Real-space Renormalization Group Approach Research Articles

Integer quantum Hall transition in the presence of a long-range-correlated quenched disorder

We theoretically study the effect of long-ranged inhomogeneities on the critical properties of the integer quantum Hall transition. For this purpose we employ the real-space renormalization-group (RG) approach to the network model of the transition. We start by testing the accuracy of the RG approach in the absence of inhomogeneities, and infer the correlation length exponent $\ensuremath{\nu}=2.39$ from a broad conductance distribution. We then incorporate macroscopic inhomogeneities into the RG procedure. Inhomogeneities are modeled by a smooth random potential with a correlator which falls off with distance as a power law, ${r}^{\ensuremath{-}\ensuremath{\alpha}}.$ Similar to the classical percolation, we observe an enhancement of $\ensuremath{\nu}$ with decreasing \ensuremath{\alpha}. Although the attainable system sizes are large, they do not allow one to unambiguously identify a cusp in the $\ensuremath{\nu}(\ensuremath{\alpha})$ dependence at ${\ensuremath{\alpha}}_{c}=2/\ensuremath{\nu},$ as might be expected from the extended Harris criterion. We argue that the fundamental obstacle for the numerical detection of a cusp in the quantum percolation is the implicit randomness in the Aharonov-Bohm phases of the wave functions. This randomness emulates the presence of a short-range disorder alongside the smooth potential.

Renormalization-group magnetization of a ferrimagnetic Ising system

We employ a real-space renormalization-group (RSRG) approach to study a mixed-spin (spin- 1 2 and spin-1) antiferromagnetic Ising model on the square lattice. The model incorporates next-nearest-neighbor interactions which are relevant to describe ferrimagnetism. We present an RSRG calculation of the spontaneous magnetization and compensation temperatures of the mixed-spin model. The influence of crystal-field and next-nearest-neighbor interactions on compensation temperatures is studied; the important physical ingredient that leads to the presence of these temperatures is the next-nearest-neighbor coupling between spins 1 2 .

Role of disorder on the quantum critical point of a model for heavy fermions

A zero-temperature real-space renormalization group (RG) approach is used to investigate the role of disorder near the quantum critical point (QCP) of a Kondo necklace $(XY\ensuremath{-}KN)$ model. In the pure case this approach yields ${J}_{c}=0$ implying that any coupling $J\ensuremath{\ne}0$ between the local moments and the conduction electrons leads to a nonmagnetic phase. We also consider an anisotropic version of the model $(X\ensuremath{-}KN),$ for which there is a quantum phase transition at a finite value of the ratio between the coupling and the bandwidth, $(J/W).$ Disorder is introduced either in the on-site interactions or in the hopping terms. We find that in both cases randomness is irrelevant in the $X\ensuremath{-}\mathrm{KN}$ model, i.e., the disorder induced magnetic-nonmagnetic quantum phase transition is controlled by the same exponents of the pure case. Finally, we show the fixed point distributions ${P}_{J}(J/W)$ at the attractors of the disordered, nonmagnetic phases.

Collective surface diffusion of repulsively interacting particles on a triangular lattice: Comparison of real-space renormalization group and Monte Carlo approaches

A two-dimensional lattice-gas model with triangular symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of this method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. Using the RSRG method, we have explored phase diagrams of a two-dimensional Ising spin model and of a triangular lattice gas with pair lateral repulsive interactions. We have calculated: (i) adsorption isotherms and thermodynamic factors for different temperatures and (ii) the coverage dependence for the pair, three, and four nearest-neighboring particles correlation functions, the tracer, jump, and chemical diffusion coefficients using four different models of adparticle jumps. All these quantities have also been obtained by Monte Carlo (MC) simulations. Despite the fact that both methods, RSRG and MC, constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore we conclude that the RSRG method can be applied, at least for the systems discussed here, to characterize the thermodynamic and kinetic properties of strongly interacting adsorbates. It is also shown that drastic changes in the surface diffusion coefficients occur when (i) lateral interactions force ordering of the adatoms via second-order phase transition and (ii) different models of adparticle jumps are used.

Quantum phase transitions in random magnets

This talk will discuss novel critical behavior and “Griffiths–McCoy” singularities (which occur away from criticality) near quantum phase transitions in disordered systems with discrete (Ising) symmetry. One finds that distributions of many quantities are very broad, so it is insufficient to simply take the average. After reviewing some exact results in one dimension obtained using a real space renormalization group approach, I will discuss extensions of these results obtained by a numerical method in which the spin problem is mapped on to free fermions. In this way, whole distributions can be determined with precision. The talk will then describe quantum Monte Carlo simulations which show that similar results also occur in higher dimensions, at least for d=2.

