Renormalization Transformation Research Articles

The spin-5/2 Blume–Capel model by mean-field approach and renormalization group theory

The spin-5/2 Blume–Capel model was studied using the mean-field approximation and the Migdal–Kadanoff renormalization group method for two- and three-dimensional systems. We determined the phase diagrams in the (crystal field, temperature) plane where the system exhibited first- and second-order phase transitions as well as isolated critical, bicritical and triple points. In order to show first-order transitions at low temperature, we presented the total magnetization per site and the derivative of the free energy as a function of the crystal field. Moreover, the critical exponents of the system were calculated by linearizing the renormalization transformation at the vicinity of the second-order fixed points.

Self-Similar Groups and Holomorphic Dynamics: Renormalization, Integrability, and Spectrum

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of atomic spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

Asymptotic scaling and universality for skew products with factors in SL(2,)

AbstractWe consider skew-product maps over circle rotations $x\mapsto x+\alpha \;(\mod 1)$ with factors that take values in ${\textrm {SL}}(2,{\mathbb {R}})$ . In numerical experiments, with $\alpha $ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

Buried points in complex dynamical systems and glassy transitions

<p indent=0mm>A new interdisciplinary research direction is addressed across the fields of statistical mechanics and complex dynamical systems together with specific research topics, such as the connection between buried points and glassy transitions. First, pertinent results on complex dynamical systems are delivered. Let<italic> f</italic> be a complex analytical mapping from the Riemann sphere (or the complex plane) to itself, i.e., <italic>f</italic> is rational or entire. The stable set <italic>F</italic>(<italic>f</italic>) of this dynamical system is called the Fatou set of <italic>f</italic>, and the unstable set <italic>J</italic>(<italic>f</italic>) the Julia set. Every component of <italic>F</italic>(<italic>f</italic>) is called a Fatou component. Obviously, the boundary of each Fatou component belongs to <italic>J</italic>(<italic>f</italic>). If a point in <italic>J</italic>(<italic>f</italic>) does not belong to the boundary of any Fatou component, then it is a buried point of <italic>f</italic>, which is often used to describe the topological complexity of the Julia set. In this quest, Makienko’s conjecture is undoubtedly one of the most important problems which is stated as follows: Let <italic>R</italic> be a rational mapping and <italic>F</italic>(<italic>R</italic>) contain infinitely many components. If every Fatou component is not completely invariant, then <italic>R</italic> has buried points. The known result is the following: If <italic>J</italic>(<italic>R</italic>) is disconnected, or connected and locally connected, then Makienko’s conjecture is true; if <italic>J</italic>(<italic>R</italic>) is disconnected and <italic>R</italic> has a buried point, then it must have buried components. Concerning entire functions, we have: If <italic>F</italic>(<italic>f</italic>) is not empty, then <italic>f</italic> has a buried point if and only if <italic>F</italic>(<italic>f</italic>) is disconnected. Following the above results, interesting topological properties on the sets of buried points are deduced. However, many open problems remain unsolved in regard to these special points. In the second part, we explained the topological complexity of the set of Yang-Lee zeros. In 1952, Yang C.N. and Lee T.D. established an analytic theory in statistical mechanics, summarized as the circle theorem: For statistical models like 2-dimensional lattice gas, the zeros of the partition function condense to the unit circle. Henceforth, the importance of the distribution of Yang-Lee zeros in the complex plane is emphasized for general models. In 1983, researchers including B. Derrida, L. De Seze and C. Itzykson found that the distribution of Yang-Lee zeros can be cast into a problem on Julia sets of renormalization transformations. In the third part, we describe the connection between buried points and glass transitions. Concerning the Potts model on a generalized diamond hierarchical lattice, we present a new type of distribution of Julia sets of renormalization transformations, which is called the real Feigenbaum-type distribution of buried points. Physically, this means that the limiting set of Yang-Lee zeros could contain an interval of the real axis, all of which are singularities of the free energy function. This important observation indicates that the exploration of renormalization transformation dynamics is essential to the understanding of the phenomenon of glassy transitions. Finally, we discussed the theoretical value and practical significance of the above research topics and directions. It is well known that <italic>Science</italic> published 125 most challenging scientific questions in 2005 to celebrate the Journal’s 125th anniversary, among which Question 47 is “What is the essence of the glassy substance?”. There are many theoretical models for glassy transitions, but up to now, a complete theory remains illusive. Perhaps, the connection between complex dynamics and glassy transition established above could shed new light on this old problem with the help of powerful tools from mathematics, as ever before.

