Equation of state and dynamical properties near the Yang-Lee edge singularity

Statics and dynamics of the Yang-Lee edge singularity are investigated by field theoretic renormalization group techniques. Exploiting a continuous symmetry under a shift of the order parameter we calculate the static critical exponent to order ²=(6–d)², in accordance with previous results; in addition, we derive the equation of state and its asymptotic behaviour. The dynamic scaling exponentz is calculated to order ² from a purely relaxational model with non-conserved order parameter; joining the -expansion to an exact result ford=0 in a [2/1] Padè approximant we estimatez=1.81 ford=3.     

Critical behaviour of directed branched polymers and the dynamics at the Yang-Lee edge singularity

Relying on a field theoretic model due to Day and Lubensky we establish the one-to-one correspondence of the directed branched polymer problem ind dimensions to (relaxational) critical dynamics at the Yang-Lee edge ind–1 spatial dimensions; like their isotropic counterparts the directed polymer exponents andv are uniquely determined by the static Yang-Lee exponent whereasv requires in addition the dynamic Yang-Lee exponentz. JoiningO(²)-expansions about the upper critical dimensiond _c =7 to exact results atd=1 and 2 by Padé-interpolations we obtain good agreement with series expansion data for low dimensions.     

Finite size scaling for directed percolation and related stochastic evolution processes

Finite size scaling effects are investigated for certain evolution processes modelled by a one-component reaction-diffusion system with an absorbing state. The model possesses a non-equilibrium critical point, and the associated universality class includes directed bond percolation, cellular automata, Reggeon field theory and a stochastic version of Schlögl's first autocatalytic reaction scheme. Using renormalisation group techniques, we calculate the linear relaxation time in a cubic geometry of finite sizeL, with periodic boundary conditions imposed. The corresponding scaling behaviour toO() (=4–d,d being the spatial dimension) is presented in universal form.     

Field theory of critical behavior in a driven diffusive system

We use a field theoretic renormalization group method to study the critical properties of a diffusive system with a single conserved density subject to a constant uniform external field. A fixed point stable belowd _c=5 is found to govern the critical behavior. Scaling forms of density correlation functions are derived and critical exponents are obtained to all orders in =5–d. Spatial correlations are found to be very anisotropic with elongated correlations along the external field. Long wavelength transverse fluctuations are suppressed completely to yield mean field transverse exponents.     

Relaxation times in a finite Ising system with random impurities

Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM ²(t). In addition, universal cumulant ratios are calculated to second order in ^1/4, where =4–d andd<4 denotes the spatial dimension.     

1/d corrections for interacting spinless fermions: One-particle properties

One-particle properties of the spinless fermion model with repulsion at half filling are calculated within an approach correct to first order in the inverse of the lattice dimensiond. Continuity of the limitd requires a scaling of the nearest-neighbour hopping proportional to and of the nearest-neighbour interaction proportional to 1/d. Due to this scaling the Hartree approximation becomes exact in infinite dimensions. We show that 1/d corrections comprise the Fock diagram and the local correlation diagram in the self-consistent Dyson equation. This approach is applied to simple-cubic systems in dimensiond=1, 2 and 3. Ground state properties and the charge-density wave phase diagram are calculated. AtT=0 the inclusion of 1/d terms gives only small corrections to the leading Hartree contribution ind=2, 3. ForT>0, however, the 1/d corrections are important. They lead to a non-negligible reduction of the critical temperature. Ind=1 the 1/d corrections are very large, but they do not succeed in removing the spurious phase transition atT>0. The 1/d approach provides a good and tractable approximation ind=3 and probably ind=2, which allows also further systematic improvement.     

On the critical behavior of the magnetization in high-dimensional Ising models

We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that ind4 dimensions the magnetization is continuous atT _c and its critical exponents take the classical values=3 and=1/2, with possible logarithmic corrections atd=4. The continuity, and other explicit bounds, formally extend tod>3 1/2. Other systems to which the results apply include long-range models ind=1 dimension, with 1/|x–y| couplings, for which 2/(–1) replacesd in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.Research supported in part by NSF grant PHY-8301493 A02, and by a John S. Guggenheim Foundation fellowship (M.A.).     

Scaling behavior of permeability and conductivity anisotropy near the percolation threshold

We use the finite-size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp _c . Here is defined by the scaling lawk _l /k _t –1(p–p _c ), wherek _t andk _t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are (d=2)0.819±0.011 and (d=3)0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for due to Straley, and propose that (d=2)=t- _B , wheret is the critical exponent of the conductivity or permeability of the system, and _B is the critical exponent of the backbone of percolation clusters.     

