Renormalization Transformation Research Articles

Thermodynamics, geometrical frustration and quantum fluctuations in coupled spin chains

The linear-perturbation real space renormalization transformation (LPRG) is presented and applied to the study of quantum spin chains coupled by interchain interaction (k1) weaker than intrachain one (k). The method is examined in two exact solvable cases: Ising chains on the square and triangular lattices and quantum XY chain. For the Ising model, in the second order in the cumulant epansion, the deviation of the critical temperature from the exact value is less than 1% for 0:5k > k1 > 0:15k, but even in the case of the standard Ising model (k1 = k) we found the value of Tc which differs by 2% from the exact one. For the quantum XY chain the deviation of the free energy value found by using LPRG from the exact Katsura result is less than 1% for T=J > 1, and for rather low temperature T=J = 0:08 is about 6%. The LPRG is used to study the effects of interchain frustration on the phase transition in 2D Heisenberg spin chains with easy axis along the z direction. It is shown that contrary to the pure Ising model in systems with in-plane interactions (XY), the interchain frustration does not destroy the nite-temper ature transition. However, such a frustration changes the character of the phase transition from Ising-like to, probably, Kosterlitz-Thouless-like. We have also applied the LPRG method to the calculation of the isothermal magnetocaloric coefcient (MT) for several spin models in disordered phases. Is is demonstrated that in the presence of antiferromagnetic uctuations , MT changes sign at some value of the magnetic eld. Generally, MT is negative if magnetic eld competes with a short-range order, and consequently it can be an indicator of the change in the short-range correlation.

The renormalization transformation for two-type branching models

Cet article etudie des systemes denombrables de diffusions en interaction hierarchiques et lineaires vivant dans le quadrant positif. De tels systemes apparaissent dans la dynamique d’individus de deux types qui migrent tout en interagissant dans des colonies. Le comportement a grande echelle et temps long peut etre etudie en utilisant le programme de renormalisation. Ce programme, qui a permis de resoudre d’autres cas (principalement uni-dimensionnels) est base sur la construction et l’analyse d’une transformation de renormalisation non lineaire, agissant sur la fonction de diffusion des composants du systeme et connectant l’evolution de blocs moyennes sur le temps a differentes echelles. Nous identifions une classe generale de fonctions de diffusion dans le quadrant positif pour lequel la transformation de renormalisation est bien definie et qui, sous une conjecture de comportement aux bords, peut-etre iteree. A l’interieur de certaines sous-classes, nous identifiens les points fixes de la transformation et etudions leurs domaines d’attraction. Ces domaines d’attraction constitutent les classes d’universalite du systeme apres changement d’echelle dans le temps et l’espace.

Using the Renormalization Group to Classify Boolean Functions

This paper argues that the renormalization group technique used to characterize phase transitions in condensed matter systems can be used to classify Boolean functions. A renormalization group transformation is presented that maps an arbitrary Boolean function of N Boolean variables to one of N−1 variables. Applying this transformation to a generic Boolean function (one whose output for each input is chosen randomly and independently to be one or zero with equal probability) yields another generic Boolean function. Moreover, applying the transformation to some other functions known to be non-generic, such as Boolean functions that can be written as polynomials of degree ξ with ξ≪N and functions that depend on composite variables such as the arithmetic sum of the inputs, yields non-generic results. One can thus define different phases of Boolean functions as classes of functions with different types of behavior upon repeated application of the renormalization transformation. Possible relationships between different phases of Boolean functions and computational complexity classes studied in computer science are discussed.

On functional equations associated with the renormalization of non-commuting circle mappings

We investigate the functional equations associated with critical non-commuting circle homeomorphisms. Using Epstein's Herglotz function methods, we show that there are analytic solutions of a class of functional equations closely related to period-two points of the circle-map renormalization transformations for rotation number with continued fraction .

Performance of deterministic dynamical decoupling schemes: Concatenated and periodic pulse sequences

Dynamical decoupling can be used to preserve arbitrary quantum states despite undesired interactions with the environment, using control Hamiltonians affecting the system only. We present a system-independent analysis of dynamical decoupling based on leading order decoupling error estimates, valid for bounded-strength environments. Using as a key tool a renormalization transformation of the effective system-bath coupling Hamiltonian, we delineate the reliability domain of dynamical decoupling used for quantum state preservation, in a general setting for a single qubit. We specifically analyze and compare two deterministic dynamical decoupling schemes -- periodic and concatenated -- and distinguish between two limiting cases of fast versus slow environments. We prove that concatenated decoupling outperforms periodic decoupling over a wide range of parameters. These results are obtained for both ``ideal'' (zero-width) and realistic (finite-width) pulses This work extends and generalizes our earlier work, Phys. Rev. Lett. 95, 180501 (2005).

