Real-space Renormalization Research Articles

Entanglement entropy in random quantum spin-Schains

We discuss the scaling of entanglement entropy in the random singlet phase (RSP) of disordered quantum magnetic chains of general spin-S. Through an analysis of the general structure of the RSP, we show that the entanglement entropy scales logarithmically with the size of a block and we provide a closed expression for this scaling. This result is applicable for arbitrary quantum spin chains in the RSP, being dependent only on the magnitude S of the spin. Remarkably, the logarithmic scaling holds for the disordered chain even if the pure chain with no disorder does not exhibit conformal invariance, as is the case for Heisenberg integer spin chains. Our conclusions are supported by explicit evaluations of the entanglement entropy for random spin-1 and spin-3/2 chains using an asymptotically exact real-space renormalization group approach.

Wilson-Like Real-Space Renormalization Group and Low-Energy Effective Spectrum of the XXZ Chain in the Critical Regime

We present a novel real-space renormalization group(RG) for the one-dimensional XXZ model in the critical regime, reconsidering the role of the cut-off parameter in Wilson's RG for the Kondo impurity problem. We then demonstrate the RG calculation for the XXZ chain with the free boundary. Comparing the hierarchical structure of the obtained low-energy spectrum with the Bethe ansatz result, we find that the proper scaling dimension is reproduced as a fixed point of the RG transformation.

Surface diffusion of particles adsorbed on an inhomogeneous lattice with two non-equivalent sites

We investigate surface diffusion in a system of particles with the nearest neighbor pairwise lateral interaction adsorbed on a two-dimensional inhomogeneous lattice of square symmetry with deep and shallow sites. General analytical expressions for the chemical and jump diffusion coefficients have been derived in case of strong inhomogeneity. The expressions are valid for the inhomogeneous lattices with different geometries and dimensionalities. We have calculated the coverage dependencies of the tracer, jump and chemical diffusion coefficients for different temperatures using the real-space renormalization group (RSRG) method and compared the data with the numerical results obtained by the MC simulations. The coincidence between the data obtained by these quite different methods is rather good.

Symmetry adapted finite-cluster solver for quantum Heisenberg model in two dimensions: a real-space renormalization approach

We present a quantum cluster solver for the spin-S Heisenberg model on a two-dimensional lattice. The formalism is based on the real-space renormalization procedure and uses the lattice point group-theoretical analysis and non-Abelian SU(2) spin symmetry technique. The exact diagonalization procedure is used twice at each renormalization group step. The method is applied to the spin-half antiferromagnet on a square lattice, and a calculation of local observables is demonstrated. A symmetry-based truncation procedure is suggested and verified numerically.

Real-space renormalization group approach to driven diffusive systems

We introduce a real-space renormalisation group procedure for driven diffusive systems which predicts both steady state and dynamic properties. We apply the method to the boundary driven asymmetric simple exclusion process and recover exact results for the steady state phase diagram, as well as the crossovers in the relaxation dynamics for each phase.

Modeling Gaussian-to-self-avoiding-walk crossover for stiff chain polymers

We explore the long-standing discrepancy between an experimentally observed sudden onset of the excluded-volume effect in stiff chains [Murakami, Macromolecules 13, 345 (1980)] and much more gradual onset seen by Monte Carlo simulations of the persistent self-avoiding walk model. We find by using various simulations and real-space renormalization methods that, while models incorporating parallel-bond inhibition enhance the excluded-volume effect, they have little effect on the abruptness of its onset. We also observe that relaxational simulations, which include reptationlike motion can, however, lead to a sudden onset much like in the experimental observation.

Ferromagnetic Potts model under an external magnetic field: An exact renormalization group approach

The q-state ferromagnetic Potts model under a non-zero magnetic field coupled with the 0^th Potts state was investigated by an exact real-space renormalization group approach. The model was defined on a family of diamond hierarchical lattices of several fractal dimensions d_F. On these lattices, the renormalization group transformations became exact for such a model when a correlation coupling that singles out the 0^th Potts state was included in the Hamiltonian. The rich criticality presented by the model with q=3 and d_F=2 was fully analyzed. Apart from the Potts criticality for the zero field, an Ising-like phase transition was found whenever the system was submitted to a strong reverse magnetic field. Unusual characteristics such as cusps and dimensional reduction were observed on the critical surface.

Eliminated corrections to scaling around a renormalization-group fixed point: Transfer-matrix simulation of an extendedd=3Ising model

Extending the parameter space of the three-dimensional (d=3) Ising model, we search for a regime of eliminated corrections to finite-size scaling. For that purpose, we consider a real-space renormalization group (RSRG) with respect to a couple of clusters simulated with the transfer-matrix (TM) method. Imposing a criterion of "scale invariance," we determine a location of the nontrivial RSRG fixed point. Subsequent large-scale TM simulation around the fixed point reveals eliminated corrections to finite-size scaling. As anticipated, such an elimination of corrections admits systematic finite-size-scaling analysis. We obtained the estimates for the critical indices as nu=0.6245(28) and y(h)=2.4709(73). As demonstrated, with the aid of the preliminary RSRG survey, the transfer-matrix simulation provides rather reliable information on criticality even for d=3, where the tractable system size is restricted severely.

