Real-space Renormalization Group Approach Research Articles

Real-space renormalization group approach of the Potts model on the octagonal quasi-periodic tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the ferromagnetic (FM) Potts model on the two-dimensional (2D) octagonal quasi-periodic tiling (OQT). The critical exponents of the correlation length in theq=1,2,3,4 cases and the crtitical surface of the Ising model are obtained. The results are discussed by comparing with previous results on the OQT and the square lattice (SQL).

Strongly disordered antiferromagnetic spin-1 chains with random anisotropy

The zero-temperature phase diagram of the anisotropic, strongly disordered antiferromagnetic quantum spin-1 chain is investigated, within the framework of a real-space renormalization-group approach. In the presence of uniform anisotropy there occurs a phase transition from a random singlet phase to an Ising antiferromagnetic phase controlled by a fixed point that does not have Heisenberg symmetry. We find evidence of a fixed point due to randomness in the anisotropy similar to that which occurs for quantum Heisenberg spin-$\frac{1}{2}$ chains.

Nonlinear optics of random metal-dielectric films

The optical properties of random metal-dielectric thin films (referred also to as semicontinuous metal films) are of great interest, in large part because of their high potential for various applications. Two-dimensional (2d) semicontinuous metal films are usually produced by thermal evaporation or sputtering of metal onto an insulating (dielectric) substrate. In the growing process small metallic grains are formed on the substrate first. As the film grows, the metal concentration increases and coalescences occur, so that irregularly shaped clusters are formed on the substrate eventually resulting in 2d fractal structures. At the percolation threshold, the sizes of the fractal structures diverge and a continuous conducting path of metal appears between the ends of the sample. At higher surface coverage, the film is mostly metallic, with voids of irregular fractal shapes. As further coverage increases, the film becomes uniform. We calculated the field distributions on a semicontinuous film at the fundamental and generated (in nonlinear optical processes) frequencies using a very effective numerical method based on the real space renormalization group (RSRG) approach. Our RSRG calculations demonstrate the large fluctuations for the intensity of the local electric fields.

Renormalization of Self‐Organized Critical Models

Abstract: Self‐organized criticality refers to a mechanism whereby complex dissipative dynamical systems naturally evolve to a self‐sustaining critical point, exhibiting correlations on all length scales and time scales. A recent theory based on a real space renormalization group approach sheds light on fundamental aspects of the phenomenon.

Diffusion on a honeycomb lattice: Real-space renormalization-group approach

A two-dimensional lattice-gas model with honeycomb symmetry is investigated by using the real-space renormalization group (RSRG) approach with a number of RSRG transformations based on clusters of different symmetries and sizes (up to 42 sites). The accuracy of calculations is found to be a nonmonotonic function of the size of a cluster, and to depend strongly on its symmetry. The highest obtained accuracy with respect to the determination of the critical value of the pair interaction parameter is 0.38%. The phase diagram of the Ising antiferromagnetic spin model and of the corresponding lattice-gas model is constructed with this accuracy. The ordered phase in the lattice system is shown to appear in the very narrow density interval, $0.447lnl0.553.$ In addition, the coverage dependence of the chemical diffusion coefficient and mean-square density fluctuations are studied at temperatures below the critical one. Both quantities are demonstrated to exhibit singularities at the critical coverages. The type of singularities (in particular, the critical slowdown of diffusion) is in agreement with the predictions of the scaling theory and also with recent results obtained for a model treating the effect of surface reconstruction on diffusion.

Random-mass Dirac fermions in doped spin-Peierls and spin-ladder systems: One-particle properties and boundary effects

Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\epsilon$, the single-particle Green function decays in real space like $G(x,\epsilon) \propto (1/x)^{3/2}$. It follows that at these energies the correlations in the disordered system are strong -- even stronger than in the pure system without the gap. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as $1/\sqrt{x}$ and the density of states saturates to its bulk value on the scale $L_\epsilon \propto \ln^2 (1/\epsilon)$. This scale is different from the Thouless localization length $\lambda_\epsilon\propto\ln (1/\epsilon)$. We also discuss some implications of these results for the spin systems and their relation to the investigations based on real-space renormalization group approach.

