Epsilon Expansion Research Articles

Stochastic magnetohydrodynamic turbulence in space dimensions d > or =2.

Interplay of kinematic and magnetic forcing in a model of a conducting fluid with randomly driven magnetohydrodynamic equations has been studied in space dimensions d > or =2 by means of the renormalization group. A perturbative expansion scheme, parameters of which are the deviation of the spatial dimension from two and the deviation of the exponent of the powerlike correlation function of random forcing from its critical value, has been used in one-loop approximation. Additional divergences have been taken into account that arise at two dimensions and have been inconsistently treated in earlier investigations of the model. It is shown that in spite of the additional divergences, the kinetic fixed point associated with the Kolmogorov scaling regime remains stable for all space dimensions d > or =2 for rapidly enough falling off correlations of the magnetic forcing. A scaling regime driven by thermal fluctuations of the velocity field has been identified and analyzed. The absence of a scaling regime near two dimensions driven by the fluctuations of the magnetic field has been confirmed. A renormalization scheme has been put forward and numerically investigated to interpolate between the epsilon expansion and the double expansion.

Polymers in long-range-correlated disorder.

We study the scaling properties of polymers in a d-dimensional medium with quenched defects that have power law correlations approximately r(-a) for large separations r. This type of disorder is known to be relevant for magnetic phase transitions. We find strong evidence that this is true also for the polymer case. Applying the field-theoretical renormalization group approach we perform calculations both in a double expansion in epsilon=4-d and delta=4-a up to the one-loop order and second in a fixed dimension (d=3) approach up to the two-loop approximation for different fixed values of the correlation parameter, 2<or=a<or=3. In the latter case the numerical results need appropriate resummation. We find that the asymptotic behavior of self-avoiding walks in three dimensions and long-range-correlated disorder is governed by a set of separate exponents. In particular, we give estimates for the nu and gamma exponents as well as for the correction-to-scaling exponent omega. The latter exponent is also calculated for the general m-vector model with m=1,2,3.

Bicritical and Tetracritical Phenomena and Scaling Properties of the SO(5) Theory

By Monte Carlo simulations on a 3D SO(5) rotator model it is shown that, in contrast to the epsilon expansions of the renormalization group, the bicritical point is stable to biquadratic perturbations of AF-SC (antiferromagnetism-superconductivity) repulsions, which are produced by quantum fluctuations originated from the Gutzwiller projection. Therefore, the present work completes the link from a physical projected SO(5) starting point to an asymptotic SO(5) symmetry point. The tetracritical point is stable for attractive AF-SC interactions. Critical exponents and ratios are evaluated by scaling analysis. Bicritical and tetracritical scaling functions are derived for the first time. Suggestions on experiments are given.

Surface critical behavior of random systems: ordinary transition.

We calculate the surface critical exponents of the ordinary transition occurring in semi-infinite, quenched dilute Ising-like systems. This is done by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation as well as in 4-epsilon dimensions. At d=4-epsilon we extend, up to the next-to-leading order, the previous first-order results of the square root of epsilon expansion by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. In both cases numerical estimates for surface exponents are computed using Padé approximants extrapolating the perturbation theory expansions. The obtained results indicate that the critical behavior of semi-infinite systems with quenched bulk disorder is characterized by the new set of surface critical exponents.

Phase transitions in generalized chiral or Stiefel's models

We study the phase transition in generalized chiral or Stiefel's models using Monte Carlo simulations. These models are characterized by a breakdown of symmetry O(N)/O(N-P). We show that the phase transition is clearly first order for N >= 3 when P=N and P=N-1, contrary to predictions based on the Renormalization Group in 4-\epsilon expansion but in agreement with a recent non perturbative Renormalization Group approach.

Tubular phase of self-avoiding anisotropic crystalline membranes.