Chapter 10 Magnetic properties of low-dimensional organic crystals of the PNNNO family

We investigate the “local structure” of the ground state of F2PNNNO (two-dimensional organic antiferromagnet of a spin S = 1) by means of real-space renormalization group (RSRG) and density-matrix renormalization group (DMRG) techniques. The results show that the system is close to a spin liquid with a singlet ground state.

Atom diffusion in a simple cubic lattice: Real-space renormalization group and Monte Carlo calculations

A three-dimensional simple cubic lattice gas is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. We have calculated absorption isotherms and density dependencies of the pair correlation function, isothermal susceptibility, chemical and jump diffusion coefficients for different temperatures. All these quantities have also been obtained by Monte Carlo (MC) simulations. We have also obtained the density dependencies for the tracer diffusion coefficient by MC simulations. Despite the fact that both methods constitute very different approaches, the correspondence of numerical data and theoretical results is rather good. We conclude that the RSRG method can be applied to the three-dimensional lattice gas system discussed here to characterize its thermodynamic and kinetic properties.

A real-space renormalization approach to the Kubo–Greenwood formula in mirror Fibonacci systems

An exact real-space renormalization method is developed to address the electronic transport in mirror Fibonacci chains at a macroscopic scale by means of the Kubo-Greenwood formula. The results show that the mirror symmetry induces a large number of transparent states in the dc conductivity spectra, contrary to the simple Fibonacci case. A length scaling analysis over ten orders of magnitude reveals the existence of critically localized states and their ac conduction spectra show a highly oscillating behaviour. For multidimensional quasiperiodic systems, a novel renormalization plus convolution method is proposed. This combined renormalization + convolution method has shown an extremely elevated computing efficiency, being able to calculate electrical conductance of a three-dimensional non-crystalline solid with 10 30 atoms. Finally, the dc and ac conductances of mirror Fibonacci nanowires are also investigated, where a quantized dc-conductance variation with the Fermi energy is found, as observed in gold nanowires.

Continuous phase transition of a fully frustratedXYmodel in three dimensions

We have used Monte Carlo simulations, combined with finite-size scaling and two different real-space renormalization group approaches, to study a fully frustrated three-dimensional $XY$ model on a simple cubic lattice. This model corresponds to a lattice of Josephson-coupled superconducting grains in an applied magnetic field $\mathbf{H}=({\ensuremath{\Phi}}_{0}∕{a}^{2})(1∕2,1∕2,1∕2)$. We find that the model has a continuous phase transition with critical temperature ${T}_{c}=0.681\phantom{\rule{0.3em}{0ex}}J∕{k}_{B}$, where $J$ is the $XY$ coupling constant, and critical exponents $\ensuremath{\alpha}∕\ensuremath{\nu}=0.87\ifmmode\pm\else\textpm\fi{}0.01$, $v∕\ensuremath{\nu}=0.82\ifmmode\pm\else\textpm\fi{}0.01$, and $\ensuremath{\nu}=0.72\ifmmode\pm\else\textpm\fi{}0.07$, where $\ensuremath{\alpha}$, $v$, and $\ensuremath{\nu}$ describe the critical behavior of the specific heat, helicity modulus, and correlation length. We briefly compare our results with other studies of this model, and with a mean-field approximation.

Ordering near the percolation threshold in models of two-dimensional interacting bosons with quenched dilution

Randomly diluted quantum boson and spin models in two dimensions combine the physics of classical percolation with the well-known dimensionality dependence of ordering in quantum lattice models. This combination is rather subtle for models that order in two dimensions but have no true order in one dimension, as the percolation cluster near threshold is a fractal of dimension between 1 and 2: two experimentally relevant examples are the O(2) quantum rotor and the Heisenberg antiferromagnet. We study two analytic descriptions of the O(2) quantum rotor near the percolation threshold. First a spin-wave expansion is shown to predict long-ranged order, but there are statistically rare points on the cluster that violate the standard assumptions of spin-wave theory. A real-space renormalization group (RSRG) approach is then used to understand how these rare points modify ordering of the O(2) rotor. A new class of fixed points of the RSRG equations for disordered 1D bosons is identified and shown to support the existence of long-range order on the percolation backbone in two dimensions. These results are relevant to experiments on bosons in optical lattices and superconducting arrays, and also (qualitatively) for the diluted Heisenberg antiferromagnet.

Two-stage coarsening mechanism in a kinetically constrained model of an attractive colloid

We study an attractive version of the East model using the real-space renormalization group (RG) introduced by Stella et al. The former is a kinetically constrained model with an Ising-like interaction between excitations and shows striking agreement with the phenomonology of attractive colloidal systems. We find that the RG predicts two nonuniversal dynamic exponents, which suggests that in the out-of-equilibrium regime the model coarsens via a two-stage mechanism. We explain this mechanism physically and verify this prediction numerically. In addition, we predict that the characteristic relaxation time of the model is a nonmonotonic function of attraction strength, again in agreement with numerical results.