Renormalization analysis of catalytic Wright-Fisher diffusions

Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions indexed by the hierarchical group, with a linear attractive interaction between the components. In this context, the mappings are called renormalization transformations. We consider such maps for catalytic Wright-Fisher diffusions. These are diffusions on the unit square where the first component (the catalyst) performs an autonomous Wright-Fisher diffusion, while the second component (the reactant) performs a Wright-Fisher diffusion with a rate depending on the first component through a catalyzing function. We determine the limit of rescaled iterates of renormalization transformations acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.

Strong Resonance of Light in a Cantor Set

The propagation of an electromagnetic wave in a one-dimensional fractal object, the Cantor set, is studied. The transfer matrix of the wave amplitude is formulated and its renormalization transformation is analyzed. The focus is on resonant states in the Cantor set. In Cantor sets of higher generations, some of the resonant states closely approach the real axis of the wave number, leaving between them a wide region free of resonant states. As a result, wide regions of nearly total reflection appear with sharp peaks of the transmission coefficient beside them. It is also revealed that the electromagnetic wave is strongly enhanced and localized in the cavity of the Cantor set near the resonant frequency. The enhancement factor of the wave amplitude at the resonant frequency is approximately $6/|\eta_\mathrm{r}|$, where $\eta_\mathrm{r}$ is the imaginary part of the corresponding resonant eigenvalue. For example, a resonant state of the lifetime $\tau_\mathrm{r}=4.3$ms and of the enhancement factor $M=7.8\times10^7$ is found at the resonant frequency $\omega_\mathrm{r}=367$GHz for the Cantor set of the fourth generation of length L=10cm made of a medium of the dielectric constant $\epsilon=10$.

Monte Carlo renormalization: The triangular Ising model as a test case

We test the performance of the Monte Carlo renormalization method in the context of the Ising model on a triangular lattice. We apply a block-spin transformation which allows for an adjustable parameter so that the transformation can be optimized. This optimization purportedly brings the fixed point of the transformation to a location where the corrections to scaling vanish. To this purpose we determine corrections to scaling of the triangular Ising model with nearest- and next-nearest-neighbor interactions by means of transfer-matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation appears to lie well away from the nearest-neighbor critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because the standard assumptions of real-space renormalization imply that corrections to scaling vanish at the fixed point. We avoid this inconsistency by means of the optimized transformation which shifts the fixed point back to the vicinity of the nearest-neighbor critical Hamiltonian. The results of the optimized transformation in terms of the Ising critical exponents are more accurate than those obtained with the majority rule.

Application of the Perturbative Renormalization Group Method Based on the Lie Group to Pulse Dynamics

Pulse dynamics are studied using the perturbative renormalization group method based on the Lie group. The method is explained with several examples, and its similarities and dissimilarities with other methods are discussed. An interacting pulse is composed of a superposition of a single pulse solution (that is, the zero-th order solution) and its corrections. In the present method, the corrections are assumed to be expansions in terms of the zero eigenfunctions of a linear operator, whose coefficients are time-dependent variables. Unlike in other methods, in the present method, the time-dependence of the position of a pulse is introduced a posteriori in order to avoid the divergence of the corrections obtained in the naive perturbation expansion. The time-dependence of the position of a pulse is easily obtained by taking the inner product with the zero eigenfunction of the adjoint operator. It is shown that the present method based on the Lie group is a first-order approximation of interacting pulses. However, the truncated renormalization transformation reproduces the results obtained with other methods. It is necessary to use the zero-th order solutions of the linear operator and its adjoint operator in order to perform actual calculations.

Inverse Scattering Theory: Renormalization of the Lippmann−Schwinger Equation for Quantum Elastic Scattering with Spherical Symmetry

We present a new inverse scattering series for quantum elastic scattering in three spherical dimensions. The new series, which converges absolutely, independent of the strength of the scattering interaction, results from a renormalization transformation of the Lippmann−Schwinger Fredholm integral equation to a Volterra form. A new feature of the formulation is that it does not require the determination of phase shifts and it can be applied even to integral cross-section measurements. The approach is illustrated by application to a simple example problem.

One-dimensional Potts model, Lee-Yang edges, and chaos.

It is known that the exact renormalization transformations for the one-dimensional Ising model in a field can be cast in the form of the logistic map f(x)=4x(1-x) with x a function of the Ising couplings K and h. The locus of the Lee-Yang zeros for the one-dimensional Ising model in the K,h plane is given by the Julia set of the logistic map. In this paper we show that the one-dimensional q-state Potts model for q> or =1 also displays such behavior. A suitable combination of couplings, which reduces to the Ising case for q=2, can again be used to define an x satisfying f(x)=4x(1-x). The Lee-Yang zeros no longer lie on the unit circle in the complex z=e(h) plane for q not equal 2, but their locus still maps onto the Julia set of the logistic map.

Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems

We consider a class of Hamiltonians with three degrees of freedom that can be mapped into quasi-periodically driven pendulums. The purpose of this paper is to determine the threshold of the break-up of invariant tori with a specific frequency vector. We apply two techniques: the frequency map analysis and renormalization-group methods. The renormalization transformation acting on a Hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the Hamiltonian and a rescaling of phase space around the considered torus. We give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break-up of invariant tori. Furthermore, we obtain by these methods numerical values of the threshold of the break-up of the last invariant torus.

The Attractor of Renormalization¶and Rigidity of Towers of Critical Circle Maps

We demonstrate the existence of a global attractor ? with a Cantor set structure for the renormalization of critical circle mappings. The set ? is invariant under a generalized renormalization transformation, whose action on ? is conjugate to the two-sided shift with a countable alphabet.

Linear renormalization transformation for weakly interacting spin chains

The linear renormalization-group transformation is proposed to study critical temperatures and temperature dependence of the thermodynamic values (free energy, specific heat) of the weakly interacting spin chains. The method is examined in two classical systems: the standard Ising model on a rectangular lattice and the Ising spin chains coupled by pair-pair interactions. In the latter case the transformation does not exhibit any nontrivial fixed point. This result agrees with the exactly determined free energy of the system under consideration. The method is also applied to the weakly interacting quantum XY spin chains. The critical temperature as a function of the interchain interactions and the temperature dependence of the specific heat are found.

Renormalization Multigrid (RMG): Statistically Optimal Renormalization Group Flow and Coarse-to-Fine Monte Carlo Acceleration

New renormalization-group algorithms are developed with adaptive representations of the renormalized system which automatically express only significant interactions. As the amount of statistics grows, more interactions enter, thereby systematically reducing the truncation error. This allows statistically optimal calculation of thermodynamic limits, in the sense that it achieves accuracy e in just O(e−2) random number generations. There are practically no finite-size effects and the renormalization transformation can be repeated arbitrarily many times. Consequently, the desired fixed point is obtained and the correlation-length critical exponent ν is extracted. In addition, we introduce a new multiscale coarse-to-fine acceleration method, based on a multigrid-like approach. This general (non-cluster) algorithm generates independent equilibrium configurations without slow down. A particularly simple version of it can be used at criticality. The methods are of great generality; here they are demonstrated on the 2D Ising model.

On a Mathematical Definition of a Renormalization Transformation for Lattice Gauge Theories

In lattice gauge theories, the renormalization transformation and its properties are formally defined and formally proved by making use of Dirac's δ function and its properties. In this Letter, we shall give a mathematically rigorous definition of a renormalization transformation for lattice pure gauge field theories and show the required properties, which are use to show ultraviolet stability of lattice gauge theories.

Approximate renormalization with codimension-one fixed point for the break-up of some three-frequency tori

An approximate renormalization transformation is constructed for Hamiltonian systems with three degrees of freedom to study the break-up of invariant tori with a three-dimensional frequency vector from the cubic field Q( τ), where τ 3+ τ 2−2 τ−1=0. This renormalization has two fixed points: an attracting one and a hyperbolic one with codimension-one stable manifold. The associated critical exponents that characterize the universality class for the break-up of these invariant tori are computed.

Renormalization of quantum Anosov maps: reduction to fixed boundary conditions.

A renormalization scheme is introduced to study quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number ( k) is always finite. It is shown that the quasienergy eigenvalue problem of a QAM for all k BCs is exactly equivalent to that of the renormalized QAM (with Planck's constant Planck's over 2pi(') = Planck's over 2pi/k) at some fixed BCs that can be of four types. The quantum cat maps are, up to time reversal, fixed points of the renormalization transformation. Several results at fixed BCs, in particular the existence of a complete basis of "crystalline" eigenstates in a classical limit, can then be derived and understood in a simple and transparent way in the general-BCs framework.

Dynamical decimation renormalization-group technique: kinetic gaussian model on nonbranching, branching, and multibranching koch curves

A generalizing formulation of dynamical real-space renormalization that is appropriate for arbitrary spin systems is suggested. The alternative version replaces single-spin flipping Glauber dynamics with single-spin transition dynamics. As an application, in this paper we mainly investigate the critical slowing down of the Gaussian spin model on three fractal lattices, including nonbranching, branching, and multibranching Koch curves. The dynamical critical exponent z is calculated for these lattices using an exact decimation renormalization transformation in the assumption of the magneticlike perturbation, and a universal result z=1/nu is found.

Approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom.

We construct an approximate renormalization transformation that combines Kolmogorov-Arnol'd-Moser and renormalization-group techniques, to analyze instabilities in Hamiltonian systems with three degrees of freedom. This scheme is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically that the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface. As a consequence, isoenergetically degenerate and nondegenerate Hamiltonians belong to the same universality class, and thus the corresponding critical invariant tori have the same type of scaling properties. We numerically investigate the structure of the attracting set on the critical surface and find that it is a strange nonchaotic attractor. We compute exponents that characterize its universality class.