(4+N)-dimensional elastic manifolds in random media: A renormalization-group analysis

Motivated by the problem of weak collective pinning of vortex lattices in high-temperature superconductors, we study the model system of a four-dimensional elastic manifold with N transverse degrees of freedom (4+N-model) in a quenched disorder environment. We assume the disorder to be weak and short-range correlated, and neglect thermal effects. Using a real-space functional renormalization group (FRG) approach, we derive a RG equation for the pinning-energy correlator up to two-loop correction. The solution of this equation allows us to calculate the size R_c of collectively pinned elastic domains as well as the critical force F_c, i.e., the smallest external force needed to drive these domains. We find R_c prop. to delta_p^alpha_2 exp(alpha_1/delta_p) and F_c prop. to delta_p^(-2 alpha_2) exp(-2 alpha_1/delta_p), where delta_p <<1 parametrizes the disorder strength, alpha_1=(2/pi)^(N/2) 8 pi^2/(N+8), and alpha_2=2(5N+22)/(N+8)^2. In contrast to lowest-order perturbation calculations which we briefly review, we thus arrive at determining both alpha_1 (one-loop) and alpha_2 (two-loop).

Conductivity fluctuations in polymer's networks

A Polymer network is treated as an anisotropic fractal with fractional dimensionality D = 1 + ε close to one. Percolation model on such a fractal is studied. Using real space renormalization group approach of Migdal and Kadanoff, we find the threshold value and all the critical exponents in the percolation model to be strongly nonanalytic functions of ε, e.g. the critical exponent of the conductivity was obtained to be ε −2 exp (−1 − 1/ε). The main part of the finite-size conductivities distribution function at the threshold was found to be universal if expressed in terms of the fluctuating variable which is proportional to a large power of the conductivity, but with ε-dependent low-conductivity cut-off. Its reduced central momenta are of the order of e −1/ε up to a very high order.

Percolation and Conductivity in Nearly-1D Systems

A polymer network is treated as an anisotropic fractal with fractional dimensionality D = 1 + ε close to one. The percolation model on such a fractal is studied. Using the real space renormalization group approach of Migdal and Kadanoff we find the threshold value and all the critical exponents in the percolation model to be strongly nonanalytic functions of ε, e.g. the critical exponent of the conductivity was obtained to be ε—2 exp (—1 — 1/ε). The main part of the finite size conductivity distribution function at the threshold was found to be universal if expressed in terms of the fluctuating variable which is proportional to a large power of the conductivity, but with ε-dependent low-conductivity cut-off. Its reduced central momenta are of the order of e—1/ε up to very high orders.

Percolation and Conductivity in Nearly-1D Systems

A polymer network is treated as an anisotropic fractal with fractional dimensionality D = 1 + ε close to one. The percolation model on such a fractal is studied. Using the real space renormalization group approach of Migdal and Kadanoff we find the threshold value and all the critical exponents in the percolation model to be strongly nonanalytic functions of ε, e.g. the critical exponent of the conductivity was obtained to be ε—2 exp (—1 — 1/ε). The main part of the finite size conductivity distribution function at the threshold was found to be universal if expressed in terms of the fluctuating variable which is proportional to a large power of the conductivity, but with ε-dependent low-conductivity cut-off. Its reduced central momenta are of the order of e—1/ε up to very high orders.

Localization and conductance fluctuations in the integer quantum Hall effect: Real-space renormalization-group approach

We consider the network model of the integer quantum Hall effect transition. By generalizing the real-space renormalization-group (RG) procedure for the classical percolation to the case of the quantum percolation, we derive a closed RG equation for the universal distribution of conductance of the quantum Hall sample at the transition. We find the numerical solution of the RG equation and use it to calculate the critical exponent of the localization length and the central moments of the conductance distribution. The results obtained are compared with the results of recent numerical simulations.

Density-matrix renormalization group for fermions: Convergence to the infinite-size limit

The density-matrix real-space renormalization-group approach is applied to the one-dimensional Hubbard model, to investigate the properties of convergence of the algorithm to a fixed orbit as the infinite-size (thermodynamic) limit is approached. We find that this convergence, which depends on the physical parameters of the model, is generally good. The interplay between the truncation of the Hilbert space and the thermodynamic limit, as well as the reasons and consequences of the existence of a fixed orbit are discussed.

