Renormalization of one-dimensional avalanche models

We investigate a renormalization group (RG) scheme for avalanche automata introduced recently by Pietroneroet al. to explain universality in self-organized criticality models. Using a modified approach, we construct exact RG equations for a one-dimensional model whose detailed dynamics is exactly solvable. We then investigate in detail the effect of approximations inherent in a practical implementation of the RG transformation where exact dynamical information is unavailable.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

Phase transition in the 3d random field Ising model

We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,d _l for the theory isd _l 2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.     

Migdal-Kadanoff renormalization group for theO(n) model

We perform a Migdal-Kadanoff renormalization group calculation on anO(n) symmetric model on ad-dimensional hypercubic lattice,d=2, 3. We find that in two dimensions the critical fixed point disappears asn=n _KT1.96, which is in good agreement with the exact valuen _KT=2. In three dimensions the fixed point persists much longer, albeit not all the way up to infinity. Surface critical phenomena in a semiinfiniteO(n) model are also considered.     

The field-theoretic description of dynamics of interfaces in disordered media

The time evolution of an interface in a disordered media is described by using the propagator method. The method enables one to represent the perturbation expansions of different quantities characterizing the interface by means of diagrams which are familiar from the field theory. By the analysis of the divergences in the vicinity of the critical dimension d_c = 4 we found that the regularization of the theory demands the renormalization of the mobility and all moments of the disorder correlator excepting the zero one. The renormalization group (RG) calculations of the average velocity of the interface, the roughness, and the functional RG equation of the disorder correlator are presented to order ? = 4 - d. The latter coincides with the result obtained by D. S. Fisher in the equilibrium case. The RG equations have a pole at the value of the driving force, which coincides with the value of the threshold below which the interface becomes pinned as predicted by Bruinsma and Aeppli. The behavior of the mobility in the vicinity of the pole is discussed.     

Epsilon expansion for multicritical fixed points and exact renormalisation group equations

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the nonlinear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε². An exact marginal operator for the full RG equations is also constructed.     

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

Variational approximations for renormalization group transformations

Approximate recursion relations which give upper and lower bounds on the free energy are described. Optimal calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, a particularly simple lower bound approximation which preserves the symmetry of the Hamiltonian (the one-hypercube approximation) is described. The approximation is applied to both the Ising model and the Wilson-Fisher model. At the fixed point a parameter is set variationally and critical indices are calculated. For the Ising model the agreement with the exact results atd = 2 is surprisingly good, 0.1%, and is good atd=3 and evend=4. For the Wilson-Fisher model the recursion relation is reduced to a one-dimensional integral equation which can be solved numerically givingv=0.652 atd=3, or by expansion in agreement with the results of Wilson and Fisher to leading order in . The method is also used to calculate thermodynamic functions for thed = 2 Ising model; excellent agreement with the Onsager solution is found.Supported in part by the National Science Foundation under Grants Nos. MPS73-04886A01 and GH-41512 and by the Brown University Materials Research Laboratory supported by the National Science Foundation. M.C.Y. was supported by a grant from the Scientific and Technical Research Council of Turkey.     

Renormalization Group Study of Ising Transition in Double－Frequency sine－Gordon Model

We utilize the renormallzation group (RG) technique to analyze the Ising critical behavior in the double frequency Sine-Gordon model. The one-loop RG equations obtained show unambiguously that there exist two Ising critical points besides the trivial Gaussian fixed point. The topology of the RG flows is obtained as well.     

Renormalization Group Study of Ising Transition in Double-Frequency sine-Gordon Model

We utilize the renormalization group (RG) technique to analyze the Ising critical behavior in the doublefrequency sine-Gordon model. The one-loop RG equations obtained show unambiguously that there exist two Isingcritical points besides the trivial Gaussian fixed point. The topology of the RG flows is obtained as well.     

Nontrivial fixed points and screening in the hierarchical two-dimensional Coulomb gas

We show the existence and asymptotic stability of two fixed points of the renormalization group transformation for the hierarchical two-dimensional Coulomb gas in the sine-Gordon representation and temperatures slightly greater than the critical one. We prove also that the correlations at the fixed points decay as in the hierarchical massive scalar free theory, that is, asd _xy ^–4 . We argue that this is the natural definition of screening in the hierarchical approximation.     

Field Theory of Critical Behavior in Driven Diffusive Systems with Quenched Disorder

We present a field-theoretic renormalization-group study for the critical behavior of a uniformly driven diffusive system with quenched disorder, which is modeled by different kinds of potential barriers between sites. Due to their symmetry properties, these different realizations of the random potential barriers lead to three different models for the phase transition to transverse order and to one model for the phase transition to longitudinal order all belonging to distinct universality classes. In these four models, which have different upper critical dimensions d _c, we find the critical scaling behavior of the vertex functions in spatial dimensions d<d _c. The deviation from purely diffusive behavior is characterized by the anomaly exponent , which we calculate at first and second order, respectively, in =d _c–d. In each model turns out to be positive, which means superdiffusive spread of density fluctuations in the driving force direction.     

