Real-space Renormalization Group Approach Research Articles

Entanglement Entropy of Random Quantum Critical Points in One Dimension

For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization-group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.

Phase Transition in a System of One-Dimensional Bosons with Strong Disorder

We study one-dimensional disordered bosons at large commensurate filling. Using a real space renormalization group approach, we find a new random fixed point which controls a phase transition from a superfluid to an incompressible Mott glass. The transition can be tuned by changing the disorder distribution even with vanishing interactions. We derive the properties of the transition, which suggest that it is in the Kosterlitz-Thouless universality class.

Real space renormalization group with effective interactions: applications to 2-D spin lattices

The Bloch’s theory of effective Hamiltonians has been used to improve the Real Space Renormalization Group approach. The effective interactions between elementary blocks of a periodic lattice can be extracted from the knowledge of the spectrum of the dimers or trimers of blocks. The potentialities of the method are illustrated on a series of quasi 1-D and 2-D problems. The spin gap of two-leg ladders is calculated and an estimate of the impact of ferromagnetic couplings between two-leg ladders on the gap is presented. The method satisfactorily identifies the phase transitions in the 1/5-depleted square lattice as well as in the spin-frustrated Shastry-Sutherland lattice. The J 2/J 1 checkerboard lattice is studied and a location of the phase transition between the Neel phase and the dimer phase is proposed.

Large-scale numerical simulations of ultrametric long-range depinning.

The depinning of an elastic line interacting with a quenched disorder is studied for long-range interactions, applicable to in-plane crack propagation or wetting. An ultrametric distance is introduced instead of the Euclidean distance, allowing for a drastic reduction of the numerical complexity of the problem. Based on large-scale simulations, two to three orders of magnitude larger than previously considered, we obtain a very precise determination of critical exponents which are shown to be indistinguishable from their Euclidean metric counterparts. Moreover, the scaling functions are shown to be unchanged. The choice of an ultrametric distance thus does not affect the universality class of the depinning transition and opens the way to an analytic real-space renormalization-group approach.

The anisotropic Kondo necklace model

A real-space renormalization-group approach is used to investigate the zero temperature phase diagram of the Kondo necklace ( KN) model in the presence of an Ising-like anisotropy parameter ( δ). In the full anisotropic case, δ=1, this approach yields a quantum critical point (QCP) separating an antiferromagnetic from a non-magnetic, Kondo-like, phase. The critical frontier of the phase diagram is presented as a function of the ratio between the coupling and the bandwith ( J/ W), and the anisotropic parameter ( δ). We find for the KN model that a critical value δ c of the anisotropy is required for the appearance of long range magnetic order at zero temperature.

Electronic spectra of GaAs/Ga xAl 1− xAs superlattice with impurities arranged according to a Fibonacci sequence

We study the electronic density of states (DOS) of a GaAs/Ga x Al 1− x As superlattice with impurities arranged according to the Fibonacci sequence. Our theory uses a Green function method based on Dyson’s equation within the real-space renormalization-group approach in the framework of the tight-binding Hamiltonian for the Fibonacci quasiperiodic lattice. An exact decimation transformation has been done for the physical parameters characterizing the Fibonacci chain considering the scaling factor, under which the system is self-similar, equal to the golden mean τ=(1+ 5 )/2 . The electronic energy spectrum are then used to determine the electronic specific heat which, due to the localization of the electronic states, exhibits strong Schottky anomalies when the temperature goes to zero together with a log-periodic oscillation.

Potts spin glass: a renormalization group approach

The phase transition of the q-state Potts spin-glass model is studied with the real-space renormalization group approach on general diamond hierarchical lattices. The model, with q=2 up to six Potts states, is investigated for the particular cases of lattices with scaling factor b=2 and dimensions d=2 up to 6. For all cases studied, the Gaussian, the uniform, the exponential, and the delta-bimodal probability distributions are considered as the initial probability distribution of coupling constants. For q⩾2 the critical temperatures, for the phase transition from the high-temperature paramagnetic to a condensed phase, are estimated for lattices with fractal dimensions ⩾ d l ( q).

Quantum spins and quasiperiodicity: a real space renormalization group approach.

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.