The anisotropic Ashkin–Teller model: a renormalization group study

The two-dimensional ferromagnetic anisotropic Ashkin–Teller model is investigated through a real-space renormalization-group approach. The critical frontier, separating five distinct phases, recovers all the known exact results for the square lattice. The correlation length ( ν T ) and crossover ( φ) critical exponents are also calculated. With the only exception of the four-state Potts critical point, the entire phase diagram belongs to the Ising universality class.

Renormalization of two-dimensional Ising systems with irrelevant, marginal and relevant aperiodic (dis)order

We introduce a class of 2d Ising models with aperiodic modulations of the coupling constants following a 2d substitution rule. The relevance of the aperiodicity to the phase transition is examined through an approximative real space renormalization group approach known as the cumulant approximation. We find that the relevance of the aperiodic disorder is determined by the fluctuations of the mean coupling as predicted by the Harris–Luck criterion.

Adatom diffusion on a square lattice. Theoretical and numerical studies

A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The minimal relative error in determining the critical values of the interaction parameters is equal to \(\). Using the RSRG method, we have explored phase diagrams of both a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We also have investigated the influence of the attractive and repulsive interactions on both the thermodynamic properties of the lattice gas and the diffusion of adsorbed particles over surface. We have calculated adsorption isotherms and coverage dependences of the pair correlation function, isothermal susceptibility and the chemical diffusion coefficient. In addition, we have included in our analysis the interaction of the activated particle in the saddle point with its nearest neighbors. We have also used Monte Carlo (MC) technique to calculate these dependences. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG approach can be applied to characterize the thermodynamic and kinetic properties of systems of particles with strong lateral interactions.

Real-space renormalization group study of the anisotropic antiferromagnetic Heisenberg model on a square lattice

In this work we apply two different real-space renormalization-group (RSRG) approaches to the anisotropic antiferromagnetic spin-1/2 Heisenberg model on the square lattice. Our calculations allow for an approximate evaluation of the $T$ vs. $\Delta$ phase-diagram: the results suggest the existence of a critical value of $\Delta>0$, at which the critical temperature goes to zero, and the presence of reentrant behavior on the critical line between the ordered and disordered phases. This whole critical line is found to belong to the same universality class as the Ising model. Our results are in accordance with previous RSRG approaches but not with numerical simulations and spin-wave calculations.

Real-space renormalization for reaction-diffusion systems

The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real-space renormalization group approach, originally developed for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d = 1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalization.

Non perturbative renormalization group approach to surface growth

We present a recently introduced real space renormalization group (RG) approach to the study of surface growth. The method permits us to obtain the properties of the KPZ strong coupling fixed point, which is not accessible to standard perturbative field theory approaches. Using this method, and with the aid of small Monte Carlo calculations for systems of linear size 2 and 4, we calculate the roughness exponent in dimensions up to d=8. The results agree with the known numerical values with good accuracy. Furthermore, the method permits us to predict the absence of an upper critical dimension for KPZ contrarily to recent claims. The RG scheme is applied to other growth models in different universality classes and reproduces very well all the observed phenomenology and numerical results. Intended as a sort of finite size scaling method, the new scheme may simplify in some cases from a computational point of view the calculation of scaling exponents of growth processes.

Diffusion of Interacting Adsorbates on a Square Lattice

A two-dimensional lattice gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach. Using the RSRG transformation with a cluster of 20 spins, we have investigated the influence of pairwise nearest and next nearest neighbor pair interactions as well as three- and four-particle interactions on the chemical adatom diffusion coefficient. In addition we have analyzed adsorption isotherms and adatom density fluctuations. We have also studied how interactions in the activated state influence surface diffusion.

Blume-Emery-Griffiths model in a random crystal field

We study the Blume-Emery-Griffiths model in a random crystal field in two and three dimensions, through a real-space renormalization-group approach and a mean-field approximation, respectively. According to the two-dimensional renormalization-group calculation, non-symmetry-breaking first-order phase transitions are eliminated and symmetry-breaking discontinuous transitions are replaced by continuous ones, when disorder is introduced. On the other hand, the mean-field calculation predicts that first-order transitions are not eliminated by disorder, although some changes are introduced in the phase diagrams. We make some comments on the consequences of a degeneracy parameter, which may be relevant in martensitic transitions.