Real-Space Renormalization Group Approach for the Corner Hamiltonian

We present a real-space renormalization group approach for the corner Hamiltonian, which is relevant to the reduced density matrix in the density matrix renormalization group. A set of self-consistent equations that the renormalized Hamiltonian should satisfy in the thermodynamic limit is also derived from the fixed point of the recursion relation for the corner Hamiltonian. We demonstrate the renormalization group algorithm for the $S=1/2$ XXZ spin chain and show that the results are consistent with the exact solution. We further examine the renormalization group for the S=1 Heisenberg spin chain and then discuss the nature of the eigenvalue spectrum of the corner Hamiltonian for the non-integrable model.

Long-range interaction between Stone-Wales defects in zigzag single-walled carbon nanotubes

Electronic properties of ($n$,0) single-walled carbon nanotubes with Stone-Wales (SW) defects are investigated by use of the tight-binding Green function method with real-space renormalization techniques. It is found that the interaction between two SW defects is long-range, with a characteristic length of several nanometers. The obtained characteristic length, reflecting the effective functional range of topologic defects, coincides with recent observations on defects in carbon nanotubes via scanning tunneling spectroscopy. By means of a monatomic chain model, the long-range interaction is revealed to be general in one-dimensional defected semiconducting systems, which can be equivallently described by a low-barrier quantum tunneling.

Dynamics of weakly coupled random antiferromagnetic quantum spin chains

We study the low-energy collective excitations and dynamical response functions of weakly coupled random antiferromagnetic spin-1/2 chains. The interchain coupling leads to Neel order at low temperatures. We use the real-space renormalization group technique to tackle the intrachain couplings and treat the interchain couplings within the Random Phase Approximation (RPA). We show that the system supports collective spin wave excitations, and calculate the spin wave velocity and spectra weight within RPA. Comparisons will be made with inelastic neutron scattering experiments quasi-one-dimensional disordered spin systems such as doped CuGeO$_3$

Dynamics of end-linked star-polymer structures

In this work we focus on the dynamics of macromolecular networks formed by end-linking identical polymer stars. The resulting macromolecular network can then be viewed as consisting of spacers which connect branching points (the cores of the stars). We succeed in analyzing exactly, in the framework of the generalized Gaussian model, the eigenvalue spectrum of such networks. As applications we focus on several topologies, such as regular networks and dendrimers; furthermore, we compare the results to those found for regular hyperbranched structures. In so doing, we also consider situations in which the beads of the cores differ from the beads of the spacers. The analytical procedure which we use involves an exact real-space renormalization, which allows to relate the star network to a (much simpler) network, in which each star is reduced to its core. It turns out that the eigenvalue spectrum of the star-polymer structure consists of two parts: one follows in terms of polynomial equations from the relaxation spectrum of the corresponding renormalized structure, while the second part involves the motion of the spacer chains themselves. Finally, we show exemplarily the situation for copolymeric dendrimers, calculate their spectra, and from them their storage and the loss moduli.

Renormalization-Group Transformations and Correlations of Seismicity

The effect of transformations analogous to those of the real-space renormalization group are analyzed for the temporal occurrence of earthquakes. A recently reported scaling law for the distribution of recurrence times implies that these distributions must be invariant under such transformations, for which the role of the correlations between the magnitudes and the recurrence times are fundamental. This approach puts the study of the temporal structure of seismicity in the context of critical phenomena.

REAL-SPACE RENORMALIZATION-GROUP APPROACH TO THE INTEGER QUANTUM HALL EFFECT

We review recent results based on an application of the real-space renormalization group (RG) approach to a network model for the integer quantum Hall (QH) transition. We demonstrate that this RG approach reproduces the critical distribution of the power transmission coefficients, i.e., two-terminal conductances, Pc(G), with very high accuracy. The RG flow of P(G) at energies away from the transition yields a value of the critical exponent ν that agrees with most accurate large-size lattice simulations. A description of how to obtain other relevant transport coefficients such as RLand RHis given. From the non-trivial fixed point of the RG flow we extract the critical level-spacing distribution (LSD). This distribution is close, but distinctively different from the earlier large-scale simulations. We find that the LSD obeys scaling behavior around the QH transition with ν = 2.37±0.02. Away from the transition it crosses over towards the Poisson distribution. We next investigate the plateau-to-insulator transition at strong magnetic fields. For a fully quantum coherent situation, we find a quantized Hall insulator with RH≈h/e2up to RL~20h/e2when interpreting the results in terms of most probable value of the distribution function P(RH). Upon further increasing RL→∞, the Hall insulator with diverging Hall resistance [Formula: see text] is seen. The crossover between these two regimes depends on the precise nature of the averaging procedure for the distributions P(RL) and P(RH). We also study the effect of long-ranged inhomogeneities on the critical properties of the QH transition. Inhomogeneities are modeled by a smooth random potential with a correlator which falls off with distance as a power law r-α. Similar to the classical percolation, we observe an enhancement of ν with decreasing α. These results exemplify the surprising fact that a small RG unit, containing only five nodes, accurately captures most of the correlations responsible for the localization-delocalization transition.