Real Space Renormalization Group Scheme Research Articles

8-16-4 graphyne: Square-lattice two-dimensional nodal line semimetal with a nontrivial topological Zak index

An unprecedented graphyne allotrope with square symmetry and nodal line semimetallic behavior has been proposed in the two-dimensional (2D) realm. The emergence of the Dirac loop around the high-symmetry points in the presence of both the inversion and time-reversal symmetries is a predominant feature of the electronic band structure of this system. Besides, the structural stability in terms of the dynamic, thermal, and mechanical properties has been critically established for the system. Following the exact analytical model based on the real-space renormalization group scheme and tight-binding approach, we have inferred that the family of 2D nodal line semimetals with square symmetry can be reduced to a universal four-level system in the low-energy limit. This renormalized lattice indeed explains the underlying mechanism responsible for the fascinating emergence of 2D square nodal line semimetals. Besides, the analytical form of the generic dispersion relation of these systems is well supported by our density-functional theory results. Finally, the nontrivial topological properties have been explored for the predicted system without breaking the inversion and time-reversal symmetry of the lattice. We have obtained that the edge states are protected by the nonvanishing topological index, i.e., Zak phase.

Nature of protected zero-energy states in Penrose quasicrystals

The electronic spectrum of the Penrose rhombus quasicrystal exhibits a macroscopic fraction of exactly degenerate zero energy states. In contrast to other bipartite quasicrystals, such as the kite-and-dart one, these zero energy states cannot be attributed to a global mismatch $\Delta n$ between the number of sites in the two sublattices that form the quasicrystal. Here, we argue that these zero energy states are instead related to a local mismatch $\Delta n(\bf r)$. Although $\Delta n(\bf r)$ averages to zero, its staggered average over self-organized domains gives the correct number of zero energy states. Physically, the local mismatch is related to a hidden structure of nested self-similar domains that support the zero energy states. This allows us to develop a real space renormalization-group scheme, which yields the scaling law for the fraction of zero energy states, $Z$, versus size of their support domain, $N$, as $Z\propto N^{-\eta}$ with $\eta =1-\ln 2/\ln(1+\tau) \approx 0.2798$ (where $\tau$ is the golden ratio). It also reproduces the known total fraction of the zero energy states, $81-50\tau\approx 0.0983$. We also show that the exact degeneracy of these states is protected against a wide variety of local perturbations, such as irregular or random hopping amplitudes, magnetic field, random dilution of the lattice, etc. We attribute this robustness to the hidden domain structure and speculate about its underlying topological origin.

Critical properties of the ground-state localization-delocalization transition in the many-particle Aubry-André model

As opposed to random disorder, which localizes single-particle wave-functions in 1D at arbitrarily small disorder strengths, there is a localization-delocalization transition for quasi-periodic disorder in the 1D Aubry-Andr\'e model at a finite disorder strength. On the single-particle level, many properties of the ground-state critical behavior have been revealed by applying a real-space renormalization-group scheme; the critical properties are determined solely by the continued fraction expansion of the incommensurate frequency of the disorder. Here, we investigate the many-particle localization-delocalization transition in the Aubry-Andr\'e model with and without interactions. In contrast to the single-particle case, we find that the critical exponents depend on a Diophantine equation relating the incommensurate frequency of the disorder and the filling fraction which generalizes the dependence, in the single-particle spectrum, on the continued fraction expansion of the incommensurate frequency. This equation can be viewed as a generalization of the resonance condition in the commensurate case. When interactions are included, numerical evidence suggests that interactions may be irrelevant at at least some of these critical points, meaning that the critical exponent relations obtained from the Diophantine equation may actually survive in the interacting case.

Flux-driven and geometry-controlled spin filtering for arbitrary spins in aperiodic quantum networks

We demonstrate that an aperiodic array of certain quantum networks comprising magnetic and non-magnetic atoms can act as perfect spin filters for particles with arbitrary spin state. This can be achieved by introducing minimal quasi-one dimensionality in the basic structural units building up the array, along with an appropriate tuning of the potential of the non-magnetic atoms, the tunnel hopping integral between the non-magnetic atoms and the backbone, and, in some cases, by tuning an external magnetic field. This latter result opens up the interesting possibility of designing a flux controlled spin demultiplexer using quantum networks. The proposed networks have close resemblance with a family of recently developed photonic lattices, and the scheme for spin filtering can thus be linked, in principle, to a possibility of suppressing any one of the two states of polarization of a single photon, almost at will. We use transfer matrices and a real space renormalization group scheme to unravel the conditions under which any aperiodic arrangement of such topologically different structures will filter out any given spin projection. Our results are analytically exact, and corroborated by extensive numerical calculations of the spin polarized transmission and the density of states of such systems.

