Renormalization Technique Research Articles

Renormalization Method in p-Adic λ-Model on the Cayley Tree

In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic $\l$-model on the Cayley tree of order two. We consider generalized $p$-adic quasi Gibbs measures depending on parameter $\r\in\bq_p$, for the $\l$-model. Such measures are constructed by means of certain recurrence equations. These equations define a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed points, which predicts the existence of a phase transition. In the second regime the system has two attractive and one neutral fixed points, which predicts the existence of a quasi phase transition. A main point of this paper is to verify (i.e. rigorously prove) and confirm that the indicated predictions (via dynamical systems point of view) are indeed true. To establish the main result, we employ the methods of $p$-adic analysis, and therefore, our results are not valid in the real setting.

Critical Probabilities and Convergence Time of Percolation Probabilistic Cellular Automata

This paper considers a class of probabilistic cellular automata undergoing a phase transition with an absorbing state. Denoting by \({\mathcal {U}}(x)\) the neighbourhood of site \(x\), the transition probability is \(T(\eta _x = 1 | \eta _{{\mathcal {U}}(x)}) = 0\) if \(\eta _{{\mathcal {U}}(x)}= \mathbf {0}\) or \(p\) otherwise, \(\forall x \in \mathbb {Z}\). For any \(\mathcal {U}\) there exists a non-trivial critical probability \(p_c( {\mathcal {U}})\) that separates a phase with an absorbing state from a fluctuating phase. This paper studies how the neighbourhood affects the value of \(p_c( {\mathcal {U}})\) and provides lower bounds for \(p_c( {\mathcal {U}})\). Furthermore, by using dynamic renormalization techniques, we prove that the expected convergence time of the processes on a finite space with periodic boundaries grows exponentially (resp. logarithmically) with the system size if \(p > p_c\) (resp. \(p<p_c\)). This provides a partial answer to an open problem in Toom et al. (Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, pp. 1–182. Manchester University Press, Manchester, 1990; Topics in Contemporary Probability and its Applications, pp. 117–157. CRC Press, Boca Raton, 1995).

Lagrange Spectra in Teichmüller Dynamics via Renormalization

We introduce Lagrange Spectra of closed-invariant loci for the action of SL(2, $${\mathbb{R}}$$ ) on the moduli space of translation surfaces, generalizing the classical Lagrange Spectrum, and we analyze them with renormalization techniques. A formula for the values in such spectra is established in terms of the Rauzy–Veech induction and it is used to show that any invariant locus has closed Lagrange spectrum and values corresponding to pseudo-Anosov elements are dense. Moreover we show that Lagrange spectra of arithmetic Teichmüller discs contain an Hall’s ray, giving an explicit bound for it via a second formula which uses the classical continued fraction algorithm. In addition, we show the equivalence of several definitions of bounded Teichmüller geodesics and bounded type interval exchange transformations and we prove quantitative estimates on excursions to the boundary of moduli space in terms of norms of positive matrices in the Rauzy–Veech induction.

Renormalizability of Yang-Mills theory with Lorentz violation and gluon mass generation

We show that pure Yang-Mills theories with Lorentz violation are renormalizable to all orders in perturbation theory. To do this, we employ the algebraic renormalization technique. Specifically, we control the breaking terms with a suitable set of external sources which, eventually, attain certain physical values. The Abelian case is also analyzed as a starting point. The main result is that the renormalizability of the usual Maxwell and Yang-Mills sectores are both left unchanged. Furthermore, in contrast to Lorentz violating QED, the odd CPT violation sector of Yang-Mills theories renormalizes independently. Moreover, the method induces, in a natural way, mass terms for the gauge field while the photon remains massless (at least n the sense of a Proca-like term). The entire analysis is carried out at the Landau gauge.

From the necessary to the possible: the genesis of the spin-statistics theorem

The spin-statistics theorem, which relates the intrinsic angular momentum of a single particle to the type of quantum statistics obeyed by a system of many such particles, is one of the central theorems in quantum field theory and the physics of elementary particles. It was first formulated in 1939/40 by Wolfgang Pauli and his assistant Markus Fierz. This paper discusses the developments that led up to this first formulation, starting from early attempts in the late 1920s to explain why charged matter particles obey Fermi-Dirac statistics, while photons obey Bose-Einstein statistics. It is demonstrated how several important developments paved the way from such general philosophical musings to a general (and provable) theorem, most notably the use of quantum field theory, the discovery of new elementary particles, and the generalization of the notion of spin. It is also discussed how the attempts to prove a spin-statistics connection were driven by Pauli from formal to more physical arguments, culminating in Pauli’s 1940 proof. This proof was a major success for the beleaguered theory of quantum field theory and the methods Pauli employed proved essential for the renaissance of quantum field theory and the development of renormalization techniques in the late 1940s.

One-dimensional Chern–Simons theory and theÂgenus

We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X, as such a Chern-Simons theory. Our main result is that the partition function of this theory is naturally identified with the A-genus of X. From the perspective of derived geometry, our quantization construct a volume form on the derived loop space which can be identified with the A-class.