Golden mean renormalization for the almost Mathieu operator and related skew products

Considering SL(2,R) skew-product maps over circle rotations, we prove that a renormalization transformation associated with the golden mean α* has a nontrivial periodic orbit of length 3. We also present some numerical results, including evidence that this period 3 describes scaling properties of the Hofstadter butterfly near the top of the spectrum at α* and scaling properties of the generalized eigenfunction for this energy.

Dynamics of a family of rational maps concerning renormalization transformation

Considering a family of rational maps Tnλ concerning renormalization transformation, we give a perfect description about the dynamical properties of Tnλ and the topological properties of the Fatou components F(Tnλ). Furthermore, we discuss the continuity of the Hausdorff dimension HD(J(Tnλ)) about real parameter λ.

Renormalization and universality of the Hofstadter spectrum

We consider a renormalization transformation for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of determine universal constants analogous to the critical exponents in the theory of phase transitions. Restricting to skew-product maps over circle-rotations by the golden mean, we find several periodic orbits for , and we conjecture that there are infinitely many. Interestingly, all scaling factors that have been determined to high accuracy appear to be algebraically related to the circle-rotation number. We present evidence that these values describe (among other things) local scaling properties of the Hofstadter spectrum.

Lee–Yang–Fisher Zeros for the DHL and 2D Rational Dynamics, II. Global Pluripotential Interpretation

In a classical work of the 1950s, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee–Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee–Yang–Fisher (LYF) zeros. In this paper, we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $$R$$ in two variables (the Migdal–Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the LYF zeros are equidistributed with respect to a dynamical (1, 1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal–Kadanoff renormalization transformation of various other hierarchical lattices.

The Canonical Ensemble of Open Self-Avoiding Strings

Statistical models of a single open string avoiding self-intersections in the d-dimensional Euclidean space ℝd, 2 ≤ d < 4, and the ensemble of strings are considered. The presentation of these models is based on the Darwin–Fowler method, used in statistical mechanics to derive the canonical ensemble. The configuration of the string in space ℝd is described by its contour length L and the spatial distance R between its ends. We establish an integral equation for a transformed probability density W(R, L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations. This allows us using the renormalization group method to investigate the asymptotic behavior of this density in the case where R → ∞ and L → ∞. For the model of an ensemble of M open strings with the mean string contour length over the ensemble given by $$ \overline{L} $$ , we obtain the most probable distribution of strings over their lengths in the limit as M → ∞. Averaging the probability density W(R, L) over the canonical ensemble eventually gives the sought density 〈W(R, L)〉.

Synchronization of Phase Oscillators on the Hierarchical Lattice

Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction betweenthe oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.

Momentum-Space Renormalization Group Transformation in Bayesian Image Modeling by Gaussian Graphical Model

A new Bayesian modeling method is proposed by combining the maximization of the marginal likelihood with a momentum-space renormalization group transformation for Gaussian graphical models. Moreover, we present a scheme for computint the statistical averages of hyperparameters and mean square errors in our proposed method based on a momentumspace renormalization transformation.

Dynamics of a Polymer Network Modeled by a Fractal Cactus.

In this paper, we focus on the relaxation dynamics of a polymer network modeled by a fractal cactus. We perform our study in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. By performing real-space renormalization transformations, we determine analytically the whole eigenvalue spectrum of the connectivity matrix, thereby rendering possible the analysis of the Rouse-dynamics at very large generations of the structure. The evaluation of the structural and dynamical properties of the fractal network in the Rouse type-approach reveals that they obey scaling and the dynamics is governed by the value of spectral dimension. In the Zimm-type approach, the relaxation quantities show a strong dependence on the strength of the hydrodynamic interaction. For low and medium hydrodynamic interactions, the relaxation quantities do not obey power law behavior, while for slightly larger interactions they do. Under strong hydrodynamic interactions, the storage modulus does not follow power law behavior and the average displacement of the monomer is very low. Remarkably, the theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results from the literature.

Relaxation dynamics of Sierpinski hexagon fractal polymer: Exact analytical results in the Rouse-type approach and numerical results in the Zimm-type approach

In this paper, we focus on the relaxation dynamics of Sierpinski hexagon fractal polymer. The relaxation dynamics of this fractal polymer is investigated in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. In the Rouse-type approach, by performing real-space renormalization transformations, we determine analytically the complete eigenvalue spectrum of the connectivity matrix. Based on the eigenvalues obtained through iterative algebraic relations we calculate the averaged monomer displacement and the mechanical relaxation moduli (storage modulus and loss modulus). The evaluation of the dynamical properties in the Rouse-type approach reveals that they obey scaling in the intermediate time/frequency domain. In the Zimm-type approach, which includes the hydrodynamic interactions, the relaxation quantities do not show scaling. The theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results.

