An Analytic Equation of State for Ising-like Models

Using an Environmentally Friendly Renormalization we derive, from an underlying field theory representation, a formal expression for the equation of state, $y=f(x)$, that exhibits all desired asymptotic and analyticity properties in the three limits $x\to 0$, $x\to \infty$ and $x\to -1$. The only necessary inputs are the Wilson functions $\gamma_\lambda$, $\gamma_\phi$ and $\gamma_{\phi^2}$, associated with a renormalization of the transverse vertex functions. These Wilson functions exhibit a crossover between the Wilson-Fisher fixed point and the fixed point that controls the coexistence curve. Restricting to the case N=1, we derive a one-loop equation of state for $2< d<4$ naturally parameterized by a ratio of non-linear scaling fields. For $d=3$ we show that a non-parameterized analytic form can be deduced. Various asymptotic amplitudes are calculated directly from the equation of state in all three asymptotic limits of interest and comparison made with known results. By positing a scaling form for the equation of state inspired by the one-loop result, but adjusted to fit the known values of the critical exponents, we obtain better agreement with known asymptotic amplitudes.Comment: 10 pages, 2 figure

Fractal and Transfractal Recursive Scale-Free Nets

We explore the concepts of self-similarity, dimensionality, and (multi)scaling in a new family of recursive scale-free nets that yield themselves to exact analysis through renormalization techniques. All nets in this family are self-similar and some are fractals - possessing a finite fractal dimension - while others are small world (their diameter grows logarithmically with their size) and are infinite-dimensional. We show how a useful measure of "transfinite" dimension may be defined and applied to the small world nets. Concerning multiscaling, we show how first-passage time for diffusion and resistance between hub (the most connected nodes) scale differently than for other nodes. Despite the different scalings, the Einstein relation between diffusion and conductivity holds separately for hubs and nodes. The transfinite exponents of small world nets obey Einstein relations analogous to those in fractal nets.Comment: Includes small revisions and references added as result of readers' feedbac

Novel glassy behavior in a ferromagnetic p-spin model

Recent work has suggested the existence of glassy behavior in a ferromagnetic model with a four-spin interaction. Motivated by these findings, we have studied the dynamics of this model using Monte Carlo simulations with particular attention being paid to two-time quantities. We find that the system shares many features in common with glass forming liquids. In particular, the model exhibits: (i) a very long-lived metastable state, (ii) autocorrelation functions that show stretched exponential relaxation, (iii) a non-equilibrium timescale that appears to diverge at a well defined temperature, and (iv) low temperature aging behaviour characteristic of glasses.Comment: 6 pages, 5 figure

Critical Fluctuations and Disorder at the Vortex Liquid to Crystal Transition in Type-II Superconductors

We present a functional renormalization group (FRG) analysis of a Landau-Ginzburg model of type-II superconductors (generalized to $n/2$ complex fields) in a magnetic field, both for a pure system, and in the presence of quenched random impurities. Our analysis is based on a previous FRG treatment of the pure case [E.Br\'ezin et. al., Phys. Rev. B, {\bf 31}, 7124 (1985)] which is an expansion in $\epsilon = 6-d$. If the coupling functions are restricted to the space of functions with non-zero support only at reciprocal lattice vectors corresponding to the Abrikosov lattice, we find a stable FRG fixed point in the presence of disorder for $1<n<4$, identical to that of the disordered $O(n)$ model in $d-2$ dimensions. The pure system has a stable fixed point only for $n>4$ and so the physical case ($n = 2$) is likely to have a first order transition. We speculate that the recent experimental findings that disorder removes the apparent first order transition are consistent with these calculations.Comment: 4 pages, no figures, typeset using revtex (v3.0

Random field Ising systems on a general hierarchical lattice: Rigorous inequalities

Random Ising systems on a general hierarchical lattice with both, random fields and random bonds, are considered. Rigorous inequalities between eigenvalues of the Jacobian renormalization matrix at the pure fixed point are obtained. These inequalities lead to upper bounds on the crossover exponents $\{\phi_i\}$.Comment: LaTeX, 13 pages, figs. 1a,1b,2. To be published in PR

Charge Transport in the Dense Two-Dimensional Coulomb Gas

The dynamics of a globally neutral system of diffusing Coulomb charges in two dimensions, driven by an applied electric field, is studied in a wide temperature range around the Berezinskii-Kosterlitz-Thouless transition. I argue that the commonly accepted ``free particle drift'' mechanism of charge transport in this system is limited to relatively low particle densities. For higher densities, I propose a modified picture involving collective ``partner transfer'' between bound pairs. The new picture provides a natural explanation for recent experimental and numerical findings which deviate from standard theory. It also clarifies the origin of dynamical scaling in this context.Comment: 4 pages, RevTeX, 2 eps figures included; some typos corrected, final version to be published in Phys. Rev. Let

