RG

We study the stability of Hamiltonian systems in classical mechanics with two degrees of freedom by renormalization-group methods. One of the key mechanisms of the transition to chaos is the break-up of invariant tori, which plays an essential role in the large scale and long-term behavior. The aim is to determine the threshold of break-up of invariant tori and its mechanism. The idea is to construct a renormalization transformation as a canonical change of coordinates, which deals with the dominant resonances leading to qualitative changes in the dynamics. Numerical results show that this transformation is an efficient tool for the determination of the threshold of the break-up of invariant tori for Hamiltonian systems with two degrees of freedom. The analysis of this transformation indicates that the break-up of invariant tori is a universal mechanism. The properties of invariant tori are described by the renormalization flow. A trivial attractive set of the renormalization transformation characterizes the Hamiltonians that have a smooth invariant torus. The set of Hamiltonians that have a non-smooth invariant torus is a fractal surface. This critical surface is the stable manifold of a single strange set encompassing all irrational frequencies. This hyperbolic strange set characterizes the Hamiltonians that have an invariant torus at the threshold of the break-up. From the critical strange set, one can deduce the critical properties of the tori (self-similarity, universality classes).

We study a recently proposed quantum action depending on temperature. At zero temperature the quantum action is obtained analytically and reproduces the exact ground state energy and wave function. This is demonstrated for a number of cases with parity symmetric confining potentials. In the case of the hydrogen atom, it also reproduces exactly energy and wave function of a subset of excited state (those of lowest energy for given angular momentum l) and the quantum action is consistent with O(4) symmetry. In the case of a double-well potential, the quantum action generates the ground state of double-hump shape. In all cases we observe a coincidence (in position) of minima of the quantum potential with maxima of the wave function. The semi-classical WKB formula for the ground state wave function becomes exact after replacing the parameters of the classical action by those of quantum action.

The renormalization group has played an important role in the physics of the second half of the 20th century both as a conceptual and a calculational tool. In particular, it provided the key ideas for the construction of a qualitative and quantitative theory of the critical point in phase transitions and started a new era in statistical mechanics. Probability theory lies at the foundation of this branch of physics and the renormalization group has an interesting probabilistic interpretation as it was recognized in the middle 1970s. This paper intends to provide a concise introduction to this aspect of the theory of phase transitions which clarifies the deep statistical significance of critical universality.

We review the use of an exact renormalization group equation in quantum field theory and statistical physics. It describes the dependence of the free energy on an infrared cutoff for the quantum or thermal fluctuations. Non-perturbative solutions follow from approximations to the general form of the coarse-grained free energy or effective average action. They interpolate between the microphysical laws and the complex macroscopic phenomena. Our approach yields a simple unified description for O(N)-symmetric scalar models in two, three or four dimensions, covering in particular the critical phenomena for the second-order phase transitions, including the Kosterlitz¯Thouless transition and the critical behavior of polymer chains. We compute the aspects of the critical equation of state which are universal for a large variety of physical systems and establish a direct connection between microphysical and critical quantities for a liquid¯ gas transition. Universal features of first-order phase transitions are studied in the context of scalar matrix models. We show that the quantitative treatment of coarse graining is essential for a detailed estimate of the nucleation rate. We discuss quantum statistics in thermal equilibrium or thermal quantum field theory with fermions and bosons and we describe the high-temperature symmetry restoration in quantum field theories with spontaneous symmetry breaking. In particular, we explore chiral symmetry breaking and the high-temperature or high-density chiral phase transition in quantum chromodynamics using models with effective four-fermion interactions.

The material presented in this invaluable textbook has been tested in two courses. One of these is a graduate-level survey of statistical physics; the other, a rather personal perspective on critical behavior. Thus, this book defines a progression starting at the book-learning part of graduate education and ending in the midst of topics at the research level. To supplement the research-level side the book includes some research papers. Several of these are classics in the field, including a suite of six works on self-organized criticality and complexity, a pair on diffusion-limited aggregation, some papers on correlations near critical points, a few of the basic sources on the development of the real-space renormalization group, and several papers on magnetic behavior in a plain geometry. In addition, the author has included a few of his own papers.

