We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.

The emergence of generic scale invariance, i.e., algebraic behavior without tuning to special critical points, appears to be remarkably common in systems that are settled in a non-equilibrium steady state. Prototypical examples are simple non-linear Langevin equations that describe driven diffusive systems and driven interfaces or growth models far from thermal equilibrium, whose distinct phases are characterized by non-trivial RG fixed points and hence universal scaling exponents. We start with driven lattice gases with particle exclusion that are described by generalizations of the one-dimensional noisy Burgers equation for fluid hydrodynamics. Symmetries and conservation laws completely determine the ensuing stationary power laws, as well as the intermediate aging scaling regime and even the large-deviation function for the particle current fluctuations. Next we address the non-equilibrium critical point for driven Ising lattice gases, whose critical exponents can again be computed exactly. We then turn our attention to the prominent Kardar–Parisi–Zhang equation, originally formulated to describe growing crystalline surfaces and the dynamics of driven interfaces, but also closely related to the noisy Burgers equation and even to the equilibrium statistical mechanics of directed lines in disordered environments. After introducing the scaling theory for interface fluctuations, we proceed to a renormalization group analysis at fixed dimension d. For d > 2, a non-trivial unstable RG fixed point separates a phase with Gaussian or Edwards–Wilkinson scaling exponents from a strong-coupling rough phase that is inaccessible by perturbative methods.

Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n-component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = (d + 2 − η)/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O(n)-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets (n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d/2 for all slow modes.

In the second part of this book, we consider the dynamics of systems far away from thermal equilibrium. This departure from equilibrium may be caused by an external driving force, as is the case for driven diffusive systems or growing interfaces considered in Chapter 11; there also exist fundamentally open athermal systems which never reach equilibrium, as is true for some of the reaction-diffusion systems considered in Chapter 9. In both instances, the constraints imposed by detailed balance and the ensuing fluctuation-dissipation theorem on the form of phenomenological equations describing the temporal evolution of such systems are absent. Even at the fundamental level of quantum mechanics, the dynamical description of such open dissipative systems is still quite poorly understood. Generally, it may thus seem a hopeless task to derive coarse-grained equations of motion for only a few mesoscopic degrees of freedom from such an unsatisfactory foundation.

Fortunately, this conclusion is too pessimistic, at least if we are interested in systems whose non-equilibrium steady state is tuned close to a critical point, or displays generic scale invariance. For, in these situations, we may appeal to the concept of universality to allow us to constrain through basic symmetry and conservation arguments the terms which must be retained in an effective dynamical description. Similarly, we may hope that the universal properties of drastically simplified models which happen to be exactly solvable may extend to more realistic and technologically relevant systems.

In this chapter, we develop the basic tools for our study of dynamic critical phenomena. We introduce dynamic correlation, response, and relaxation functions, and explore their general features. In the linear response regime, these quantities can be expressed in terms of equilibrium properties. A fluctuation-dissipation theorem then relates dynamic response and correlation functions. Under more general non-equilibrium conditions, we must resort to the theory of stochastic processes. The probability P1(x, t) of finding a certain physical configuration x at time t is governed by a master equation. On the level of such a ‘microscopic’ description, we discuss the detailed-balance conditions which guarantee that P1(x, t) approaches the probability distribution of an equilibrium statistical ensemble as t → ∞. Taking the continuum limit for the variable(s) x, we are led to the Kramers–Moyal expansion, which often reduces to a Fokker–Planck equation. Three important examples elucidate these concepts further, and also serve to introduce some calculational methods; these are biased one-dimensional random walks, a simple population dynamics model, and kinetic Ising systems. We then venture towards a more ‘mesoscopic’ viewpoint which focuses on the long-time dynamics of certain characteristic, ‘relevant’ quantities. Assuming an appropriate separation of time scales, the remaining ‘fast’ degrees of freedom are treated as stochastic noise. As an introduction to these concepts, the Langevin–Einstein theory of free Brownian motion is reviewed, and the associated Fokker–Planck equation is solved explicitly.

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.