Universal Critical Behavior of Aperiodic Ferromagnetic Models

We investigate the effects of geometric fluctuations, associated with aperiodic exchange interactions, on the critical behavior of $q$-state ferromagnetic Potts models on generalized diamond hierarchical lattices. For layered exchange interactions according to some two-letter substitutional sequences, and irrelevant geometric fluctuations, the exact recursion relations in parameter space display a non-trivial diagonal fixed point that governs the universal critical behavior. For relevant fluctuations, this fixed point becomes fully unstable, and we show the apperance of a two-cycle which is associated with a novel critical behavior. We use scaling arguments to calculate the critical exponent $\alpha$ of the specific heat, which turns out to be different from the value for the uniform case. We check the scaling predictions by a direct numerical analysis of the singularity of the thermodynamic free-energy. The agreement between scaling and direct calculations is excellent for stronger singularities (large values of $q$). The critical exponents do not depend on the strengths of the exchange interactions.Comment: 4 pages, 1 figure (included), RevTeX, submitted to Phys. Rev. E as a Rapid Communicatio

Nonuniversal Correlations and Crossover Effects in the Bragg-Glass Phase of Impure Superconductors

The structural correlation functions of a weakly disordered Abrikosov lattice are calculated in a functional RG-expansion in $d=4-\epsilon$ dimensions. It is shown, that in the asymptotic limit the Abrikosov lattice exhibits still quasi-long-range translational order described by a {\it nonuniversal} exponent $\eta_{\bf G}$ which depends on the ratio of the renormalized elastic constants $\kappa ={c}_{66}/ {c}_{11}$ of the flux line (FL) lattice. Our calculations clearly demonstrate three distinct scaling regimes corresponding to the Larkin, the random manifold and the asymptotic Bragg-glass regime. On a wide range of {\it intermediate} length scales the FL displacement correlation function increases as a power law with twice the manifold roughness exponent $\zeta_{\rm RM}(\kappa)$, which is also {\it nonuniversal}. Correlation functions in the asymptotic regime are calculated in their full anisotropic dependencies and various order parameters are examined. Our results, in particular the $\kappa$-dependency of the exponents, are in variance with those of the variational treatment with replica symmetry breaking which allows in principle an experimental discrimination between the two approaches.Comment: 17 pages, 10 figure

Critical behaviour of the Random--Bond Ashkin--Teller Model, a Monte-Carlo study

The critical behaviour of a bond-disordered Ashkin-Teller model on a square lattice is investigated by intensive Monte-Carlo simulations. A duality transformation is used to locate a critical plane of the disordered model. This critical plane corresponds to the line of critical points of the pure model, along which critical exponents vary continuously. Along this line the scaling exponent corresponding to randomness $\phi=(\alpha/\nu)$ varies continuously and is positive so that randomness is relevant and different critical behaviour is expected for the disordered model. We use a cluster algorithm for the Monte Carlo simulations based on the Wolff embedding idea, and perform a finite size scaling study of several critical models, extrapolating between the critical bond-disordered Ising and bond-disordered four state Potts models. The critical behaviour of the disordered model is compared with the critical behaviour of an anisotropic Ashkin-Teller model which is used as a refference pure model. We find no essential change in the order parameters' critical exponents with respect to those of the pure model. The divergence of the specific heat $C$ is changed dramatically. Our results favor a logarithmic type divergence at $T_{c}$, $C\sim \log L$ for the random bond Ashkin-Teller and four state Potts models and $C\sim \log \log L$ for the random bond Ising model.Comment: RevTex, 14 figures in tar compressed form included, Submitted to Phys. Rev.

Short-Range Ising Spin Glass: Multifractal Properties

The multifractal properties of the Edwards-Anderson order parameter of the short-range Ising spin glass model on d=3 diamond hierarchical lattices is studied via an exact recursion procedure. The profiles of the local order parameter are calculated and analysed within a range of temperatures close to the critical point with four symmetric distributions of the coupling constants (Gaussian, Bimodal, Uniform and Exponential). Unlike the pure case, the multifractal analysis of these profiles reveals that a large spectrum of the $\alpha$-H\"older exponent is required to describe the singularities of the measure defined by the normalized local order parameter, at and below the critical point. Minor changes in these spectra are observed for distinct initial distributions of coupling constants, suggesting an universal spectra behavior. For temperatures slightly above T_{c}, a dramatic change in the $F(\alpha)$ function is found, signalizing the transition.Comment: 8 pages, LaTex, PostScript-figures included but also available upon request. To be published in Physical Review E (01/March 97