Real Space Renormalization Group Theory Research Articles

Real space renormalization group theory of disordered models of glasses

We develop a real space renormalization group analysis of disordered models of glasses, in particular of the spin models at the origin of the random first-order transition theory. We find three fixed points, respectively, associated with the liquid state, with the critical behavior, and with the glass state. The latter two are zero-temperature ones; this provides a natural explanation of the growth of effective activation energy scale and the concomitant huge increase of relaxation time approaching the glass transition. The lower critical dimension depends on the nature of the interacting degrees of freedom and is higher than three for all models. This does not prevent 3D systems from being glassy. Indeed, we find that their renormalization group flow is affected by the fixed points existing in higher dimension and in consequence is nontrivial. Within our theoretical framework, the glass transition results in an avoided phase transition.

Migdal–Kadanoff solution of the mixed spin-1 and spin-3/2 Blume–Capel model with different single-ion anisotropies

Using the real space renormalization group theory, we investigate the alternated mixed Blume–Capel model with spins S=1 and S=3/2, on the hyper cubic lattice. First, the ground state phase diagram of the system at zero temperature is obtained on the single-ion anisotropies plane showing the same topology obtained early by other approximations.According to the different values of the single-ion anisotropies, three types of phase diagrams are obtained, with first and second order transitions; tricritical points are highlighted. Also, we note the absence of dimensional crossover between d=2 and d=3.

Density matrix renormalization group study of random dimerized antiferromagnetic Heisenberg chain

The effect of dimerization on the random antiferomagnetic Heisenberg chain with spin ½ is studied using the density matrix renormalization group method. The ground-state energy, the energy gap distribution and the string-order parameter are calculated and analyzed using the finite size scaling method. The ground-state energy gain due to dimerization behaves as u a with a ≳ 2, where u is the degree of dimerization, suggesting the absence of spin-Peierls instability. The string order is found to behave as u 2β with β ∼ 0.35 in contrast to the prediction of real space renormalization group theory.

Density Matrix Renormalization Group Study of Random Dimerized Antiferromagnetic Heisenberg Chains

The effect of dimerization on the random antiferomagnetic Heisenberg chain with spin 1/2 is studied by the density matrix renormalization group method. The ground state energy, the energy gap distribution and the string order parameter are calculated. Using the finite size scaling analysis, the dimerization dependence of the these quantities are obtained. The ground state energy gain due to dimerization behaves as u a with a > 2 where u denotes the degree of dimerization, suggesting the absence of spin-Peierls instability. It is explicitly shown that the string long range order survives even in the presence of randomness. The string order behaves as u 2β with β∼0.37 in agreement with the recent prediction of real space renormalization group theory (\(\beta =(3-\sqrt{5})/2 \simeq 0.382\)). The physical picture of this behavior in this model is also discussed.

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.

Numerical calculations of the thermal conductivities of composites: a 2-D model

SUMMARY A 2-D numerical finite-difference model has been developed to calculate the thermal conductivity of a composite material. The method is general in that the distribution of components within the material can be ordered or random. Results from the calculations are compared with those based on a geometric model in which the conductivity of the composite depends on the volume fractions of the different components and with conductivities calculated using real-space renormalization group theory (RSRG) and the effective medium approximation (EMA). The results from the present work compare well with the geometric model and the EMA, whereas the RSRG results generally indicate lower conductivities. Possible reasons for the differences between the RSRG theory and the other methods are discussed.

Numerical model calculations of the effects of grain sizes and orientations on the thermal conductivities of composites

A numerical method is used to investigate the effects of particle sizes and orientations on the thermal conductivities of two-component composites. The method is based on a finite-difference simulation of the divided-bar thermal conductivity measuring device, and solves the steady-state heatflow equation in two dimensions throughout a region representing the divided bar with included sample. Calculations are made for two-component particle sizes for two orientations and for random distributions. The results are compared with those from a geometric model in which the conductivity of the composite depends on the volume fractions of the different components and with conductivities calculated using realspace renormalization group theory (RSRG) and the effective medium approximation (EMA). The results show substantial differences between samples with different particle orientation and sizes. For the particular distributions examined here, differences of up to a factor of one and one-half occur in some instances.