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, “normal,” and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small “H” cells. Our results are of the formN=AL D SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.

Critical behavior of nonlinear random conductance networks: Real-space renormalization group approach

The real-space renormalization group theory is used to study the critical behavior of nonlinear random conductance networks in two and three dimensions, in which the components obey a current-voltage ( I – V) relation of the form I = χ μ V γ with γ > 0 and μ = i, m where i, m represent the inclusion of volume fraction p and the host medium of volume fraction q ( p + q = 1), respectively. Two important limits are worth studying: (1) normal conductor-insulator ( N/ I) mixture ( χ m = 0) and (2) superconductor-normal conductor ( S/ N) mixture ( χ m = ∞). As the percolation threshold p c (or q c ) is approached, the effective nonlinear response χ e is found to behave as χ e ≈ ( p − p c ) t in the N/ I limit while χ e ≈ ( q c − q) − s in the S/ N limit. We calculate the critical exponents t and s as a function of γ. A more general duality relation is found to obey in two dimensions. By a connection with the links-nodes-blobs picture, t and s can be related to critical exponents ζ R and ζ s , which describe the geometry of the percolating network. The results are compared with those of the series analysis and excellent agreements are found over a wide range of γ.

Renormalization group approach as an approximate solution of diophantine system of equations: Application to the Ising model

It is demonstrated that the real-space renormalization group approach is equivalent to an appropriate diophantine system of equations. The method is applied on several examples of the Ising model.

Random spin-1 quantum chains

We study disordered spin-1 quantum chains with random exchange and biquadratic interactions using a real space renormalization group approach. We find that the dimerized phase of the pure biquadratic model is unstable and gives rise to a random singlet phase in the presence of weak disorder. In the Haldane region of the phase diagram we obtain a quite different behavior.

The bond-dilutedZ(q) ferromagnetic model: criticality and break-collapse method

Within a real-space renormalization group (RG) which preserves two-site correlation functions, we study, on the square lattice, the criticality of the bond-diluted Z(q) ferromagnetic model. We generalize the `break-collapse method' which simplifies greatly the exact calculation of arbitrary Z(q) two-terminal clusters (commonly appearing in RG approaches) mainly for a large value of q. We reproduce, in the pure case, several known exact results. The structure of the phase diagrams, for all the values of q, is obtained with a good precision. The massless spin - wave-like phase, which evolves into the Kosterlitz - Touless phase for , occurs around (in agreement with the well known exact result). The structure of the phase diagrams in the diluted case is qualitatively similar to that obtained from the pure model. The massless spin - wave-like phase resists in an interval of concentration p which increases for the large values of q.

Phonon properties of one-dimensional nanocrystalline solids.

We study phonon properties of one-dimensional nanocrystalline solids that are associated with a model nanostructured sequence. A real-space renormalization-group approach, connected with a series of renormalization-group transformations, is developed to calculate numerically the local phonon Green's function at an arbitrary site, and then the phonon density of states of these kinds of nanocrystalline chains. Some interesting phonon properties of nanocrystalline chains are obtained that are in qualitative agreement with the experimental results for the optical-absorption spectra of nanostructured solids.

Character Expansion, Zeroes of Partition Function and -Term in U(1) Gauge Theory

Character expansion developed in real space renormalization group (RSRG) approach is applied to U(1) lattice gauge theory with $\th$-term in 2 dimensions. Topological charge distribution $P(Q)$ is shown to be of Gaussian form at any $\b$(inverse coupling constant). The partition function $Z(\th)$ at large volume is shown to be given by the elliptic theta function. It provides the information of the zeros of partition function as an analytic function of $\ze= e^{i \th}$ ($\th$ = theta parameter). These partition function zeros lead to the phase transition at $\th=\pi$. Analytical results will be compared with the MC simulation results. In MC simulation, we adopt (i)``set method" and (ii)``trial function method".

CORE — A new computational method for lattice systems

The contractor renormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is introduced. The method combines contraction and variational techniques with the real-space renormalization group approach. It applies to lattice systems of infinite extent and is suitable for studying phase structure and critical phenomena. The CORE approximation is simple to implement and is systematically improvable. The method is illustrated using the 1+1-dimensional Ising model.