Percolation on infinitely ramified fractals

We present a family of exact fractals with a wide range of fractal and fracton dimensionalities. This includes the case of the fracton dimensionality of 2, which is critical for diffusion. This is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal. These fractals include the cases of finite and infinite ramification characterized by a ramification exponentp. The infinite ramification makes the problem of percolation on these lattices a nontrivial one. We give numerical evidence for a percolation transition on these fractals. This transition is tudied by a real-space renormalization group technique on lattices with fractal dimensionality ¯d between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.     

Breakdown of Fermi liquid behavior near the hot spots in a two-dimensional model: A two-loop renormalization group analysis

Motivated by a recent experimental observation of a nodal liquid on both single crystals and thin films of Bi₂Sr₂CaCu₂O_8 + δ by Chatterjee et al. [Nature Phys. 6 (2010) 99], we perform a field-theoretical renormalization group (RG) analysis of a two-dimensional model such that only eight points located near the “hot spots” on the Fermi surface are retained, which are directly connected by spin density wave ordering wavevector. We derive RG equations up to two-loop order describing the flow of renormalized couplings, quasiparticle weight, several order-parameter response functions, and uniform spin and charge susceptibilities of the model. We find that while the order-parameter susceptibilities investigated here become non-divergent at two loops, the quasiparticle weight vanishes in the low-energy limit, indicating a breakdown of Fermi liquid behavior at this RG level. Moreover, both uniform spin and charge susceptibilities become suppressed in the scaling limit which indicate gap openings in both spin and charge excitation spectra of the model.     

Critical Exponents of the Diluted Ising Model between Dimensions 2 and 4

Within the massive field-theoretic renormalization-group approach the expressions for the and functions of the anisotropic mn-vector model are obtained for general space dimension d in three-loop approximation. Resumming corresponding asymptotic series, critical exponents for the case of the weakly diluted quenched Ising model (m = 1, n = 0), as well as estimates for the marginal order parameter component number m _c of the weakly diluted quenched m-vector model, are calculated as functions of d in the region 2 d < 4. Conclusions concerning the effectiveness of different resummation techniques are drawn.     

Bulk singularities at critical end points: a field-theory analysis

A class of continuum models with a critical end point is considered whose Hamiltonian [φ,ψ] involves two densities: a primary order-parameter field, φ, and a secondary (noncritical) one, ψ. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity ∼ | t|^{2 - α} of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity ∼ | t|^{1 - α} or ∼ | t|^β of the secondary density <ψ> are derived. It is clarified how the renormalization group (RG) scenario found in position-space RG calculations, in which the critical end point and the critical line are mapped onto two separate fixed points _CEP ^* and _λ ^*, translates into field theory. The critical RG eigenexponents of _CEP ^* and _λ ^* are shown to match. _CEP ^* is demonstrated to have a discontinuity eigenperturbation (with eigenvalue y = d), tangent to the unstable trajectory that emanates from _CEP ^* and leads to _λ ^*. The nature and origin of this eigenperturbation as well as the role redundant operators play are elucidated. The results validate that the critical behavior at the end point is the same as on the critical line. Received 18 January 2001     

Exact renormalization group and running Newtonian coupling in higher-derivative gravity

We discuss exact renormalization group (RG) in R ² gravity using the effective average action formalism. The truncated evolution equation for such a theory against the de Sitter background leads to a system of nonperturbative RG equations for cosmological and gravitational coupling constants. An approximate solution of these RG equations shows that antiscreening or screening behavior of the Newtonian coupling arises, depending on the higher-derivative coupling constants. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 8, 571–575 (25 April 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Hierarchical Spherical Model from a Geometric Point of View

A continuous version of the hierarchical spherical model at dimension d=4 is investigated. Two limit distributions of the block spin variable X ^γ , normalized with exponents γ=d+2 and γ=d at and above the critical temperature, are established. These results are proven by solving certain evolution equations corresponding to the renormalization group (RG) transformation of the O(N) hierarchical spin model of block size L ^d in the limit L ↓1 and N→∞. Starting far away from the stationary Gaussian fixed point the trajectories of these dynamical system pass through two different regimes with distinguishable crossover behavior. An interpretation of this trajectories is given by the geometric theory of functions which describe precisely the motion of the Lee–Yang zeroes. The large-N limit of RG transformation with L ^d fixed equal to 2, at the criticality, has recently been investigated in both weak and strong (coupling) regimes by Watanabe (J. Stat. Phys. 115:1669–1713, 2004) . Although our analysis deals only with N=∞ case, it complements various aspects of that work. D.H.U. Marchetti partially supported by CNPq and FAPESP. W.R.P. Conti supported by FAPESP under grant 05/57416-8.     

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.