Electron–phonon interaction in quantum transport through quantum dots and molecular systems

The quantum transport and effects of decoherence properties are studied in quantum dots systems and finite homogeneous chains of aromatic molecules connected to two semi-infinite leads. We study these systems based on the tight-binding approach through Green's function technique within a real space renormalization and polaron transformation schemes. In particular, we calculate the transmission probability following the Landauer–Büttiker formalism, the I−V characteristics and the noise power of current fluctuations taken into account the decoherence. Our results may explain the inelastic effects through nanoscopic systems.

Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder

In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal–Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the renormalization transformation). We prove that R is partially hyperbolic on an invariant cylinder C. The Lee–Yang zeros are organized in a transverse measure for the central-stable foliation of R|C. Their distribution is absolutely continuous. Its density is C∞ (and non-vanishing) below the critical temperature. Above the critical temperature, it is C∞ on a open dense subset, but it vanishes on the complementary set of positive measure.

The Hausdorff dimension of the Julia sets concerning renormalization transformation

We consider a family of rational maps Tn, λ of λ-state diamond-like hierarchical Potts models concerning renormalization transformation. We give an asymptotic formula of the Hausdorff dimension of the Julia sets J(Tn, λ) for sufficiently large λ.

Phase Transitions in One-Dimensional Translation Invariant Systems: A Ruelle Operator Approach

We consider a family of potentials f, derived from the Hofbauer potentials, on the symbolic space Omega=\{0,1\}^\mathbb{N} and the shift mapping $\sigma$ acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pressure is not analytic, there are multiple eigenprobabilities for the dual of the Ruelle operator, the DLR-Gibbs measure is not unique and finally the Thermodynamic Limit is not unique. Additionally, we explicitly calculate the critical points for these phase transitions. Some examples which are not of Hofbauer type are also considered. The non-uniqueness of the Thermodynamic Limit is proved by considering a version of a Renewal Equation. We also show that the correlations decay polynomially and that each one of these Hofbauer potentials is a fixed point for a certain renormalization transformation.

A renormalization approach to the universality of scaling in phyllotaxis

Phyllotaxis, i.e. the arrangement of plant organs like leaves, florets, scales, bracts etc. around a shoot, stem, or cone, is often highly regular. Across the plant kingdom phyllotaxis shows not only qualitatively, but also quantitatively identical features, like the occurrence of divergence angles close to noble irrationals. In a previous study (Reick, 2012) a mechanism has been identified that explains the selection of these particular divergence angles on the basis of self-similarity and scaling, numerically found in the bifurcation diagrams of simple dynamical models of phyllataxis. In the present paper, by constructing a renormalization theory, the universality of this scaling is proved for a whole class of models, prototypically represented by Thornley’s model of phyllotaxis (Thornley, 1975). The renormalization is constructed from another self-similarity found numerically for the Fourier transform of the abstract potential governing the mutual inhibition of primordia. Surprisingly, the resulting renormalization transformation is already known from the treatment of the quasiperiodic transition to chaos but operates here on a different function space. It turns out that the fixed points of the renormalization transformation are characterized by divergences of the form Θ(κ)=1/τ(κ), where, written as continued fraction, τ(κ)=[κ;κ,κ,…], κ∈N+. To show the universality of the scaling, it is demonstrated that the fixed points are unstable and that the associated scaling factors α(κ)=−(τ(κ))2 and β(κ)=τ(κ) are exactly those that were numerically found in (Reick, 2012) to rule the selfsimilarity of the bifurcation structure. Thereby, the present paper puts forward an explanation for the universal appearance of certain phyllotactic patterns that is independent of physiological detail of plant growth.

On buried points and phase transition points in the Julia sets concerning renormalization transformation

Considering a family of rational maps concerning renormalization transformation, we give a perfect description of buried points and phase transition points in the Julia set . Furthermore, we prove that contains an open interval where all points are buried points for some parameters n and λ, which is according to the problem that Curry and Mayer proposed. MSC:37F10, 37F45.

A d-dimensional model of the canonical ensemble of open strings

We propose a d-dimensional model of the canonical ensemble of open self-avoiding strings. We consider the model of a solitary open string in the d-dimensional Euclidean space ℝ d, 2 ≤ d < 4, where the string configuration is described by the arc length L and the distance R between string ends. The distribution of the spatial size of the string is determined only by its internal physical state and interaction with the ambient medium. We establish an equation for a transformed probability density W(R,L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations; this allows using the renormalization group method to investigate the asymptotic behavior of this density in the case where R→∞ and L→∞. We consider the model of an ensemble of M open strings with the mean string length over the ensemble given by $\bar L$ , and we use the Darwin-Fowler method to obtain the most probable distribution of strings over their lengths in the limit as M →∞. Averaging the probability density W(R,L) over the canonical ensemble eventually gives the sought density 〈W(R, $\bar L$ )〉.