Strange attractor in the Potts spin glass on hierarchical lattices

The spin-glass q-state Potts model on d-dimensional diamond hierarchical lattices is investigated by an exact real space renormalization group scheme. Above a critical dimension dl(q) for q>2, the coupling constants probability distribution flows to a low-temperature strange attractor or to the high-temperature paramagnetic fixed point, according to the temperature is below or above the critical temperature Tc(q,d). The strange attractor was investigated considering four initial different distributions for q=3 and d=5 presenting strong robustness in shape and temperature interval suggesting a condensed phase with algebraic decay.

Random walks in a random environment on a strip: a renormalization group approach

We present a real space renormalization group scheme for the problem of random walks in a random environment on a strip, which includes the one-dimensional random walk in a random environment with bounded non-nearest-neighbour jumps. We show that the model renormalizes to an effective one-dimensional random walk with nearest-neighbour jumps and conclude that Sinai scaling is valid in the recurrent case, while in the sub-linear transient phase, the displacement grows as a power of the time.

Critical properties of the model on diamond-type hierarchial lattices

We study the critical properties of the S-4 model on diamond-type hierarchical lattices in the presence of an external magnetic field. It is assumed that for this type of inhomogenous fractal lattice, the Gaussian distribution constant, the four-spins interaction parameter and the external magnetic field in the S4 model depend on the coordination number of the site on the fractal lattices. By combining the real-space renormalization-group scheme with the cumulative expansion method, we obtain the critical points and further calculate critical exponents according to the scaling theory. The results show that on diamond-type hierarchical lattices with branches m > 4 of 2 bonds, the critical point of the S4 model is just the Gaussian fixed point, and therefore critical exponents are in full agreement with those of the Gaussian model, and that on those with m <= 4, the Wilson-Fisher fixed point as well as the Gaussian fixed point is obtained. The Wilson-Fisher fixed point has a decisive influence on the critical behavior of the system, and critical exponents are related to the fractal dimensionality. We found that the S4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality. (c) 2005 Elsevier B.V. All rights reserved.

Electronic transport in a Cantor stub waveguide network

We theoretically investigate the character of electronic eigenstates and transmission properties of a one-dimensional array of stubs with Cantor geometry. Within the framework of real space renormalization group (RSRG) and transfer matrix methods we analyze the resonant transmission and extended wave functions in a Cantor array of stubs, which lack translational order. Apart from resonant states with high transmittance we unravel a whole family of wave functions supported by such an array clamped between two-infinite ordered leads, which have an extended character in the RSRG scheme but, for such states the transmission coefficient across the lead-sample-lead structure decays following a power law as the system grows in size. This feature is explained from renormalization group ideas and may lead to the possibility of trapping of electronic, optical, or acoustic waves in such hierarchical geometries.

Phase diagrams and universality classes of random antiferromagnetic spin ladders

The random antiferromagnetic two-leg and zigzag spin-1/2 ladders are investigated using the real space renormalization group scheme and their complete phase diagrams are determined. We demonstrate that the first system belongs to the same universality class of the dimerized random spin-1/2 chain. The zigzag ladder, on the other hand, is in a random singlet phase at weak frustration and disorder. Otherwise, we give additional evidence that it belongs to the universality class of the random antiferromagnetic and ferromagnetic quantum spin chains, although the universal fixed point found in the latter system is never realized. We find, however, a new universal fixed point at intermediate disorder.

Electronic transport properties of Sierpinski lattices in a magnetic field

We have studied the electronic transport properties of open Sierpinski gasket systems in the presence of a magnetic field. The real-space renormalization-group scheme combined with the generalized eigenfunction method is used to calculate the transmission and reflection coefficients. Three kinds of electronic exits are investigated upto the thirtieth generation Sierpinski lattice. Some resonant-transmission features and the symmetry of the transmission coefficient T to the magnetic flux Φ are observed and discussed.

Rare-event induced binding transition of heteropolymers.

Sequence heterogeneity broadens the binding transition of a polymer onto a linear or planar substrate. This effect is analyzed in a real-space renormalization group scheme designed to capture the statistics of rare events. In the strongly disordered regime, binding initiates at an exponentially rare set of "good contacts." Renormalization of the contact potential yields a Kosterlitz--Thouless-type transition in any dimension. This and other predictions are confirmed by extensive numerical simulations of a directed polymer interacting with a columnar defect.

Vibrational properties of a general aperiodic Thue-Morse lattice: Role of the pseudoinvariant of the trace map

Using the real-space renormalization group (RSRG) scheme of Ghosh and Karmakar [Phys. Rev. B 58, 2586 (1998)], we analytically determine the trace-map relation for a general spring-mass model of the aperiodic Thue-Morse (TM) lattice, and, interestingly observe that this map has a pseudoinvariant. This pseudoinvariant has a crucial role on the nature of the eigenmodes of this lattice. When the pseudoinvariant vanishes identically, as in the case of the on-site, transfer, or mixed model of the TM lattice, all normal modes are found to be delocalized, whereas the eigenmodes are critical for more general models with nonzero pseudoinvariant. Our RSRG scheme also gives the average phonon density of states $\ensuremath{\rho}(\ensuremath{\omega})$ and Lyapunov exponent $\ensuremath{\gamma}(\ensuremath{\omega})$ (= inverse localization length) of the eigenmodes.

