Field Theoretic Renormalization Group Research Articles

Polymers in media with long-range-correlated quenched disorder

Abstract We study the scaling properties of self-avoiding walks (SAWs) on a d -dimensional disordered lattice with quenched defects obeying a power law correlation ∼ r − a for large distances r . Such type of disorder is known to be relevant for magnetic phase transitions. We apply the field-theoretical renormalization group approach and perform calculations in a double expansion in e = 4 − d , δ = 4 − a . The asymptotic behaviour of SAWs on a lattice with long-range-correlated disorder is found to be governed by a new exponent ν long = 1/2 + δ/8, (e/2 pure = 1/2 + e/16, (e > 0).

Crossover behavior in the isothermal susceptibility near the 3He critical point.

We present high-resolution measurements of the isothermal susceptibility of pure 3He near the liquid-gas critical point. PVT measurements were performed in the single-phase region over the reduced temperature range 3 x 10(-5)<T/Tc-1<1.5 x 10(-1). The crossover behavior of the susceptibility along the critical isochore was analyzed using a field-theoretical renormalization-group calculation based on the phi4 model. A similar crossover analysis was performed on previously obtained Xe susceptibility measurements. A comparison of the rescaled susceptibility for 3He and Xe shows theoretically predicted universal crossover behavior.

Branching and annihilating Lévy flights.

We consider a system of particles undergoing the branching and annihilating reactions A-->(m+1)A and A+A-->Ø, with m even. The particles move via long-range Lévy flights, where the probability of moving a distance r decays as r(-d-sigma). We analyze this system of branching and annihilating Lévy flights using field theoretic renormalization group techniques close to the upper critical dimension d(c)=sigma with sigma<2. These results are then compared with Monte Carlo simulations in d=1. For sigma close to unity in d=1, the critical point for the transition from an absorbing to an active phase occurs at zero branching. However, for sigma bigger than about 3/2 in d=1, the critical branching rate moves away from zero with increasing sigma, and the transition lies in a different universality class, inaccessible to controlled perturbative expansions. We measure the exponents in both universality classes and examine their behavior as a function of sigma.

Anomalous exponents to order epsilon 3 in the rapid-change model of passive scalar advection.

Field-theoretic renormalization group is applied to the Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance <v(t,x)v(t('),x)>-<v(t,x)v(t('),x('))>~delta(t-t('))|x-x(')|(epsilon). Inertial-range anomalous exponents, related to the scaling dimensions of tensor composite operators built of the scalar gradients, are calculated to the order epsilon(3) of the epsilon expansion. The nature and the convergence of the epsilon expansion in the models of turbulence are briefly discussed.

Critical Frequency Dependence of the Shear Viscosity

Recent low-gravity measurements of the shear viscosity in xenon near its critical point are compared with theoretical results obtained within the field theoretical renormalization group (RG) theory. Nonasymptotic effects and gravity effects are included in our theoretical description, which allows a comparison outside the asymptotic region as well as with earthbound experiments affected by gravity. Comparisons are also made with the theoretical result of the mode coupling theory. In both theories no agreement with the frequency dependence of the real part of the shear viscosity within one-loop theory can be reached. The experimental value of the ratio of the imaginary part to the real part of the shear viscosity at Tc is found to be in agreement with the value calculated within the decoupled mode theory using the two-loop value for the critical exponent of the temperature dependence of the shear viscosity, but not with the one-loop value obtained in RG theory. Thus a complete two-loop calculation of the vertex function for the shear viscosity is required.

A marginal dimension of a weakly diluted quenched m-vector model

We calculate a marginal order parameter dimension $m_c$ which in a weakly diluted quenched $m$-vector model controls the crossover from a universality class of a ``pure'' model ($m>m_c$) to a new universality class ($m<m_c$). Exploiting the Harris criterion and the field-theoretical renormalization group approach allows us to obtain $m_c$ as a five-loop $\epsilon$-expansion as well as a six-loop pseudo-$\epsilon$ expansion. In order to estimate the numerical value of $m_c$ we process the series by precisely adjusted Pad\'e--Borel--Leroy resummation procedures. Our final result $m_c=1.912\pm0.004<2$ stems from the longer and more reliable pseudo-$\epsilon$ expansion, suggesting that a weak quenched disorder does not change the values of $xy$-model critical exponents as it follows from the experiments on critical properties of ${\rm He}^4$ in porous media.

Anomalous scaling of a passive scalar in the presence of strong anisotropy.

