Real-space Renormalization Group Method Research Articles

Real-space renormalisation approach to the Chalker–Coddington model revisited: Improved statistics

The real-space renormalisation group method can be applied to the Chalker–Coddington model of the quantum Hall transition to provide a convenient numerical estimation of the localisation critical exponent, ν. Previous such studies found ν∼2.39 which falls considerably short of the current best estimates by transfer matrix (ν=2.593+0.005−0.006) and exact-diagonalisation studies (ν=2.58(3)). By increasing the amount of data 500 fold we can now measure closer to the critical point and find an improved estimate ν=2.51+0.11−0.11. This deviates only ∼3% from the previous two values and is already better than the ∼7% accuracy of the classical small-cell renormalisation approach from which our method is adapted. We also study a previously proposed mixing of the Chalker–Coddington model with a classical scattering model which is meant to provide a route to understanding why experimental estimates give a lower ν∼2.3. Upon implementing this mixing into our RG unit, we find only further increases to the value of ν.

Real-space renormalization-group treatment of quadratic chains

We have recently proposed a one-dimensional nonperiodic chain with lattice positions at 02 d, 12 d, 22 d, ... with length d a constant. The spectrum is singular-continuous, and for weak potential, the states are all extended apart from a trivial set of localized states. In this study, we obtain the exact extended-state spectrum of the quadratic chain in a nearest-neighbor tight-binding model where the quadratic modulation is in the onsite matrix elements. Then, a real-space renormalization-group method (RSRG) is used by decimation to reduce the transfer matrix for the chain into self-similar matrix products. The RSRG decimation scheme is used here to organize the calculation and facilitate numerical computation. The extended-state spectrum appears as minibands broken by numerous gaps. Previous work on quadratic chains shows that the structure factor is singular-continuous and given by a dense set of states with wavevectors with scaling exponent γ(k) = 2 as in periodic and quasi-periodic chains. The origin of extended states in this nonperiodic lattice appears to arise from a type of mechanism not yet identified in deterministic nonperiodic lattices, and is based on a hidden symmetry giving rise to an energy-dependent translational invariance of the transfer matrix.

Tunable caging of excitation in decorated Lieb-ladder geometry with long-range connectivity

Controlled Aharonov–Bohm caging of wave train is reported in a quasi-one-dimensional version of Lieb geometry with next-nearest-neighbor hopping integral within the tight-binding framework. This longer-wavelength fluctuation is considered by incorporating periodic, quasi-periodic or fractal kind of geometry inside the skeleton of the original network. This invites exotic eigenspectrum displaying a distribution of flat band states. Also a subtle modulation of external magnetic flux leads to a comprehensive control over those non-resonant modes. Real space renormalization group method provides us an exact analytical prescription for the study of such tunable imprisonment of excitation. The non-trivial tunability of external agent is important as well as challenging in the context of experimental perspective.

Quantum information spreading in random spin chains

We study the spreading of quantum correlations and information in a one-dimensional quantum spin chain with critical disorder as encoded in an infinite randomness fixed point. Specifically, we focus on the dynamics after a quantum quench of the R\'enyi entropies, of the mutual information and of the entanglement negativity in the prototypical XXZ spin chain with random bonds and anisotropy parameters. We provide analytic predictions in the scaling regime based on real-space renormalization group methods. We support these findings through numerical simulations in the non-interacting limit, where we can access the scaling regime.

Critical properties of the Ising ferromagnet VI[formula omitted] monolayer

Motivated by the experimental observation of two-dimensional van der Waals ferromagnets, we investigated the critical properties of Ising ferromagnet VI3 monolayer using the real-space renormalization group method. It is worth noting that a block-spin construction we choose contains an even number of sites. Thereupon, we present a scheme that combines the majority rule and central rule to determine the block-spin values. The critical temperature TC is calculated to be 46.8 K under the first-order cumulant-expansion approximation, which is in excellent agreement with the recent numerical result of TC = 46 K using the Metropolis Monte Carlo simulations based on the classical Heisenberg model, and is slightly lower than the experimental value TC = 50 K of the corresponding bulk VI3. Furthermore, the critical exponents β and ν are estimated, and compared with the mean-field results and the exact solutions, respectively. Our numerical result β≅0.324 suggests that the magnetic phase transition of the VI3 monolayer appears to lie between the two-dimensional Ising universality class (β=0.125) and three-dimensional Heisenberg universality class (β=0.365). The theoretical results in this paper may stimulate further experimental studies of the VI3 monolayer.