Real space renormalization group theory of the percolation model

A cluster expansion renormalization group method in real space is-developed to determine the critical properties of the percolation model. In contrast to previous renormalization group approaches, this method considers the cluster size distribution (free energy) rather than the site or bond probability distribution (coupling constants) and satisfies the basic renormalization group requirement of free energy conservation. In the construction of the renormalization group transformation, new couplings are generated which alter the topological structure of the clusters and which must be introduced in the original system. Predicted values of the critical exponents appear to converge to presumed exact values as higher orders in the expansion are considered. The method can in principle be extended to different lattice structures, as well as to different dimensions of space.

Theory of superconducting wire networks and Josephson-junction arrays in magnetic fields.

In this paper we study the superconducting-normal phase boundaries of a variety of wire networks and Josephson-junction arrays. We have obtained the mean-field phase diagrams for a number of geometries and compared them to the corresponding experimental data. We have introduced an analytical approach to the analysis of the structures present in the phase diagrams. We have shown in great detail how the gross structure is determined by the statistical distributions of the cell areas, and how the fine structures are determined by correlations among neighboring cells in the lattices. We have also studied the effect of thermal fluctuations on the structure of the phase diagram by a cluster mean-field calculation and a real-space renormalization-group (RG) theory. The RG theory provides a natural link between the structures at fractional fluxes to those at integral ones, predicting a pronounced hierarchical behavior of the phase diagram and an infinite slope for the cusps.

Effective transport properties of disordered, multi-phase composites: Application of real-space renormalization group theory

Real-space renormalization group theory (RSRG) provides a powerful tool for the estimation of effective transport coefficients of disordered, two- and three-phase composites on the basis of the intraphase transport coefficients. In this paper, we present methods for estimating effective transport coefficients of space-filling composites modelled by square, hexagonal and cubic tessellations. These methods are particularly useful for composites whose phases differ from each other in their individual transport coefficients by two or more orders of magnitude and whose effective transport properties are predominantly influenced by the connectedness of the phases. Problems of interest could include evaluation of effective diffusion coefficients of small molecules in incompatible polymer blends, effective thermal conductivities of porous media, effective electrical conductivities of metal-filled polymers, and effective elastic moduli of disordered composites.

Application of Real-Space Renormalization Group Theory to the Antiferromagnetic Potts Model

The real-space renormalization group theory using the first order cumulant is applied to the q -state antiferromagnetic Potts model on 2-d (dimensional) and 3-d lattice. The special features of this paper are that the Potts states are represented by q unit vectors in q -1 dimensional space and the two-sublattice structure is preserved after the renormalization transformation. The critical exponents obtained for q =2 in 3-d are in fairly good agreement with the reliable values and those for q =3 and 4 are obtained for the first time.

Application of Real-Space Renormalization Group Theory to a Triangular Ising Lattice

Application of Real-Space Renormalization Group Theory to a Triangular Ising Lattice

New real-space renormalization-group calculation for the critical properties of lattice spin systems

In evaluating the critical properties of lattice spin systems in the real-space renormalization-group theory we use the cluster variation method. A configuration in the transformed system is constrained and the probability of occurrence of this configuration is calculated both in the transformed system and in the original system. By equating the two probabilities and forming ratios of two such equalities (for two or more constrained configurations) the fixed point of the renormalization transformation is evaluated. The method can avoid the trouble due to different singularities in the original and transformed systems, and hence can obviate the possible development of spurious singularities in the transformation at low temperatures. The two-dimensional triangular Ising model is treated with numerical results comparable with those obtained by the cluster treatment of Niemeijer and van Leeuwen who used more and larger cluster types than those we introduce.

Finite-temperature quantum renormalization-group theory for the one-dimensional anisotropic Heisenberg model

A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.

Spin Wave Theory of the Spin 1/2 XY Model

The spin wave theory of Kondo and Yamaji is applied to the spin 1/2 XY model in one to three dimensions. In one dimension, the value of the gap Δ which appears in the spectrum of x -component spin remains finite up to 0 K, and no phase transition occurs. The value of the nearest neighbour correlation function is in good agreement with the exact values. In three dimensions, Δ vanishes at a finite temperature, and the second order phase transition occurs. In two dimensions, as T →0, Δ becomes very small for square lattice and tends to zero as exp (- T 0 / T ) for triangular lattice, which means the non-existence of the phase transition. The susceptibility, however, becomes anomalously large for low temperatures, and the extrapolation of Curie-Weiss law gives a “fictitious” transition temperature whose value agrees with those given by the high temperature expansion and the real space renormalization group theory.