A renormalization approach to many-atom radiation processes

We present a second-order many-body perturbation approach to coherent radiation processes. The theory gives a unified view of several many-atom effects as eigenmodes of the interacting atom-field system. Two renormalization procedures are essential to perform our program; (complex) energy renormalization using a noncanonical transformation and light-velocity renormalization using Bogoliubov transformation for field operators.     

A Markov chain approach to renormalization group transformations

We aim at an explicit characterization of the renormalized Hamiltonian after decimation transformation of a one-dimensional Ising-type Hamiltonian with a nearest-neighbor interaction and a magnetic field term. To facilitate a deeper understanding of the decimation effect, we translate the renormalization flow on the Ising Hamiltonian into a flow on the associated Markov chains through the Markov–Gibbs equivalence. Two different methods are used to verify the well-known conjecture that the eigenvalues of the linearization of this renormalization transformation about the fixed point bear important information about all six of the critical exponents. This illustrates the universality property of the renormalization group map in this case.     

A renormalization approach to the band structure of superlattices

A k-space renormalization technique for evaluating the band structure of superlattices is examined. The large number of degrees of freedom of a superlattice tight-binding Hamiltonian is first reduced by exploiting the translational symmetry properties in layers perpendicular and parallel to the superlattice axis. The remaining degrees of freedom of the unit supercell are then systematically eliminated by renormalization techniques. Our combination of projective techniques on the one hand and full exploitation of symmetry properties on the other result in a very flexible and efficient algorithm. Some results for a simplified single-site model superlattice are presented as an example of our novel procedure.     

Nonperturbative renormalization group approach to turbulence

We suggest a new, renormalization group (RG) based, nonperturbative method for treating the intermittency problem of fully developed turbulence which also includes the effects of a finite boundary of the turbulent flow. The key idea is not to try to construct an elimination procedure based on some assumed statistical distribution, but to make an ansatz for possible RG transformations and to pose constraints upon those, which guarantee the invariance of the nonlinear term in the Navier-Stokes equation, the invariance of the energy dissipation, and other basic properties of the velocity field. The role of length scales is taken to be inverse to that in the theory of critical phenomena; thus possible intermittency corrections are connected with the outer length scale. Depending on the specific type of flow, we find different sets of admissible transformations with distinct scaling behaviour: for the often considered infinite, isotropic, and homogeneous system K41 scaling is enforced, but for the more realistic plane Couette geometry no restrictions on intermittency exponents were obtained so far. Received: 28 December 1997 / Accepted: 6 August 1998     

A stochastic approach to Wilson loops

Wilson loops exp (i A (x) dx) are investigated in two-dimensional Euclidean space-time. The electromagnetic vector potential A is regarded as a generalized random field given by the stochastic partial differential equation A = F where is a first-order differential operator and F is white noise. We give a rigorous definition of Wilson loops and examine the properties of the N-loop Schwinger functions.     

On position-space renormalization group approach to percolation

In a position-space renormalization group (PSRG) approach to percolation one calculates the probabilityR(p,b) that a finite lattice of linear sizeb percolates, wherep is the occupation probability of a site or bond. A sequence of percolation thresholdsp _c (b) is then estimated fromR(p _c ,b)=p _c (b) and extrapolated to the limitb to obtainp _c =p _c (). Recently, it was shown that for a certain spanning rule and boundary condition,R(p _c ,)=R _c is universal, and sincep _c is not universal, the validity of PSRG approaches was questioned. We suggest that the equationR(p _c ,b)=, where isany number in (0,1), provides a sequence ofp _c (b)'s thatalways converges top _c asb. Thus, there is anenvelope from any point inside of which one can converge top _c . However, the convergence is optimal if =R _c . By calculating the fractal dimension of the sample-spanning cluster atp _c , we show that the same is true aboutany critical exponent of percolation that is calculated by a PSRG method. Thus PSRG methods are still a useful tool for investigating percolation properties of disordered systems.     

An approach to renormalization on the n-torus

The coding theory of rotations (by inspecting closely their relation to flows) and the continued fractions algorithm (by considering even two-coloring of the integers with a given proportion of, say, blue and red) are revisited. Then, even n-coloring of the integers is defined. This allows one to code rotations on the (n-1)-torus by considering linear flows on the n-torus and yields a simple geometric approach to renormalization on tori by first return maps on the coding regions.     

Effective medium approach to real space renormalization

We use the ideas of effective medium theory to present a self-consistent decimation which yields the exact fixed point for the q-component Potts model in a square lattice and also generates the exact critical surface for anisotropic interactions.     

A variational approach to stochastic nonlinear problems

A variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities. Approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a Bernoulli oscillator driven by white noise. Comparison with exact data available for this system shows that the variational approach to such problems can be quite effective.     

A stochastic approach to open quantum systems

Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schr?dinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schr?dinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.     

Migdal renormalization group approach to quantum spin systems

The Migdal RG approximation is extended to quantum spin systems such as the Heisenberg and XY-models. This yields the non-existence of phase transition in the two-dimensional Heisenberg model. The phase transition of the two-dimensional XY-model is also studied.     

Migdal renormalization group approach to deconfinement phase transition

The Migdal renormalization group approach is applied to a finite temperature lattice gauge theory. Imposing the periodic boundary condition in the timelike orientation, the phase structure of the finite temperature lattice gauge system with a gauge groupG in (d+1)-dimensional space is determined by two kinds of recursion equations, describing spacelike and timelike correlations, respectively. One is the recursion equation for ad-dimensional gauge system with the gauge groupG, and the other corresponds to ad-dimensional spin system for which the effective theory is described by the nearest neighbor interaction of the Wilson lines. Detailed phase structure is investigated for theSU(2) gauge theory in (3+1)-dimensional space. Deconfinement phase transition is obtained. Using the recursion equation for the three dimensional spin system of the Wilson lines, it is shown that the flow of the renormalization group trajectories leads to a phase transition of the three dimensional Ising model.     

A new renormalization approach to the coherent state in the Kondo lattice

We develop a new theoretical approach to study the coherent state in the Kondo lattice. A renormalization operator is introduced to reflect the many-body interactions. A characteristic temperature, below which the system enters a coherent state (i.e., heavy-fermion state), is obtained, and this is in agreement with the recent theoretical result derived from a different method.     

A stochastic approach to hopping transport in semiconductors

Generalized charge carrier equations for hopping transport in semiconductors are derived which include also the widely used Van Roosbroeck equations. The approach is based on a microscopic stochastic interacting particle system which models the hopping of electrons on a random set of states.     

A stochastic approach to simulations of fermionic systems

We introduce a Langevin equation approach to the analysis of fermionic theories. We find a Langevin type equation, depending on a parameter τ, such that its equilibrium distributions are those of the original fermionic system in the limit of τ going to zero. We explicitly treat a simple example, proving the exactness of the method in the limit τ → 0, and we estimate the error induced by the method at first order in τ.