Phase diagrams, quantum correlations and critical phenomena of antiferromagnetic Heisenberg model on diamond-type hierarchical lattices

The spin-1/2 antiferromagnetic (AF) Heisenberg systems are studied on three typical diamond-type hierarchical lattices (systems A, B and C) with fractal dimensions 1.63, 2 and 2.58, respectively, and the phase diagrams, critical phenomena and quantum correlations are calculated by a combination of the equivalent transformation and real-space renormalization group methods. We find that there exist a reentrant behavior for system A and a finite temperature phase transition in the isotropic Heisenberg limit for system C (not for system B). Unlike the ferromagnetic case, the Néel temperatures of AF systems A and B are inversely proportional to (when Δ → Δc) and ln Δ (when Δ → 0), respectively. And we also find that there is a turning point of quantum correlation in the isotropic Heisenberg limit Δ = 0 where there is a ‘peak’ of the contour and no matter how large the size of system is, quantum correlation will change to zero in the Ising limit for the three systems. The quantum correlation decreases with the increase of lattice size L and it is almost zero when L ⩾ 30 for system A, and for systems B and C, they still exist when L is larger than that of system A. Moreover, we discuss the effects of quantum fluctuation and analyze the errors of results in the above three systems, which are induced by the noncommutativity.

Perturbative instability of nonergodic phases in non-Abelian quantum chains

An important challenge in the field of many-body quantum dynamics is to identify nonergodic states of matter beyond many-body localization (MBL). Strongly disordered spin chains with non-Abelian symmetry and chains of non-Abelian anyons are natural candidates, as they are incompatible with standard MBL. In such chains, real space renormalization group methods predict a partially localized, nonergodic regime known as a quantum critical glass (a critical variant of MBL). This regime features a treelike hierarchy of integrals of motion and symmetric eigenstates with entanglement entropy that scales as a logarithmically enhanced area law. We argue that such tentative nonergodic states are perturbatively unstable using an analytic computation of the scaling of off-diagonal matrix elements and accessible level spacing of local perturbations. Our results indicate that strongly disordered chains with non-Abelian symmetry display either spontaneous symmetry breaking or ergodic thermal behavior at long times. We identify the relevant length and timescales for thermalization: Even if such chains eventually thermalize, they can exhibit nonergodic dynamics up to parametrically long timescales with a nonanalytic dependence on disorder strength.

Time and place of the maximum for one-dimensional diffusion bridges and meanders

For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and generalized Bessel meanders. We discuss the limits of the application of this method for other diffusion processes.

A real-space renormalization-group calculation for the quantum gauge theory on a square lattice

We revisit Fradkin and Raby’s real-space renormalization-group method to study the quantum gauge theory defined on links forming a two-dimensional square lattice. Following an old suggestion of theirs, a systematic perturbation expansion developed by Hirsch and Mazenko is used to improve the algorithm to second order in an intercell coupling, thereby incorporating the effects of discarded higher energy states. A careful derivation of gauge-invariant effective operators is presented in the Hamiltonian formalism. Renormalization group equations are analyzed near the nontrivial fixed point, reaffirming old work by Hirsch on the dual transverse field Ising model. In addition to recovering Hirsch’s previous findings, critical exponents for the scaling of the spatial correlation length and energy gap in the electric free (deconfined) phase are compared. Unfortunately, their agreement is poor. The leading singular behavior of the ground state energy density is examined near the critical point: we compute both a critical exponent and estimate a critical amplitude ratio.