Critical behavior of short range Potts glasses

We study by means of Monte Carlo simulations and the numerical transfer matrix technique the critical behavior of the short rangep=3 state Potts glass model in dimensionsd=2,3,4 with both Gaussian and bimodal (±J) nearest neighbor interactions on hypercubic lattices employing finite size scaling ideas. Ind=2 in addition the degeneracy of the glass ground state is computed as a function of the number of Potts states forp=3, 4, 5 and compared to that of the antiferromagnetic ground state. Our data indicate a transition into a glass phase atT=0 ind=2 with an algebraic singularity, aT=0 transition ind=3 with an essential singularity of the form exp(const.T ^–2), and an algebraic singularity atT0.25 ind=4. We conclude that the lower critical dimension of the present model isd _c =3 or very close to it. Some of the critical exponents are estimated and their respective values discussed.     

Crossover scaling function for the free energy

A cubic field, coupling tos|s|², inn-component spin models induces a bicritical crossover fromn-isotropic to Ising like (m=1) critical behaviour for 1<n<, but to classical behaviour in the limitn. By following the analysis of Nelson and Domany, the bicritical scaling function for the free energy ind dimensions is obtained correct to order =4–d and for general (m,n). The mechanism responsible for the breakdown of hyperscaling in the classical behaviour is discussed.     

Field theory of critical behaviour in driven diffusive systems

We present a field theoretic renormalisation group study for the critical behaviour of a diffusive system with a single conserved density subjected to an external driving force. The anisotropies induced by the external field require the introduction of two critical parameters associated with transverse and longitudinal order. The transition to transverse order is governed by a fixed point which is infrared stable below five dimensions. With the help of Ward-Takahashi identities based on Galilei invariance, we derive scaling forms for density correlation functions, critical exponents to all orders in =5–d, and the equation of state, taking care of a dangerous irrelevant composite operator. The transition is continuous and of mean-field type, with anomalous long-wavelength and long-time correlations in the longitudinal direction only. For the transition to longitudinal order, no infrared stable fixed point is found. An analysis of the mean-field equations indicates that the transition is discontinuous.     

Diffusion in layered random flows,polymers, electrons in random potentials,and spin depolarization in random fields

Several related models are studied in a common framework. We first reconsider the model of Matheron and de Marsilly for (anomalous) tracer dispersion in a stratified porous medium. In each horizontal layer the flow velocity is constant, parallel to the layer, and depends randomly on the vertical coordinate z. This model is mapped onto ad=1 localization problem in a random potential and, equivalently, onto ad=1 polymer. At larget theaveraged distribution of horizontal displacementsx takes the scaling form [P(x, t, z=0)]=at ^–5/4 Q(bxt ^–3/4), whereQ(y) is independent of the details of the model.Q(y),a, andb are obtained exactly for a large class of models. From the Lifschitz tails of the localization problem we find in the regionxt ^3/4, i.e.,y, thatQ(y)¦y¦ exp(–C¦y¦^4/3). We also obtain exactly ind=1 the scaling functions for the local and total average magnetization of spins diffusing in a random magnetic field, by mapping onto a polymer problem, as well as the average local concentration for diffusion in the presence of random sources and sinks. These mappings are then used to study higher-dimensional extensions of these models.     

O(N) vector model with twisted boundary conditions

Prompted by a recent article of Chakravarty, we reexamine theO(N) vector model with twisted boundary conditions ind dimensions in the various frameworks of the =d–2 expansion, the =4–d expansion, and the large-N expansion. These continuum models describe the physics below the critical temperatureT _c and nearT _c of a latticeO(N) spin model. We determine the effect of the twisting on finite-size scaling functions, for various geometries.On leave from G. Nadjakov Institute of Solid State Physics, 1784 Sofia, Bulgaria.     

On scaling properties of cluster distributions in Ising models

Scaling relations of cluster distributions for the Wolff algorithm are derived. We found them to be well satisfied for the Ising model ind=3 dimensions. Using scaling and a parametrization of the cluster distribution, we determine the critical exponent/=0.516(6) with moderate effort in computing time.     