We analyze the tubular phase of self-avoiding anisotropic crystalline membranes. A careful analysis using renormalization group arguments together with symmetry requirements motivates the simplest form of the large-distance free energy describing fluctuations of tubular configurations. The non-self-avoiding limit of the model is shown to be exactly solvable. For the full self-avoiding model we compute the critical exponents using an epsilon expansion about the upper critical embedding dimension for general internal dimension D and embedding dimension d. We then exhibit various methods for reliably extrapolating to the physical point (D=2,d=3). Our most accurate estimates are nu=0.62 for the Flory exponent and zeta=0.80 for the roughness exponent.

What is the Right Form of the Probability Distribution of the Conductance at the Mobility Edge?

The probability distribution of the conductance Pc(g) at the Anderson critical point is calculated. It is find that Pc(g) has a dip at small g in agreement with epsilon expansion results. The Pc(g) for the 3d system is quite different from the 2d quantum critical point of the integer quantum Hall effect. The universality or not of these distributions is of central importance to the field of disordered systems.

Modified Borel summation of divergent series and critical-exponent estimates for anN-vector cubic model in three dimensions from five-loop ε expansions

A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the exact asymptotic parameters to be known. The method is tested on functions expanded in their asymptotic power series. It is applied to estimating the critical exponent values for an N-vector field model, describing magnetic and structural phase transitions in cubic and tetragonal crystals, from five-loop \epsilon expansions.

Onset of propagation of planar cracks in heterogeneous media

The dynamics of planar crack fronts in hetergeneous media near the critical load for onset of crack motion are investigated both analytically and by numerical simulations. Elasticity of the solid leads to long range stress transfer along the crack front which is non-monotonic in time due to the elastic waves in the medium. In the quasistatic limit with instantaneous stress transfer, the crack front exhibits dynamic critical phenomenon, with a second order like transition from a pinned to a moving phase as the applied load is increased through a critical value. At criticality, the crack-front is self-affine, with a roughness exponent $\zeta =0.34\pm 0.02$. The dynamic exponent $z$ is found to be equal to $ 0.74\pm 0.03$ and the correlation length exponent $\nu =1.52\pm 0.02$. These results are in good agreement with those obtained from an epsilon expansion. Sound-travel time delays in the stress transfer do not change the static exponents but the dynamic exponent $z$ becomes exactly one. Real elastic waves, however, lead to overshoots in the stresses above their eventual static value when one part of the crack front moves forward. Simplified models of these stress overshoots are used to show that overshoots are relevant at the depinning transition leading to a decrease in the critical load and an apparent jump in the velocity of the crack front directly to a non-zero value. In finite systems, the velocity also shows hysteretic behaviour as a function of the loading. These results suggest a first order like transition. Possible implications for real tensile cracks are discussed.

Scaling and Renormalization in Statistical Physics

This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.

Critical Exponents for the Landau-De Gennes Model of the Nematic-Isotropic Phase Transition

In this paper we report the application of Wilson's recursion relations to the numerical calculation of critical indices for the d = 3 Landau-de Gennes model near the isolated critical point on the nematic-isotropic phase transition line. We find that the calculated critical exponents namely y = 1.277 and v = 0.638 in fair agreement with the best epsilon expansion results.

FLIPPING OF MAGNETIZATION IN ISING MODELS AT Tc

The two-dimensional simulations of Majumdar, Bray, Cornell, and Sire are confirmed and generalized to three to five dimensions. They agree with their epsilon expansion.

Epsilon expansion of the conductivity at the superconductor-Mott-insulator transition.

We study the critical behavior of the conductivity $\sigma(\omega)$ at the zero temperature superconductor-Mott insulator transition in $d$ space-time dimensions for a model of bosons with short-range interaction and no disorder. We obtain $\sigma(\omega_n ) = (4e^2/\hbar) \sigma_{\epsilon} \omega_n^{1-\epsilon}$, as predicted by the scaling theory, and the prefactor $\sigma_{\epsilon}$ is calculated in the $\epsilon$-expansion, to order $\epsilon ^2$ ($\epsilon = 4-d$). In two spatial dimensions, ($d=3$), we find a value of the universal conductance $\sigma^\star =0.315 (4e^2/h)$, in good agreement with the known Monte Carlo results.