Real-space renormalization-group methods for hierarchical spin glasses

We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: a sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method (Angelini M C, Parisi G and Ricci-Tersenghi F 2013 Ensemble renormalization group for disordered systems Phys. Rev. B 87 134201) in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical exponent related to the divergence of the correlation length, while it does not predict the correct qualitative behavior of below the upper critical dimension. On the other hand, the ERG procedure has been claimed to predict the correct behavior of both above and below the upper critical dimension. Here, we straighten out the reasons for the discrepancy between the two methods above, by demonstrating that the ERG method predicts a marginally stable critical fixed point, thus implying a prediction for the critical exponent given by . This prediction disagrees, on a qualitative and quantitative level, both with the mean-field value of above the critical dimension, and with numerical estimates of below the upper critical dimension. Therefore, our results show that finding a real-space RG method for spin glasses which yields the correct prediction for universal quantities below the upper critical dimension is still an open problem, for which our analysis may provide some general guidance for future studies.

Controlled trapping of single particle states on a periodic substrate by deterministic stubbing

Abstract A periodic array of atomic sites, described within a tight binding formalism is shown to be capable of trapping electronic states as it grows in size and gets stubbed by an ‘atom’ or an ‘atomic’ clusters from a side in a deterministic way. We prescribe a method based on a real space renormalization group method, that unravels a subtle correlation between the positions of the side coupled atoms and the energy eigenvalues for which the incoming particle finally gets trapped. We discuss how, in such conditions, the periodic backbone gets transformed into an array of infinite quantum wells in the thermodynamic limit. We present a case here, where the wells have a hierarchically distribution of widths, hosting standing wave solutions in the thermodynamic limit.

Entanglement entropy distribution in the strongly disordered one-dimensional Anderson mode

The entanglement entropy (EE) distribution of strongly disordered one dimensional spin chains, which are equivalent to spinless fermions at half-filling on a bond (hopping) disordered one-dimensional Anderson model, has been shown to exhibit very distinct features such as peaks at integer multiplications of , essentially counting the number of singlets traversing the boundary. Here we show that for a canonical Anderson model with box distribution on-site disorder and repulsive nearest-neighbor interactions the EE distribution also exhibits interesting features, albeit different than the distribution seen for the bond disordered Anderson model. The canonical Anderson model shows a broad peak at low entanglement values and one narrower peak at . Density matrix renormalization group calculations reveal this structure and the influence of the disorder strength and the interaction strength on its shape. A modified real space renormalization group method was used to get a better understanding of this behavior. As might be expected the peak centered at low values of EE has a tendency to shift to lower values as disorder is enhanced. A second peak appears around the EE value of , this peak is broadened and no additional peaks at higher integer multiplications of are seen. We attribute the differences in the distribution between the canonical model and the broad hopping disorder to the influence of the on-site disorder which breaks the symmetry across the boundary.

Density of states of two dimensional quasiperiodic lattices in the thermodynamic limit

A scheme based on real-space renormalization group method to find density of states of two dimensional lattices in the thermodynamic limit is introduced in this paper. The main importance of this method is its applicability in two dimensional quasiperiodic systems for which no analytic solution is possible. Previously there were many works with real-space renormalization scheme in one dimensional lattices, but its application in two dimensional systems was not successful. In the present scheme, we have mapped the two dimensional system into a number of one dimensional systems, and applying real-space renormalization method to those new one dimensional systems we have found out the density of states of the entire two dimensional system.

Spectral engineering and tunable thermoelectric behavior in a quasiperiodic ladder network

Double-stranded quasiperiodic copper mean arrangement has been studied in respect of its electronic property and thermoelectric signature. The two-arm network is demonstrated by a tight-binding Hamiltonian. The eigenspectrum of such aperiodic mesh that does not convey translational invariance, is significantly dependent on the parameters of the Hamiltonian. It is observed that specific correlation between the parameters obtained from the commutation relation between the on-site energy and overlap integral matrices can eventually modify the spectral nature and generate absolutely continuous energy spectrum. This part is populated by atypical extended states that has a large localization length substantiated by the flow of the hopping integral under successive real space renormalization group method steps. This sounds delocalization of single particle energy states in such non-translationally invariant networks. Further this can be engineered at will by selective choice of the relative strengths of the parameters. This precise correlation has a crucial impact on the thermoelectric behavior. Anomalous nature of thermoelectric coefficient may inspire the experimentalists to frame tunable thermo-devices. Specific correlations can help us to tune the continuous band and determine the band position at will.