Finite-size dependence of the helicity modulus within the mean spherical model

The validity of the finite-size scaling prediction about the existence of logarithmic corrections in the helicity modulus of three-dimensional O(n)-symmetric order parameter systems in confined geometries is studied for the three-dimensional mean spherical model of geometryL ^3/s-d×^d, 0d<3. For a fully finite geometry the general case ofd _p0 periodic,d _a0 antiperiodic,d ₀0 free, andd ₁0 fixed (d _p+d_a+d₀+d₁=d, d=3) boundary conditions is considered, whereas for film (d=2) and cylinder (d=1) geometries only the case of antiperiodic and/or periodic boundary conditions is investigated. The corresponding expressions for the finite-size scaling function of the helicity modulus and its asymptotics in the vicinity, below, and above the bulk critical temperatureT _c and the shifted critical temperatureT _c,L are derived. The obtained results are not in agreement with the hypothesis of the existence of a log(L) correction term to the finite-size behavior of the helicity modulus in the finite-size critical region if d=3. In the case of film and cylinder geometries there are no logarithmic corrections. In the case of a fully finite geometry a universal logarithmic correction term –[(d ₀ –d ₁)/4–2^da–1/²] lnL/L is obtained only for (T _c-T) LlnL.     

Restoration of universality for the rod-to-coil transition scaling in the infinite-dimensionality limit: Exact results for directed walks

We derive scaling forms for the thermodynamic and correlation quantities for the turn-weighted fully and partially directed self-avoiding walks on the hypercubic lattices ind2. In the grand canonical (fixed fugacity per step) ensemble, the conformational rod-to-coil transition sets up in the regimew¯N=O(1), wherew is the weight of each 90° turn and¯N is the (fugacity-dependent) average number of steps. Contrary to the conventional critical phenomena wisdom, the scaling functions for the two different walk models, directed and partially directed, become universal only in the limitd.     

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.     

Possible symmetrization of Fisher's droplet picture nearT c : A bubble-droplet formula

An expression for the pressure is proposed which leads to a symmetric equation of state for liquid and gas near the critical point. Our bubble-droplet formula is similar to Fisher's cluster expansion but contains an additional term due to the density dependence of the surface tension. Also, it assumes the density difference between a droplet (or bubble) and the surrounding medium to be proportional tol ^{– 1/} and not to be independent of the droplet sizel. Then, the scaling homogeneity assumption and some scaling laws, includingd=2–, can be derived (d is dimensionality). The additional assumption of spherical droplets and bubbles leads to a new scaling law 1+=(d–1), which is only slightly violated in the lattice gas ford=2, 3, 4.Work partially supported by Research Corporation, a Frederick Gardner Cottrell grant aid.     

Scaling exponents for kinetic roughening in higher dimensions

We discuss the results of extensive numerical simulations in order to estimate the scaling exponents associated with kinetic roughening in higher dimensions, up tod=7 + l. To this end, we study the restricted solid-on-solid growth model, for which we employ a novel fitting ansatz for the spatially averaged height correlation function¯G(t)t ² to estimate the scaling exponent. Using this method, we present a quantitative determination of ind=3 + 1 and 4+1 dimensions. To check the consistency of these results, we also compute the interface width and determine andx from it independently. Our results are in disagreement with all existing theories and conjectures, but in four dimensions they are in good agreement with recent simulations of Forrest and Tang for a different growth model. Above five dimensions, we use the time dependence of the width to obtain lower bound estimates for. Within the accuracy of our data, we find no indication of an upper critical dimension up tod = 7 + 1.     

Order parameters forn-vector spin glasses ind dimensions

We report Monte Carlo simulations of the time-dependent behavior of Edwards-Anderson spin glasses with Gaussian nearest-neighbor exchange, for both spin dimensionalityn and space dimensionalityd from 2 up to 6. A (nearly) logarithmic decay of the Edwards-Anderson order parameter with time is observed for alln and alld, similar to earlier studies forn=1. But the Monte Carlo data forn>1 suggest stronger than those forn=1 that all order parameters considered vanish in thermal equilibrium for nonzero temperature, because the decay forn>1 is faster at the temperatures of interest. For Heisenberg spins (n=3) no significant dependence of the Edwards-Anderson order parameterq on the size of the lattice was observed ford=2,3 and 4, whereas ford=5 and 6,q was smaller for smaller systems (in contrast to thed=5 Ising case). These results are the first Monte Carlo indication of a change in the bulk behavior of Heisenberg spin glasses at dimensionalityd=4. Quenching the system to zero temperature and then applying a field we find that the order parameter , measuring the alignment with respect to the state at zero field, is destroyed by a sufficiently strong magnetic field, for all observedn andd.Sonderforschungsbereich 125 Aachen-Jülich-Köln, FRG