Edge states in topological magnon insulators

For magnons, the Dzyaloshinskii-Moriya interaction accounts for spin-orbit interaction and causes a nontrivial topology that allows for topological magnon insulators. In this theoretical investigation we present the bulk-boundary correspondence for magnonic kagome lattices by studying the edge magnons calculated by a Green function renormalization technique. Our analysis explains the sign of the transverse thermal conductivity of the magnon Hall effect in terms of topological edge modes and their propagation direction. The hybridization of topologically trivial with nontrivial edge modes enlarges the period in reciprocal space of the latter, which is explained by the topology of the involved modes.

One dimensional acoustic direct nonlinear inversion using the Volterra inverse scattering series

Direct inversion of acoustic scattering problems is nonlinear. One way to treat the inverse scattering problem is based on the reversion of the Born–Neumann series solution of the Lippmann–Schwinger equation. An important issue for this approach is the radius of convergence of the Born–Neumann series for the forward problem. However, this issue can be tackled by employing a renormalization technique to transform the Lippmann–Schwinger equation from a Fredholm to a Volterra integral form. The Born series of a Volterra integral equation converges absolutely and uniformly in the entire complex plane. We present a further study of this new mathematical framework. A Volterra inverse scattering series (VISS) using both reflection and transmission data is derived and tested for several acoustic velocity models. For large velocity contrast, series summation techniques (e.g., Cesàro summation, Euler transform, etc) are employed to improve the rate of convergence of VISS. It is shown that the VISS method with summation techniques can provide a relatively good estimation of the velocity profile. The method is fully data-driven in the respect that no prior information of the model is required. Besides, no internal multiple removal is needed. This one dimensional VISS approach is useful for inverse scattering and serves as an important step for studying more complicated and realistic inversions.

Renormalization group approach to 1D cellular automata with large updating neighborhoods

We study self‐similarity in one‐dimensional probabilistic cellular automata (PCA) by applying a real‐space renormalization technique to PCA with increasingly large updating neighborhoods. By studying the flow about the critical point of the renormalization, we may produce estimates of the spatial scaling properties of critical PCA. We find that agreement between our estimates and experimental values are improved by resolving correlations between larger blocks of spins, although this is not sufficient to converge to experimental values. However, applying the technique to PCA with larger neighborhoods, and, therefore, more renormalization parameters, results in further improvement. Our most refined estimate produces a spatial scaling exponent, found at the critical point of the five‐neighbor PCA, of ν = 1.056 which should be compared to the experimental value of ν = 1.097. © 2014 Wiley Periodicals, Inc. Complexity 21: 206–213, 2015

Gauge fields on non‐commutative spaces and renormalization

AbstractConstructing renormalizable models on non‐commutative spaces constitutes a big challenge. Only few examples of renormalizable theories are known, such as the scalar Grosse‐Wulkenhaar model. Gauge fields are even more difficult, since new renormalization techniques are required which are compatible with the inherently non‐local setting on the one hand, and also allow to properly treat the gauge symmetry on the other hand. In this proceeding (which is based on my talk given at the “Workshop on Noncommutative Field Theory and Gravity” in Corfu/Greece, September 8–15, 2013), I focus on this last point and present new extensions to existing renormalization schemes (which are known to work for gauge field theories in commutative space) adapted to non‐commutative Moyal space.

Parameters estimation based adaptive Generalized Projective Synchronization (GPS) of chaotic Chua’s circuit with application to chaos communication by parametric modulation

In this paper we study the design and implementation of adaptive Generalized Projective Synchronization (GPS) problem between two chaotic circuits (master and slave) via a scalar transmitted signal and by a new method not requiring the same structure of master and slave circuits. Starting from a simple synchronization method, a renormalization technique enables us to design an adaptive observer (slave) of the known Chua’s circuit using simple and standard electronic components. This observer identifies some parameters proportional to values of components of the master with which it synchronizes through GPS when values of these parameters vary with time. Both numerical simulation and Pspice simulation of GPS for chaos communication by parametric modulation between the two circuits are performed.

Linear response within the projector-based renormalization method: many-body corrections beyond the random phase approximation

The explicit evaluation of linear response coefficients for interacting many-particle systems still poses a considerable challenge to theoreticians. In this work we use a novel many-particle renormalization technique, the so-called projector-based renormalization method, to show how such coefficients can systematically be evaluated. To demonstrate the prospects and power of our approach we consider the dynamical wave-vector dependent spin susceptibility of the two-dimensional Hubbard model and also determine the subsequent magnetic phase diagram close to half-filling. We show that the superior treatment of (Coulomb) correlation and fluctuation effects within the projector-based renormalization method significantly improves the standard random phase approximation results.