FORCE-INDUCED DESORPTION OF SEMI-FLEXIBLE POLYMERS IN NON-HOMOGENEOUS MEDIA

Force-induced desorption of semi-flexible linear polymers in good solvents and non-homogeneous media is studied. We apply the model of self-avoiding walk with bending rigidity on 3-simplex fractal lattice. One boundary of the lattice represents an adsorbing wall, while an external force (perpendicular to the wall) pulls the polymer from the wall. Due to the hierarchical structure of the lattice under study, the phase diagram of polymer critical behavior can be exactly studied by real-space renormalization group method. Temperature dependence of the critical force, at which desorption occurs, is obtained and discussed for various values of the bending rigidity. Influence of the force and the polymer rigidity on the nature of desorption phase transition is also analyzed. All obtained results are compared with the related previous ones.

Healing of defects in random antiferromagnetic spin chains

We study the effects of a weakened link in random antiferromagnetic spin chains. We show that healing occurs, and that homogeneity is restored at low energy, in a way that is qualitatively similar to the fate of impurities in clean ferromagnetic spin chains, or in Luttinger liquids with attractive interactions. Healing in the random case occurs even without interactions, and is characteristic of the random singlet phase. Using real-space renormalization group and exact diagonalization methods, we characterize this universal healing crossover by studying the entanglement across the weak link. We identify a crossover healing length that separates a regime where the system is cut in half by the weak link from a fixed point where the spin chain is healed. Our results open the way to the study of impurity physics in disordered spin chains.

Renormalization of trace distance and multipartite entanglement close to the quantum phase transitions of one- and two-dimensional spin-chain systems

We investigate the quantum phase transitions of spin systems in one and two dimensions by employing trace distance and multipartite entanglement along with the real-space quantum renormalization group method. As illustration examples, a one-dimensional and a two-dimensional XY models are considered. It is shown that the quantum phase transitions of these spin-chain systems can be revealed by the singular behaviors of the first derivatives of renormalized trace distance and multipartite entanglement in the thermodynamics limit. Moreover, we find that the renormalized trace distance and multipartite entanglement obey certain universal exponential-type scaling laws in the vicinity of the quantum critical points.

Multiscale Monte Carlo equilibration: Pure Yang-Mills theory

We present a multiscale thermalization algorithm for lattice gauge theory, which enables efficient parallel generation of uncorrelated gauge field configurations. The algorithm combines standard Monte Carlo techniques with ideas drawn from real space renormalization group and multigrid methods. We demonstrate the viability of the algorithm for pure Yang-Mills gauge theory for both heat bath and hybrid Monte Carlo evolution, and show that it ameliorates the problem of topological freezing up to controllable lattice spacing artifacts.

Entanglement spectrum and entangled modes of highly excited states in randomXXspin chains

We examine the real space renormalization group method of finding \textit{excited eigenstate} (RSRG-X) of the XX spin-1/2 chain, from entanglement perspectives. Eigenmodes of entanglement Hamiltonian, especially the maximally entangled mode and corresponding entanglement energies are studied and compared with predictions of RSRG-X. Our numerical results demonstrate the accuracy of the RSRG-X method in the strong disorder limit, and quantify its error when applied to weak disorder regime. Overall, our results validate the RSRG-X method qualitatively, but also show that its accuracy decreases with increasing temperature.

Semiflexible crossing-avoiding trails on plane-filling fractals

We have studied the statistics of semiflexible polymer chains modeled by crossing-avoiding trails (CAT) situated on the family of plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension df is equal to 2 for all members of the fractal family. By applying the exact and Monte Carlo real-space renormalization group method we have calculated the critical exponent ν, which governs the scaling behavior of the end-to-end distance of the polymer, as well as the entropic critical exponent γ, for a large set of fractals, and various values of polymer flexibility. Our results, obtained for CAT model on PF fractals, show that both critical exponents depend on the polymer flexibility, in such a way that less flexible polymer chains display enlarged values of ν, and diminished values of γ. We have compared the obtained results for CAT model with the known results for the self-avoiding walk and self-avoiding trail models and discussed the influence of excluded volume effect on the values of semiflexible polymer critical exponents, for a large set of studied compact fractals.