On predicting the effective elastic properties of polymer bonded explosives using the recursive cell method

Polymer bonded explosives are particulate composites containing elastic particles in a viscoelastic binder. The particles occupy an extremely high fraction of the volume, often greater than 90%. Under low strain rate loading (∼0.001–1 s −1) and at room temperature and higher, the elastic modulus of the particles can be four orders of magnitude higher than that of the binder. Rigorous bounds and analytical estimates for the effective elastic properties of these materials have been found to be inaccurate. The computational expense of detailed numerical simulations for the determination of effective properties of these composites has led to the search for a faster technique. In this work, one such technique based on a real-space renormalization group approach is explored as an alternative to direct numerical simulations in determining the effective elastic properties of PBX 9501. The method is named the recursive cell method (RCM). The differential effective medium approximation, the finite element method, and the generalized method of cells (GMC) are investigated with regard to their suitability as homogenizers in the RCM. Results show that the RCM overestimates the effective properties of particulate composites and PBX 9501 unless large blocks of subcells are renormalized and the particles in a representative volume element are randomly distributed. The GMC homogenizer is found to provide better estimates of effective elastic properties than the finite element based homogenizer for composites with particle volume fractions less than 0.80.

Microwave conductivity of ad-wave superconductor disordered by extended impurities: A real-space renormalization group approach

Using a real-space renormalization group (RSRG) technique, we compute the microwave conductivity of a d-wave superconductor disordered by extended impurities. To do this, we invoke a semiclassical approximation which naturally accesses the Andreev bound states localized near each impurity. Tunneling corrections (which are captured using the RSRG) lead to a delocalization of these quasiparticles and an associated contribution to the microwave conductivity.

Real space renormalization group approach to spin-glass dynamics

The slow non-equilibrium dynamics of the Edwards-Anderson spin glass model on a hierarchical lattice is studied by means of a coarse-grained description based on renormalization concepts. We evaluate the isothermal aging properties and show how the occurrence of temperature chaos is connected to a gradual loss of memory when approaching the overlap length. This leads to rejuvenation effects in temperature shift protocols and to rejuvenation--memory effects in temperature cycling procedures with a pattern of behavior parallel to experimental observations.

Spin waves in random spin chains

We study quantum spin-1/2 Heisenberg ferromagnetic chains with dilute random antiferromagnetic impurity bonds with the modified spin-wave theory. By describing thermal excitations in the language of spin waves, we observe a low-temperature Curie susceptibility due to the formation of large spin clusters first predicted by the real-space renormalization-group approach, as well as a crossover to a pure ferromagnetic spin-chain behavior at intermediate and high temperatures. We compare our results of the modified spin-wave theory to quantum Monte Carlo simulations.

Diffusion of particles adsorbed on a triangular lattice: pairwise and three-particle interactions

In the present paper, the influence of both pairwise and three-particle interactions on the mobility of adsorbed particles diffusing on a lattice with triangular symmetry has been studied. Two different techniques has been used for describing the surface diffusion phenomenon. On one hand, explicit expressions for the chemical and jump diffusion coefficients have been calculated by using real-space renormalization group (RSRG) approach. A number of the RSRG transformations with blocks of different sizes and symmetries have been investigated. In particular, it has been shown that the precision of the method depends strongly not only on the number of sites in the RSRG blocks but also on their composition and structure. On the other hand, numerical simulations by using the Monte Carlo scheme has been used to simulate the process of particle migration. Using both methods, adsorption isotherms for different temperatures and the coverage dependencies for the thermodynamic factor and the chemical diffusion coefficient have been calculated. The behavior of the above mentioned quantities has been compared when the adparticles interact via only either pairwise or three-particle interactions. Despite the fact that both methods constitute very different approaches, the correspondence of numerical data with analytical results is surprisingly good. Therefore, it can be concluded that the RSRG method can be successfully applied for lattice gas systems to characterize the thermodynamic and kinetic properties of strongly interacting adsorbates.

Renormalization group for evolving networks.

We propose a renormalization group treatment of stochastically growing networks. As an example, we study percolation on growing scale-free networks in the framework of a real-space renormalization group approach. As a result, we find that the critical behavior of percolation on the growing networks differs from that in uncorrelated networks.

Renormalization group approach to the energy level statistics at the integer quantum Hall transition

We apply the real-space renormalization group (RG) approach to investigate the energy level statistics at the integer quantum Hall (QH) transition. Within the RG approach the macroscopic array of saddle points of the Chalker–Coddington network is replaced by a fragment consisting of only five saddle points. Previously, we have demonstrated that the RG approach reproduces the distribution of the conductance at the transition, P( G), with very high accuracy. To assess the level statistics we analyze the phases of the transmission coefficients of the saddle points. We find that, at the transition, the nearest-neighbor energy level spacing distribution (LSD) exhibits well-pronounced level repulsion. We emphasize that a metal-like LSD emerges when the fixed point distribution P c of G is used. Studying the change of the LSD around the QH transition we observe scaling behavior. Using a one-parameter finite-size scaling analysis we are able to extract a critical exponent ν=2.38±0.04 of the localization length.