Temperature-Dependent ac Conductivity of the Fibonacci Lattice

A new practical equation is derived from the Miller-Abrahams theory for different site-energies, and the temperature-dependent conductivity of Fibonacci lattice is calculated by a real-space renormalization-group approach. It is shown that there exist two types of temperature-dependent conductivity at low- and high-frequencies. Furthermore, it is found that the low-frequency conductivity oscillates dramatically with temperature.

Real-space renormalization group approach to the Potts model on the two-dimensional Penrose tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the q-state Potts model on the two-dimensional (2D) Penrose tiling (PT). The critical exponents of the correlation length ν( q) in the region between q=1 and q=4 are obtained. The bond percolation (BP) threshold and Ising critical temperature in the isotropic case are also obtained. The comparison of the results with previous results on the PT and the exact results on square lattice (SQL) indicates that the universal classes of q-state Potts model on PT and SQL may be the same.

Adatom diffusion on a square lattice: Comparison of real-space renormalization group and Monte Carlo approaches

A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The most accurate results have been obtained for the largest cluster containing 34 sites. The minimal relative error in determining the critical values of interaction parameter is equal to 0.13%. It has been shown that the ``antiferromagnetic majority rule'' for the choice values of block spins gives much better results for the repulsion between the adsorbed particles, as compared to the ordinary ``majority rule.'' Using the RSRG method, we have explored phase diagrams of a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We have calculated: (a) adsorption isotherms for different temperatures, (b) the coverage and temperature dependences of both the pair correlation function for the nearest neighboring adparticles and the chemical diffusion coefficient, (c) the temperature dependence of the specific heat, and (d) the coverage dependences of the isothermal susceptibility for different temperatures. All these quantities have also been obtained by Monte Carlo simulations. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG method can be applied at least for the systems discussed here to characterize the thermodynamic and kinetic properties of interacting adsorbates.

Adsorption and diffusion in a square lattice gas with strong attraction between particles: The real-space renormalization group approach

A two-dimensional lattice gas with an attractive nearest neighbor (NN) pairwise lateral interaction on a lattice of square symmetry and its equivalent two-dimensional Ising spin model have been investigated by using the real-space renormalization group (RSRG) approach with blocks of different sizes and symmetries. We have calculated (a) adsorption isotherms for different temperatures and (b) the spontaneous magnetization for the Ising spin ferromagnet. The RSRG data are compared with the well-known exact expression. The coincidence between the exact and RSRG data is rather good. It is shown that the precision of the RSRG method strongly depends not only on the number of sites in the block but also on its symmetry.In addition, we have calculated the coverage dependences of the pair correlation function for nearest neighbor adparticles, the isothermal susceptibility and the chemical adparticle diffusion coefficient at different temperatures. Furthermore, we have calculated the critical behavior of the mean square adparticle density fluctuations and of the chemical diffusion coefficient. The critical exponent is close to the value obtained by other methods. The RSRG data have been compared with results from Monte Carlo simulations. The coincidence is very good and, therefore, we conclude that the RSRG method can be applied at least for the systems discussed here to investigate the thermodynamic and kinetic properties of interacting adsorbates.

HOPPING CONDUCTIVITY OF NANOSTRUCTURED CHAIN: REAL-SPACE RENORMALIZATION GROUP APPROACH

In terms of Miller-Abrahams' theory, a real-space renormalization group approach is developed to calculate the hopping conductivity of one-dimensional nanostructured chain.It is found that the kinds and the sizes,as well as the interface structures and lattice distortion,of nano-grains have a notable effect on the hopping conductivity of nanostructured systems.

Real-space renormalization-group approaches for two-dimensional Gaussian Ising spin glass

Two-dimensional Ising spin glass with Gaussian couplings is studied through different real-space renormalization-group procedures. The stiffness exponent y, associated with the evolution of the coupling distribution at zero temperature, is computed. Our results are discussed and compared with other estimates available.

A real-space renormalization group approach to the ferromagnetic Potts model on the two-dimensional octagonal quasi-periodic tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the ferromagnetic (FM) q-state Potts model on the two-dimensional (2D) octagonal quasi-periodic tiling (OQT). The critical exponents of the correlation length for different values of q and the critical temperature of the Ising model are obtained. The results are shown to be not sensitive to the choice of parameters. The comparison of the results with previous results for the OQT and the square lattice (SQL) seems to show that the universal classes of the q-state Potts models on the OQT and the SQL are the same for the range from q = 1 to q = 3, in accordance with previous research.