States on a Sierpinski triangle are described using a formally exact and general Hamiltonian renormalization . The spectra of new (as well as previously examined) models are characterized . Numerical studies based on the renormalization suggest that the only models which exhibit absolutely continuous specta are effectively one-dimensional in the limit of large distances .

We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann¯Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.

In this work, we present a review of recently achieved progress in the field of soft condensed matter physics, and in particular on the study of the static properties of solutions or suspensions of colloidal particles. The latter are macromolecular entities with typical sizes ranging from 1 nm to 1 small mu, Greekm and their suspension typically contain, in addition to the solvent, smaller components such as salt ions or free polymer chains. The theoretical tool introduced is the effective Hamiltonian which formally results by a canonical trace over the smaller degrees of freedom for a fixed, frozen configuration of the large ones. After presenting the formal definitions of this effective Hamiltonian, we proceed with the applications to some common soft matter systems having a variable softness and ranging from free polymer chains to hard colloidal particles. We begin from the extreme case of nondiverging effective interactions between ultrasoft polymer chains and derive an exact criterion to determine the topology of the phase diagrams of such systems. We use star polymers with a variable arm number f as a hybrid system in order to interpolate between these two extremes. By deriving an effective interaction between stars we can monitor the change in the phase behavior of a system as the steepness of the repulsion between its constituent particles increases. We also review recent results on the nature and the effects of short-range attractions on the phase diagrams of spherical, nonoverlapping colloidal particles.

Crossover phenomena are both ubiquitous and important. In this report, we review the foundations of renormalization group theory as applied to crossover behavior and consider several paradigmatic applications. We confine ourselves to situations where the crossover can be described in terms of an effective field theory, in particular concentrating on the prototypical example of an O(N) model in a constrained geometry or at finite temperature. Calculation of universal crossover scaling functions is considered where we show how the renormalization group can in principle be applied to the latter to obtain expressions as accurate as those of standard universal quantities, such as critical exponents and amplitude ratios. Particular attention is paid to the scaling equation of state for an O(N) model.

Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d1 of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d-d1=1 (for which we find nonphysical random fixed points, suggesting the existence of nonperturbative fixed distributions) and d-d1>1 (for which we do find acceptable perturbartive random fixed points), in agreement with previous numerical calculations by Andelman and Aharony. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.

We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality classes, including the Nrightwards arrow0 limit that describes the critical behavior of self-avoiding walks. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau¯Ginzburg¯Wilson Hamiltonians, such as N-component systems with cubic anisotropy, O(N)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau¯Ginzburg¯Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the small beta, Greek-functions. Finally, we consider a Hamiltonian with symmetry O(n1)plus sign in circleO(n2) that is relevant for the description of multicritical phenomena.

Renormalization group, and in particular its quantum field theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized [small phi, Greek] 34 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N -vector model (Le Guillou and Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95; Phys. Rev. B 21 (1980) 3976; Guida and Zinn-Justin, J. Phys. A 31 (1998) 8103; cond-mat/9803240) and of the equation of state of the 3D Ising model (Guida and Zinn-Justin, Nucl. Phys. B 489 [FS] (1997) 626, hep-th/9610223). These results are among the most precise available probing field theory in a non-perturbative regime. Precise calculations first require enough terms of the perturbative expansion. However perturbation series are known to be divergent. The divergence has been characterized by relating it to instanton contributions. The information about large-order behaviour of perturbation series has then allowed to develop efficient summation techniques, based on Borel transformation and conformal mapping (Le Guillou and Zinn-Justin (Eds.), Large Order Behaviour of Perturbation Theory, Current Physics, vol. 7, North-Holland, Amsterdam, 1990). We first discuss exponents and describe our recent results (Guida and Zinn-Justin, 1998). Compared to exponents, the determination of the scaling equation of state of the 3D Ising model involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order dependent mapping method. From the knowledge of the equation of state a number of ratio of critical amplitudes can also be derived. Finally we emphasize that few physical quantities which are predicted by renormalization group to be universal have been determined precisely, and much work remains to be done. Considering the steady increase in the available computer resources, many new calculations will become feasible. In addition to the infinite volume quantities, finite size universal quantities would also be of interest, to provide a more direct contact with numerical simulations. Let us also mention dynamical observables, a largely unexplored territory.