Field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta(t,x), advected by the Gaussian strongly anisotropic velocity field with the covariance infinity delta(t-t('))/x-x(')/(epsilon). Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the structure functions S(n)(r) identical with<[straight theta(t,x+r)-straight theta(t,x)](n)> are obtained. They are represented by superpositions of power laws; the corresponding anomalous exponents, which depend explicitly on the anisotropy parameters, are calculated to the first order in epsilon in any space dimension d. In the limit of vanishing anisotropy, the exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. For the finite anisotropy, the exponents cannot be associated with individual operators (which are essentially "mixed" in renormalization), but the aforementioned hierarchy survives for all the cases studied. The second-order structure function S2 is studied in more detail using the renormalization group and zero-mode techniques; the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon, 1/d, and in the anisotropy parameters. If the anisotropy of the velocity is strong enough, the skewness factor S(3)/S(3/2)(2) increases going down towards the depth of the inertial range; the higher-order odd ratios increase even if the anisotropy is weak.

Asymptotic properties of Levy flights in quenched random fields

Long-time asymptotic behavior of the probability distribution function of Levy flights in quenched random fields is analyzed with the use of field-theoretic renormalization group. This problem has been recently studied with the aid of a dynamic renormalization group based on the momentum-shell integration method [H.C. Fogedby, Phys. Rev. E 58, 1690 (1998)]. While a great deal of the results of the quoted paper are confirmed by the present analysis, it is also shown that random field with long-range spatial correlations gives rise to asymptotic behavior with the dynamic critical exponent z less than the step index f of the Levy flights, for a finite range of values of f contrary to the conjecture that always z=f. In particular, in divergenceless random field z=d/2+1-alpha<f, when alpha<1+d/2-f and d<2+2alpha, and correlations fall off as r(-d+2alpha). The physical content of a new critical dimension proposed in the aforementioned paper in connection with the anomalous scaling of the diffusion coefficient is also discussed.

Two-species reaction-diffusion system with equal diffusion constants: anomalous density decay at large times

We study a two-species reaction-diffusion model where A+A-->, A+B-->, and B+B-->, with annihilation rates lambda(0), delta(0)>lambda(0), and lambda(0), respectively. The initial particle configuration is taken to be randomly mixed with mean densities n(A)(0)>n(B)(0), and with the two species A and B diffusing with the same diffusion constant. A field-theoretic renormalization group analysis suggests that, contrary to expectation, the large-time density of the minority species decays at the same rate as the majority when d</=2. Monte Carlo data support the field theory prediction in d=1, while in d=2 the logarithmically slow convergence to the large-time asymptotics makes a numerical test difficult.

Anomalous scaling of a passive scalar advected by the synthetic compressible flow

The field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a Gaussian nonsolenoidal velocity field with finite correlation time in the presence of large-scale anisotropy. The energy spectrum of the velocity in the inertial range has the form E( k)∝ k 1− ε , and the correlation time at the wave number k scales as k −2+ η . It is shown that depending on the values of the exponents ε and η, the model exhibits various types of inertial-range scaling regimes with nontrivial anomalous exponents. Explicit asymptotic expressions for the structure functions and other correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy) anomalous exponents, calculated to the first-order in ε and η in any space dimension. These anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. The anomalous exponents depend explicitly on the degree of compressibility.

Renormalized field theory of driven lattice gases under infinitely fast drive

We use field theoretic renormalization group methods to study the critical behavior of a recently proposed Langevin equation for driven lattice gases under infinitely fast drive. We perform an expansion around the upper critical dimension, d(c)=4, and obtain the critical exponents to one-loop order. The main features of the two-loop calculation are also outlined. The renormalized theory is shown to exhibit a behavior different from the standard field theory for the driven lattice gas with finite driving, i.e., it is not mean-field-like.

Crossover Analyses of Heat Capacity and Susceptibility Measurements near the 3He Liquid-Gas Critical Point

An experiment called MISTE (Microgravity Scaling Theory Experiment) is being developed for a future mission on the International Space Station. The main objective of this flight experiment is to perform in-situ PVT, heat capacity at constant volume, C V , and isothermal susceptibility, XT, measurements in the asymptotic region of the 3 He liquid-gas critical point. On the ground, gravity induces a density gradient that does not permit an accurate test of theoretical predictions within the asymptotic region. In preparation for this flight experiment, precision ground-based measurements are now being performed in the crossover region away from the critical point to determine the crossover parameters. Recent C V and XT measurements along the critical isochore have been analyzed using a new crossover parametric equation-of-state and a field theoretical Renormalization Group calculation based upon the o 4 model. A description of the experimental techniques and preliminary results of the theoretical analyses will be presented.

Structure factor of thin films near continuous phase transitions

The two-point correlation function in thin films is studied near the critical point of the corresponding bulk system. Based on fieldtheoretic renormalization group theory the dependences of this correlation function on the lateral momentum, the two distances normal to the free surfaces, temperature, and film thickness are determined. The corresponding scattering cross section of X-rays and neutrons under grazing incidence is calculated. It reveals the various singularities of the two-point correlation function.

Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field.

The field theoretic renormalization group (RG) is applied to the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient. The energy spectrum in the inertial range has the form E(k) proportional to (1-epsilon), and the correlation time at the wave number k scales as k(-2+eta). It is shown that, depending on the values of the exponents epsilon and eta, the model in the inertial-convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the RG equations: diffusive-type regimes for which the advection can be treated within ordinary perturbation theory, and three nontrivial convection-type regimes for which the correlation functions exhibit anomalous scaling behavior. The explicit asymptotic expressions for the structure functions and other correlation functions are obtained; the anomalous exponents, determined by the scaling dimensions of the scalar gradients, are calculated to the first order in epsilon and eta in any space dimension. For the first nontrivial regime the anomalous exponents are the same as in the rapid-change version of the model; for the second they are the same as in the model with time-independent (frozen) velocity field. In these regimes, the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator. For the last regime the exponents are nonuniversal (they can depend also on the amplitudes); however, the nonuniversality can reveal itself only in the second order of the RG expansion. A brief discussion of the passive advection in the non-Gaussian velocity field governed by the nonlinear stochastic Navier-Stokes equation is also given.

Influence of compressibility on scaling regimes of strongly anisotropic fully developed turbulence

A statistical model of strongly anisotropic fully developed turbulence of the weakly compressible fluid is considered by means of the field theoretic renormalization group. The corrections due to compressibility to the infrared form of the kinetic energy spectrum have been calculated in the leading order in the Mach number expansion. Furthermore, in this approximation the validity of the Kolmogorov hypothesis on the independence of the dissipation length of velocity correlation functions in the inertial range has been proved.

Correlated initial condition simulations in directed percolation

We investigate the influence of correlated initial conditions on the temporal evolution of a (d+1)-dimensional critical directed percolation process. Generating initial states with correlations 〈sisi+r〉∼rσ−d we observe that the density of active sites in Monte Carlo simulations evolves as ρ(t)∼tκ. The exponent κ depends continuously on σ and varies in the range −β/ν∥≤κ≤η. Our numerical results agree with that of an exact field-theoretical renormalization group calculation.

Universal critical coupling constants for the three-dimensional n-vector model from field theory.

The field-theoretical renormalization group (RG) approach in three dimensions is used to estimate the universal critical values of renormalized coupling constants g6 and g8 for the O(n)-symmetric model. The RG series for g6 and g8 are calculated in the four- and three-loop approximations, respectively, and then resummed by means of the Padé-Borel-Leroy technique. Under the optimal value of the shift parameter b providing the fastest convergence of the iteration procedure, numerical estimates for g*6 are obtained with an accuracy no worse than 0.3%. The RG expansion for g8 demonstrates a stronger divergence, and results in considerably cruder numerical estimates.

Multifractality of Brownian motion near absorbing polymers.

We characterize the multifractal behavior of Brownian motion in the vicinity of an absorbing star polymer. We map the problem to an O(M)-symmetric phi(4)-field theory relating higher moments of the Laplacian field of Brownian motion to corresponding composite operators. The resulting spectra of scaling dimensions of these operators display the convexity properties that are necessarily found for multifractal scaling but unusual for power of field operators in field theory. Using a field-theoretic renormalization group approach we obtain the multifractal spectrum for absorption at the core of a polymer star as an asymptotic series. We evaluate these series using resummation techniques.

Boundary effects in reaction-diffusion processes

The effects of a boundary on reaction systems are examined in the framework of the general single-species reaction/coalescence process. The boundary naturally represents the reactants' container, but is applicable to exciton dynamics in a doped TMMC crystal. We show that a density excess, which extends into the system diffusively from the boundary, is formed in two dimensions and below. This implies a surprising result for the magnetization near a fixed spin in the coarsening of the one-dimensional critical Ising model. The universal, dimensionally dependent functional forms of this density excess are given by an exact solution and the field-theoretic renormalization group.

The correction-to-scaling exponent in dilute systems

The leading correction-to-scaling exponent $\omega$ for the three-dimensional dilute Ising model is calculated in the framework of the field theoretic renormalization group approach. Both in the minimal subtraction scheme as well as in the massive field theory (resummed four loop expansion) excellent agreement with recent Monte Carlo calculations [Ballesteros H G, et al Phys. Rev. B 58, 2740 (1998)] is achieved. The expression of $\omega$ as series in a $\sqrt{\epsilon}$-expansion up to ${\cal O}(\epsilon^2)$ does not allow a reliable estimate for $d=3$.