Critical point dominance in one dimension

The renormalized, dimensionless 4-point coupling constant of scalar one dimensional field theories is maximized uniquely by the critical point theories (obtainable as the scaling limit of ⁴ models). The renormalized coupling constant of certain scalar one dimensional lattice field theories is maximized uniquely (for fixed correlation length) by the corresponding spin-1/2 model.Alfred P. Sloan Research Fellow, on leave from Indiana University. Research supported in part by NSF Grant MCS 77-20683 and by the U.S.-Israel Binational Science Foundation     

Random shearing direction models for isotropic turbulent diffusion

Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With the spectral parameter, for –<<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for <2 to anomalous behavior for with 2<<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for with <2 to an explicit family with a non-Gaussian covariance for with 2<<4. These non-Gaussian distributions have tails that are broader than Gaussian as varies with 2<<4 and behave for large values like exp(–C _c |x|^4–), withC _c an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.     

Proof of the Bogoliubov-Parasiuk theorem on renormalization

A new proof is given that the subtraction rules ofBogoliubov andParasiuk lead to well-defined renormalized Green's distributions. A collection of the counter-terms in trees removes the difficulties with overlapping divergences and allows fairly simple estimates and closed expressions for renormalized Feynman integrals. The renormalization procedure, which also applies to conventionally non-renormalizable theories, is illustrated in the ⁴-theory.Research supported by the National Science Foundation.     

A theorem on asymptotic expansion of Feynman amplitudes

For any Feynman amplitude, where any subset of invariants and/or squared masses is scaled by a real parameter going to zero or infinity, the existence of an expansion in powers of and ln is proved, and a method is given for determining such an expansion. This is shown quite generally in euclidean metric, whatever the external momenta (exceptional or not) and the internal masses (vanishing or not) may be, and for some simple cases in minkowskian metric, assuming only finiteness of the — eventually renormalized — amplitude before scaling. The method uses what is called Multiple Mellin representation, the validity of which is related to a generalized power-counting theorem.On leave of absence from University of Bahia (Brazil). Fellow of CAPES, Brazil     

Averaging for diffusions with fine-grained boundaries

This article investigates the limiting behavior of a diffusion in a half space with a complicated boundary condition. The boundary condition implements a reflection condition everywhere except a number of small sets or holes that meet Dirichlet or mixed boundary conditions. Probabilistic methods associated with the Feynman-Kac formula are used to find the limiting behavior of the diffusion equations as the number of holes gets large and the size of each hole is reduced. With particular scaling homogenization occurs, and we see that the complicated boundary condition is replaced by a simple mixed boundary condition depending on the capacitance and distribution of the holes.This work was partly supported by ARO Grant DAAL03-92-@-0219The author acknowledges the direction and encouragement provided by his advisor, Mark Freidlin     

Exact solution of the totally asymmetric simple exclusion process: Shock profiles

The microscopic structure of macroscopic shocks in the one-dimensional, totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles-first and second class. This nonequilibrium steady state factorizes about any second-class particle, which implies factorization in the one-component system about the (random) shock position. It also exhibits several other interesting features, including long-range correlations in the limit of zero density of the second-class particles. The solution also shows that a finite number of second-class particles in a uniform background of first-class particles form a weakly bound state.     

First-order transitions in spherical models: Finite-size scaling

Finite-size behavior near the first-order phase boundary of ferromagnetic spherical models is investigated for block- and cylinder-shaped systems ind dimensions. The bulk thermodynamic singularities are rounded and, asymptotically for large size, obey appropriate scaling laws. Both short-range interactions and long-range couplings, decaying like 1/r^d+ with >0, are analyzed: the short-range results agree precisely with a recently developed scaling theory forO(n) symmetric systems in the limitn. More generally, the scaling functions are universal, depending only on . Explicit aspects of the shape and interactions enter only in the spin wave or Goldstone mode contributions which appear, technically, as corrections to scaling. An appendix analyzes the truncation error in the approximation, by many-fold sums, of multivariate integrals with integrands diverging like [_ja_{j j} ² ]^- as 0.     

The spherical-model limit in a random field

The spherical-model limitn of then-vector model in a random field, with either a statistically independent distribution or with long-range correlated random fields, is studied to demonstrate the correctness of the replica method in which then and replica limits limits are interchanged, provided the replica and thermodynamic limits are taken in the right order, in the case of long-range correlated random fields. A scaling form for the two-point correlation function relevant to the first-order phase transition below the lower critical dimensionality of the random system is also obtained.     

Semi-classical asymptotics in solid state physics

This article studies the Schrödinger equation for an electron in a lattice of ions with an external magnetic field. In a suitable physical scaling the ionic potential becomes rapidly oscillating, and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from homogenization. For the time-dependent Schrödinger equation this construction yields wave packets which follow the trajectories of the semiclassical model. For the time-independent equation one gets asymptotic eigenfunctions (or quasimodes) for the energy levels predicted by Onsager's relation.     

On the large order behavior of Φ 4 4

We continue the rigorous study of the large order behavior of the perturbation series for the ⁴ model in 4 dimensions started in [1]. In this paper we prove a result announced in [1]. We show that the exact radius of convergence of the Borel transform of the renormalized perturbation series for ₄ ⁴ is greater than or equal to the expected value given by the position of the first renormalon [2]. This result holds for any vector (²)² model withN components, and makes use of the Lipatov bound of [1]. This result is based on a partial resummation of counterterms similar to the one of [3], but in a phase-space analysis of the renormalized series.     