Critical region of the nematic-isotropic phase transition in the epsilon expansion.

Critical region of the nematic-isotropic phase transition in the epsilon expansion.

New universality classes for quantum critical behavior.

We use the epsilon expansion to explore a new universality class of second order quantum phase transitions associated with a four-dimensional Yukawa field theory coupled to a traceless Hermitean matrix scalar field. We argue that this class includes four-fermi models in $2<D<4$ dimensions with $SU(N_F)\times U(N)$ symmetry and a U(N) scalar, $SU(N_F)$ iso-vector 4-fermi coupling. The epsilon expansion indicates that there is a second order phase transition for $N\geq N^*(N_F)$, where $N^*(N_F)\simeq.27N_F$ if $N_F\to\infty$.

Probability density, diagrammatic technique, and epsilon expansion in the theory of wave turbulence

We apply the methods of field theory to study the turbulent regimes of statistical systems. First we show how one can find their probability densities. For the case of the theory of wave turbulence with four-wave interaction we calculate them explicitly and study their properties. Using those densities we show how one can in principle calculate any correlation function in this theory by means of direct perturbative expansion in powers of the interaction. Then we give the general form of the corrections to the kinetic equation and develop an appropriate diagrammatic technique. This technique, while resembling that of ϕ 4 theory, has many new distinctive features. The role of the ϵ = d − 4 parameter of ϕ 4 theory is played here by the parameter κ = β + d − α − γ where β is the dimension of the interaction, d is the space dimension, α is the dimension of the energy spectrum and γ is the “classical” wave density dimension. If κ > 0 then the Kolmogorov index is exact, and if κ < 0 then we expect it to be modified by the interaction. For κ a small negative number, α < 1 and a special form of the interaction we compute this modification explicitly. We neglect the encountered IR divergencies with the intend to study them in a later publication.

Phase transitions and mass generation in 2+1 dimensions.

The possibility that the epsilon expansion can predict the order of phase transitions in three dimensional field theories is examined. For a Hermitean matrix-valued order parameter, the epsilon expansion predicts fluctuation induced first order phase transitions. We analyze two 2+1-dimensional quantum field theories which exhibit spontaneous symmetry breaking and have martix order parameters. Using the large $N$ expansion, we show that these models exhibit second order transitions and discuss the implications for the chiral symmetry breaking transition in 2+1-dimensional QCD for a critical number of quark flavors.

A numerical study of the hierarchical Ising model: High-temperature versus epsilon expansion

We study numerically the magnetic susceptibility of the hierarchical model with Ising spins (σ=±1) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly using recursive methods which exploit the symmetries of the model. Lattices with up to 218 sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form (1-β/β 0)g for thewhole temperature range considered. This approximate law implies a simple approximate formula for the coefficients of the high-temperature expansion, and, more importantly, approximate relations among the coefficients themselves. We found that some of these approximate relations hold with errors less then 2%. On the other hand,g differs significantly from the critical exponent γ calculated with the epsilon expansion, even when the fit is restricted to intervals closer toβ c. We discuss this discrepancy in the context the renormalization group analysis of the hierarchical model.

Anderson localization for sublattice models

A new universality class for the Anderson-localization behaviour is established. It is represented by a particular class of the Anderson model where all random variables or regular matrix transfer elements connect only points of different sublattices. For the band centre an effective field theory based on the replica trick with manifold GL(4m, C)/ U(4m) is derived. From an epsilon expansion a trivial beta function indicating extended states at the band centre in two dimensions is obtained. Due to the nonvanishing zeta function the density of states is enhanced for d > 2 and diverges as d approaches 2. For nonzero energies the sublattice restriction does not modify the symmetry group. The energy dependence of several quantities related to one- and two-particle Green functions is given in a crude approximation.

Finite-Size scaling of the interfacial tension

The interfacial free energy appropriate to a finite system in the Ising universality class is calculated to one-loop order. Consideration is given to the behavior in both three and four dimensions via an epsilon expansion. The results corresponding to three dimensions are compared with the simulation data of Mon and Jasnow.