Renormalization group approach to energy level statistics at the integer quantum Hall transition

We extend the real-space renormalization group (RG) approach to the study of the energy level statistics at the integer quantum Hall (QH) transition. Previously it was demonstrated that the RG approach reproduces the critical distribution of the {\em power} transmission coefficients, i.e., two-terminal conductances, $P_{\text c}(G)$, with very high accuracy. The RG flow of $P(G)$ at energies away from the transition yielded the value of the critical exponent, $\nu$, that agreed with most accurate large-size lattice simulations. To obtain the information about the level statistics from the RG approach, we analyze the evolution of the distribution of {\em phases} of the {\em amplitude} transmission coefficient upon a step of the RG transformation. From the fixed point of this transformation we extract the critical level spacing distribution (LSD). This distribution is close, but distinctively different from the earlier large-scale simulations. We find that away from the transition the LSD crosses over towards the Poisson distribution. Studying the change of the LSD around the QH transition, we check that it indeed obeys scaling behavior. This enables us to use the alternative approach to extracting the critical exponent, based on the LSD, and to find $\nu=2.37\pm0.02$ very close to the value established in the literature. This provides additional evidence for the surprising fact that a small RG unit, containing only five nodes, accurately captures most of the correlations responsible for the localization-delocalization transition.

Real-Space Renormalization Group Approach to the Quantum Hall Transition

The integer quantum Hall (QH) transition is described well in terms of a delocalization-localization transition of the electron wavefunction. In contrast to a usual metalinsulator transition (MIT), the QH transition is characterized by a single extended state located exactly at the center = 0 of each Landau band. When approaching = 0, the localization length ξ of the electron wavefunction diverges according to a power law −ν , where defines the distance to the MIT for a suitable control parameter, e.g., the electron energy. On the theoretical side, the value of ν has been extracted from various numerical simulations, e.g., ν = 2.5 ± 0.5, 2.4 ± 0.2, 2.35± 0.03, and 2.39± 0.01. In experiments ν ≈ 2.3 has been obtained, e.g., from the frequency or the sample size dependence of the critical behavior of the resistance in the transition region at strong magnetic field. We study the critical properties of the integer QH transition by employing the real-space renormalization-group (RG) approach to the Chalker-Coddington (CC) network model. We calculate the critical distribution Pc(G) of the conductance and the critical exponent ν of the QH transition for two different RG units. This allows to demonstrate that the quality of the results crucially depends on the choice of the RG unit. The CC model describes a single QH transition using a chiral network consisting of electron trajectories along equipotential lines (links) and saddle points (SP’s) of the potential (nodes). Each SP acts as a scatterer and relates the wavefunction amplitudes in two incoming and two outgoing channels. It can be characterized by a 2×2 S matrix, which depends only on the transmission and reflection coefficients ti and ri. The links correspond to random phases Φj and reflect the randomness of the potential disorder in a sample. We consider two previously studied, different RG units on a regular 2D square lattice as shown in Fig. 1. One is constructed from 4 SP’s, the other consists of 5 SP’s. The RG unit should be chosen in a way such that the essential properties of the network are taken into account. In the course of our RG approach an RG unit is then mapped onto a new single super-SP using the analytical dependence t′ = f ({ti, ri}, {Φj}) (1)

Effect of distortion on hopping conductivity:ThueMorse nanostructured model

A realspace renormalizationgroup approach is developed to calculate hopping conductivity of ThueMorse（ＴＭ） 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. From the effect of the degree of random on hopping conductivity, TM chain is a system between Fibonacci and periodic sequence.

Lévy flights in a quenched jump length field: a real space renormalization group approach

We study the d-dimensional walk of a particle in the presence of a quenched random field of jump lengths l( x ) drawn from an algebraic distribution p( l)∼ l −1− f with the Lévy exponent f. A real space renormalization procedure shows that the upper critical dimension is given by d c = f. Furthermore, the renormalization group approach indicates, that the dynamic z exponent characterizing the walk of a particle in such a field is changed from the annealed value z= f (usual Lévy flight) to z= f+2( f− d)/ d (quenched case). This change of the dynamical exponent z= f for an annealed Lévy walker into the exponent z> f for the same Lévy walker in a quenched jump length field is a remarkable result: an intrinsic anomalous diffusion is changed into another anomalous diffusion by an external disordered jump length field.

Real-space renormalization-group approach to field evolution equations.

An operator formalism for the reduction of degrees of freedom in the evolution of discrete partial differential equations (PDE) via real-space renormalization group is introduced, in which cell overlapping is the key concept. Applications to (1+1)-dimensional PDEs are presented for linear and quadratic equations that are first order in time.