Invariance principle for a solid-on-solid interface model

A stochastic process describing the behavior of the solid-on-solid interface in a strip of widthL is studied. The invariant and reversible measure for the process is the Gibbs state with HamiltonianH|(x)–(x+1)|. Under free boundary conditions, we show that the height of the moving interface at any site converges, when suitable renormalized, to Brownian motion with a diffusion coefficient proportional toL ^–1.     

Scaling limit of the energy variable for the two-dimensional Ising ferromagnet

The critical point limit law (scaling limit) of the suitably renormalized energy variable is explicitly calculated for the two-dimensional nearest-neighbour Ising cylinder with free edges. It is shown that the renormalization factor has to behave as (2M 2N lnN)^1/2, where 2M denotes the number of rows and 2N the number of columns. By first taking the limitM and thenN, the limit law is proven to be Gaussian.     

The use of time-dependent projection operators in the derivation of master equations

In this paper we generalize our previous work on the use of time-dependent projection operators for the derivation of master equations for general systems. Previously we had generalized the usual time-independent projection operator approach to include time-dependent projection operators, in which the relevant part of the full density operator is considered to be the uncorrelated part of the full density operator. The irrelevant part of the density operator was then the part describing the correlations between the coupled systems. In the present work we present new time-dependent projections operators which have the property that some correlations between the interacting subsystems are placed in the relevant part of the distribution function and the remaining correlations are placed in the irrelevant part of the distribution function.     

Long-range order in the XXZ model

The existence of long-range order is proved under certain conditions for the antiferromagnetic quantum spin system with anisotropic interactions (XXZ model) on the simple cubic or the square lattice. In three dimensions (the simple cubic lattice), finite long-range order exists at sufficiently low temperatures for any anisotropy(0) ifS1, and for 0<0.29 (XY-like) or>1.19 (Ising-like) ifS=1/2. In two dimensions (the square lattice), ground-state long-range order exists under the following conditions: for any anisotropy (0) ifS3/2; 0<0.032 (XY-like) or 0.67<<1.34 (almost isotropic) or>1.80 (Ising-like) ifS=1;>1.93 (Ising-like) ifS=1/2. We conjecture that the two-dimensional spin-1/2XY model (=0) has finite ground-state long-range order. Numerical evidence supporting this conjecture is given.     

Electronic surface states on Pd(110): Theory and comparison with experiment

A relativistic Green function formalism has been applied to calculate layer-projected densities of states on Pd(110). In particular, we obtained unoccupied surface states and their dispersion relations along two directions in the surface Brillouin zone. Good agreement with recent inverse photoemission data is reached by using an energy-dependent dynamical surface potential barrier, which is based on a simple electron-plasmon interaction model, instead of a static surface barrier.     

Neutron diffraction studies of the premartensitic transformation B2 → B19′ in Ti49Ni51 single crystals

We have studied structural changes in the high-temperature B2-phase in a large single crystal at temperatures near the premartensitic transformation B2 B19. We are the first to observe an extra 1/2 (110) reflection in neutron diffraction patterns taken along the [110]_B2 direction as the sample is cooled below 420 K, but still far from the martensite start temperature (M_s=180 K). This extra reflection heralds the formation of long-range order in atomic displacements with wave vectorq=(1/2±)[110]2/a. Premartensitic diffraction effects (caused by the development and correlation of lattice waves of atomic displacements with wave vectorsq ₁2/a[1/3, 1/3, 0] andq ₁2/1[1/3, 1/3] that were clearly visible in this same single crystal before the martensitic transformation B2 R, appeared at even lower temperatures with substantially lower intensities.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 56–61, January, 1995.     

Finite-size effects in a field-theoretic model with long-range exchange interaction

We present a systematic approach to the calculation of finite-size (FS) effects for anO(n) field-theoretic model with both short-range (SR) and long-range (LR) exchange interactions. The LR exchange interaction decays at large distances as 1/r ^d+2–2,0⁺,0⁺. Renormalization group calculations ind=d _u – are performed for a system with a fully finite (block) geometry under periodic boundary conditions. We calculate the FS shift of the critical temperature and the FS renormalized coupling constant of the model to one-loop order. The universal scaling variable is obtained and the FS scaling hypothesis is verified.     

Variational evaluation of correlation functions for lattice electrons in high dimensions

We present a general formalism for the diagrammatic calculation of correlation functions for Hubbard-type models in terms of projected wave functions. It is shown that in the limit of high spatial dimensionsd only diagrams with bubble-structure remain. This causes correlation functions to have an overall RPA-type form ind. Exact evaluations are performed for the Gutzwiller wave function. Nearest neighbor correlations are shown to be proportional to their value in the non-interacting case, i.e. are renormalized. However, their absolute value is only of order 1/d. Hence this wave function does not describe spin correlations adequately in high dimensions. The asymptotic behavior of the spin-correlation function is extracted and is found to have a scaling form similar tod=1. Assuming this form to hold in all dimensions we show that the Brinkman-Rice transition only occurs ind=. Finite orders of perturbation theory in 1/d around this singular point are not sufficient to remove the transition.     

Directed paths on percolation clusters

We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is geometrical in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponents and, defined by ft and xt , describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard random energy exponents (=1/3 and =2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponents=1/2 and =v/v, where v and v are the directed percolation exponents. In addition, we predict the absence of a free phase in any dimension at the percolation threshold.     

Multiatom interactions,order and stability in binary transition metal alloys

Interface delocalization or depinning transitions such as wetting or surface induced disorder are considered. At these transitions, the correlation length for transverse correlations parallel to the surface diverges. These correlations are studied in the framework of Landau theory. It is shown the t^–1/2 at all types of transitions for systems with short-range forces wheret measures the distance from bulk coexistence.