Numerical renormalization-group study of spin correlations in one-dimensional random spin chains

We calculate the ground-state two-spin correlation functions of spin-1/2 quantum Heisenberg chains with random exchange couplings using the real-space renormalization group scheme. We extend the conventional scheme to take account of the contribution of local higher multiplet excitations in each decimation step. This extended scheme can provide highly accurate numerical data for large systems. The random average of staggered spin correlations of the chains with random antiferromagnetic (AF) couplings shows algebraic decay like $1/r^2$, which verifies the Fisher's analytic results. For chains with random ferromagnetic (FM) and AF couplings, the random average of generalized staggered correlations is found to decay more slowly than a power-law, in the form close to $1/\ln(r)$. The difference between the distribution functions of the spin correlations of the random AF chains and of the random FM-AF chains is also discussed.

Stress-assisted instability in two-dimensional dislocation systems

We perform real-space renormalization-group (RG) calculations on two-dimensional dislocation systems loaded by external stress. As an extension of the existing model we find a recursion relation in the RG scheme for the coupling constant between dislocations. From our recursion relations we derive a nonlinear equation for the stress dependence of the transition temperature. Our results show significant deviation from the predictions of the earlier model depending on the core energy of the materials. Possible extensions of our model are outlined.

Monte Carlo renormalization of 2d simplicial quantum gravity coupled to Gaussian matter

We extend a recently proposed real-space renormalization group scheme for dynamical triangulations to situations where the lattice is coupled to continuous scalar fields. Using Monte Carlo simulations in combination with a linear, stochastic blocking scheme for the scalar fields we are able to determine the leading eigenvalues of the stability matrix with good accuracy both for c M = 1 and c M = 10 theories.

Real-space renormalization group technique for low-lying energy states in chain folding

We apply a real-space renormalization-group (RG) technique to study the low-lying energy states of a strongly interacting copolymer chain on a three-dimensional simple cubic lattice. The chain is 27 beads long and consists of randomly chosen hydrophobic and polar components whereby the system favors interactions between similar beads. The RG scheme is applied by dividing the chain into different parts (conformons) and applying a variety of boundary conditions to find the embedded conformons with the lowest energy. The conformations of the low-lying energy states are found by combining the conformons of different parts of the chain. We show that the 27 beads chain folds into a 3×3×3 cube for the lowest energy state. We also obtain conformations which do not fold into a cube but have structural similarities to the lowest energy state. The model is static but can help to understand folding processes of polymeric molecules and can be extended to more complex situations.

Universality and Scale Invariant Dynamics in Laplacian Fractal Growth

The individuation of the scale invariant dynamics in Laplacian fractal growth processes, like diffusion-limited aggregation (DLA), is an important problem whose solution would clarify some crucial issues concerning the origin of fractal properties and the identification of universality classes for such models. Here, we develop a real space renormalization group scheme to study the dynamic evolution of DLA in a restricted space of relevant parameters. In particular, we investigate the effect of a sticking probability P s and an effective noise reduction parameter S. The renormalization equations flow towards an attractive fixed point corresponding to the scale invariant DLA dynamics ([Formula: see text], S*≃2.0). The existence of a non-trivial fixed point value for S, shows that noise is spontaneously generated by the DLA growth process, and that screening, which is at the origin of fractal properties, persists at all scales.

Low-energy fixed points of random quantum spin chains

The one-dimensional isotropic quantum Heisenberg spin systems with random couplings and random spin sizes are investigated using a real-space renormalization group scheme. It is demonstrated that these systems belong to a universality class of disordered spin systems, characterized by weakly coupled large effective spins. In this large-spin phase the uniform magnetic susceptibility diverges as 1/T with a non-universal Curie constant at low temperatures T, while the specific heat vanishes as T^delta |ln T| for T->0. For broad range of initial distributions of couplings and spin sizes the distribution functions approach a single fixed-point form, where delta \approx 0.44. For some singular initial distributions, however, fixed-point distributions have non-universal values of delta, suggesting that there is a line of fixed points.

The generalized Hubbard model in one dimension: a renormalization group study

We have modified an existing real space renormalization group scheme to remove some physically inconsistent features and used the rectified method to study the ground state of the one-dimensional Hubbard model with nearest-neighbour repulsion and the bond-charge interaction for half filling. For the nearest-neighbour interaction we find a phase diagram which agrees nicely with the Monte Carlo results. For the case of bond-charge interaction we have reproduced some available exact results and found some interesting information in the parameter regime where no exact solution is available.

One-dimensional Ising model with long-range interactions: A renormalization-group treatment.

The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}&gt;1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value ${\mathit{T}}_{\mathit{c}}$/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/${\mathit{r}}^{2}$ model.