Baby de Sitter black holes and dS3/CFT2

Unlike three-dimensional Einstein gravity, three-dimensional massive gravity admits asymptotically de Sitter space (dS) black hole solutions. These black holes present interesting features and provide us with toy models to study the dS/CFT correspondence. A remarkable property of these black holes is that they are always in thermal equilibrium with the cosmological horizon of the space that hosts them. This invites us to study the thermodynamics of these solutions within the context of dS/CFT. We study the asymptotic symmetry group of the theory and find that it indeed coincides with the local two-dimensional conformal algebra. The charge algebra associated to the asymptotic Killing vectors consists of two copies of the Virasoro algebra with non-vanishing central extension. We compute the mass and angular momentum of the dS black holes and verify that a naive application of Cardy's formula exactly reproduces the entropy of both the black hole and the cosmological horizon. By adapting the holographic renormalization techniques to the case of dS space, we define the boundary stress tensor of the dual Euclidean conformal field theory.

Measurement of Multi-Port S-Parameters using Four-Port Network Analyzer

An efficient measurement methodology is proposed to construct the scattering parameters of a multi-port device using a four-port vector network analyzer (VNA) without the external un-terminated ports. By using the four-port VNA, the reflected waves from the un-terminated ports could be minimized. The proposed method significantly enhances the accuracy of the S-parameters with less number of measurements compared to the results of classical renormalization technique which uses twoport VNA. The proposed method is validated from the measured data with the coupled 8-port micro-strip lines.

Selection of Specific S-parameters in Multiport Measurement for the Renormalization Technique using Four-port VNA

This paper presents an algorithm to characterize scattering parameters of multi-port device with a four-port Vector Network Analyzer (VNA). By employing the renormalization of scattering matrices with different reference impedances at ports, data obtained from multi-port configuration measurements can be synthesized to build the full scattering matrix of the device-under-test (DUT). Although that procedure can be best used for the interconnect system in which the inside routing is quite apparent, we still need an appropriate algorithm to select specific S-parameters for the black-box model where the configuration inside is not known. This paper presents an algorithm to determine which S-parameters are suitable to be selected to reconstruct the full S-matrix of the system. Using the selected specific S-parameters, the renormalization of the scattering matrices could be synthesized to successfully estimate the S-parameters of a multiport interconnect system. A good agreement between the estimated and true S-parameters verifies the validness of the algorithm.

A new family of non-standard finite difference schemes for the logistic equations

This paper proposed a new family of Non-standard finite difference schemes for the Logistic equations. The technique of non-local approximation and renormalization of the denominator function was employed. The new schemes were found to possess desirable stability properties and also preserve all the monotonic properties of the logistic equations.

Collision operators for Stark line shapes: When does the unified theory fail?

It is shown that the unified theory for Stark broadening leads to incorrect results when correlated emitter–perturber collisions are present. With a kinetic theory treatment, we propose an extension of the model able to account for such correlations. The treatment presents analogies with renormalization techniques and the resulting collision operator has a structure similar to that obtained within the unified theory. We illustrate the applicability of the model through calculations of hydrogen line shapes in ideal cases.

The Brown-York mass of black holes in Warped Anti-de Sitter space

We give a direct computation of the mass of black holes in Warped Anti-de Sitter space (WAdS) in terms of the Brown-York stress-tensor at the boundary. This permits to explore to what extent the holographic renormalization techniques can be applied to such type of deformation of AdS. We show that, despite some components of the boundary stress-tensor diverge and resist to be regularized by the introduction of local counterterms, the precise combination that gives the quasilocal energy density yields a finite integral. The result turns out to be in agreement with previous computations of the black hole mass obtained with different approaches. This is seen to happen both in the case of Topologically Massive Gravity and of the so-called New Massive Gravity. Here, we focus our attention on the latter. We observe that, despite other conserved charges diverge in the near boundary limit, the finite part in the large radius expansion captures the physically relevant contribution. We compute the black hole angular momentum in this way and we obtain a result that is in perfect agreement with previous calculations.

Comparison of mass renormalization schemes for simple model systems

We discuss an alternative method to mass renormalize a quantum field Hamiltonian based on the requirement that the vacuum and single-particle sectors are not self-scattering. We illustrate the feasibility of this method for the concrete example of bosonic quadratic and quartic interactions. The results are compared with those obtained by the standard renormalization technique based on a spectral analysis of the usual field-based propagators. We also discuss a potential method for renormalizing a quantum field theory in a non-perturbative way using computational methods.

Transport properties of two finite armchair graphene nanoribbons

In this work, we present a theoretical study of the transport properties of two finite and parallel armchair graphene nanoribbons connected to two semi-infinite leads of the same material. Using a single Π-band tight binding Hamiltonian and based on Green’s function formalisms within a real space renormalization techniques, we have calculated the density of states and the conductance of these systems considering the effects of the geometric confinement and the presence of a uniform magnetic field applied perpendicularly to the heterostructure. Our results exhibit a resonant tunneling behaviour and periodic modulations of the transport properties as a function of the geometry of the considered conductors and as a function of the magnetic flux that crosses the heterostructure. We have observed Aharonov-Bohm type of interference representing by periodic metal-semiconductor transitions in the DOS and conductance curves of the nanostructures.