenomena encountered in far-from-equilibrium systems as well. Recent advances in studies of simple reaction-diffusion systems, active to absorbing state phase transitions, driven lattice gases, and scaling properties of moving interfaces and growing surfaces, among others, appear promising in this respect.

In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O(n)-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.

The Lanczos tridiagonalization method orthogonally transforms a real symmetric matrix A to symmetric tridiagonal form. Traditionally, this very simple algorithm is suitable when one needs only a few of the lower eigenvalues and the corresponding eigenvectors of very large Hermitian matrices, whose full diagonalization is technically impossible. We introduce here the basic ingredients of the recursion method based on the Lanczos tridiagonalization, and explain how calculation of the DOS as well as the dynamics of wavepackets (and related conductivity) can be performed efficiently.

Lanczos method for the density of states

The Lanczos method is a highly efficient recursive approach for calculation of the electronic structure (Lanczos, 1950). This method, first developed by Haydock, Heine, and Kelly (Haydock, Heine & Kelly, 1972, 1975), is based on an eigenvalue approach due to Lanczos. It relies on computation of Green functions matrix elements by continued fraction expansion, which can be implemented either in real or reciprocal space. These techniques are particularly well suited for treating disorder and defect-related problems, and were successfully implemented to tackle impurity-level calculations in semiconductors using a tight-binding approximation (Lohrmann, 1989), and for electronic structure investigations for amorphous semiconductors, transition metals, and metallic glasses based on linear-muffin-tin orbitals (Bose, Winer & Andersen, 1988). Recent developments include the exploration of a degenerated orbital extended Hubbard Hamiltonian of system size up to ten millions atoms, with the Krylov subspace method (Takayama, Hoshi & Fujiwara, 2004, Hoshi et al., 2012).

This section presents a brief overview of the most promising graphene applications in information and communication technologies, reflecting current activities of the scientific community and the authors' own views.

Introduction

The industrial impact of carbon nanotubes is still under debate. Carbon nanotubes exist in two complementary flavors, i.e metallic conductors and semiconductors with tunable band gap (scaled with tube diameter), both exhibiting ballistic transport. This appears ideal at first sight for creating electronic circuits, in which semiconducting nanotubes (with diameter around 1–2 nm) could be used as field effect transistors, whereas metallic single-wall tubes (or large-diameter multiwalled nanotubes), with thermal conductivity similar to diamond and superior current-carrying capacity to copper and gold, would offer ideal interconnects between active devices in microchip (Avouris, Chen & Perebeinos, 2007). Nanotube-based interconnects have been physically studied over almost a decade, with companies such as Samsung, Fujitsu, STMicroelectronics, or Intel acting significantly or encouraging academic research (Coiffic et al., 2007). The current-carrying capability of bundles of multiwalled nanotubes has been practically demonstrated to fulfill the requirements for technology and thus could replace metals (Esconjauregui et al., 2010), although a disruptive technology step remains to be achieved to integrate chemical vapor deposition (CVD) growth at the wafer-scale, a step of no defined timeline.

In this chapter we start with a presentation of the so-called Klein tunneling mechanism, which is one of the most striking properties of graphene. Later we give an overview of ballistic transport both in graphene and related materials (carbon nanotubes and graphene nanoribbons). After presenting a simple real-space mode-decomposition scheme, which can be exploited to obtain analytical results or to boost numerical calculations, we discuss Fabry-Pérot interference, contact effects, and the minimum conductivity in the 2D limit.

The Klein tunneling mechanism

The Klein tunneling mechanism was first reported in the context of quantum electrodynamics. In 1929, physicist Oskar Klein (Klein, 1929) found a surprising result when solving the propagation of Dirac electrons through a single potential barrier. In non-relativistic quantum mechanics, incident electrons tunnel a short distance through the barrier as evanescent waves, with exponential damping with the barrier depth. In sharp contrast, if the potential barrier is of the order of the electron mass, eV ~ mc2, electrons propagate as antiparticles whose inverted energy–momentum dispersion relation allows them to move freely through the barrier. This unimpeded penetration of relativistic particles through high and wide potential barriers has been one of the most counterintuitive consequences of quantum electrodynamics, but despite its interest for particle, nuclear, and astro-physics, a direct test of the Klein tunnel effect using relativistic particles still remains out of reach for high-energy physics experiments.

Carbon is a truly unique chemical element. It can form a broad variety of architectures in all dimensions, both at the macroscopic and nanoscopic scales. During the last 20+ years, brave new forms of carbon have been unveiled. The family of carbon-based materials now extends from C60 to carbon nanotubes, and from old diamond and graphite to graphene. The properties of the new members of this carbon family are so impressive that they may even redefine our era. This chapter provides a brief overview of these carbon structures.

Carbon structures and hybridizations

Carbon is one of the most versatile elements in the periodic table in terms of the number of compounds it may create, mainly due to the types of bonds it may form (single, double, and triple bonds) and the number of different atoms it can join in bonding. When we look at its ground state (lowest energy) electronic configuration, 1s22s22p2, carbon is found to possess two core electrons (1s) that are not available for chemical bonding and four valence electrons (2s and 2p) that can participate in bond formation (Fig. 1.1(a)). Since two unpaired 2p electrons are present, carbon should normally form only two bonds from its ground state.

However, carbon should maximize the number of bonds formed, since chemical bond formation will induce a decrease of the system energy. Consequently, carbon will re-arrange the configuration of these valence electrons.