Real-space Renormalization Scheme Research Articles

On the Entanglement Entropy in Gaussian cMERA

AbstractThe continuous Multi Scale Entanglement Renormalization Anstaz (cMERA) consists of a variational method which carries out a real space renormalization scheme on the wavefunctionals of quantum field theories. In this work we calculate the entanglement entropy of the half space for a free scalar theory through a Gaussian cMERA circuit. We obtain the correct entropy written in terms of the optimized cMERA variational parameter, the local density of disentanglers. Accordingly, using the entanglement entropy production per unit scale, we study local areas in the bulk of the tensor network in terms of the differential entanglement generated along the cMERA flow. This result spurs us to establish an explicit relation between the cMERA variational parameter and the radial component of a dual AdS geometry through the Ryu‐Takayanagi formula. Finally, we argue that the entanglement entropy for the half space can be written as an integral along the renormalization scale whose measure is given by the Fisher information metric of the cMERA circuit. Consequently, a straightforward relation between AdS geometry and the Fisher information metric is also established.

Bosonic entanglement renormalization circuits from wavelet theory

Entanglement renormalization is a unitary real-space renormalization scheme. The corresponding quantum circuits or tensor networks are known as MERA, and they are particularly well-suited to describing quantum systems at criticality. In this work we show how to construct Gaussian bosonic quantum circuits that implement entanglement renormalization for ground states of arbitrary free bosonic chains. The construction is based on wavelet theory, and the dispersion relation of the Hamiltonian is translated into a filter design problem. We give a general algorithm that approximately solves this design problem and provide an approximation theory that relates the properties of the filters to the accuracy of the corresponding quantum circuits. Finally, we explain how the continuum limit (a free bosonic quantum field) emerges naturally from the wavelet construction.

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.

Transport properties through an aromatic molecular wire

The quantum transport properties through a finite homogeneous chain of aromatic molecules composed by benzene rings connected to two semi-infinite leads are investigated. The study is based on the tight-binding approach using semi-analytic methods of Green's function techniques within a real space renormalization scheme. The transmission probability and the characteristic diagram I−V of an aromatic molecular wire are calculated as a function of the molecule-leads coupling and the intra-rings coupling parameters. The obtained results show a good agreement with the previous reported DFT calculations.

Shot noise and thermopower in aromatic molecules

The quantum transport properties of finite homogeneous chains of aromatic molecules composed of one, two or several benzene rings connected to two semi-infinite leads are studied. We study these molecules based on the tight-binding approach. The calculation of the transport properties is performed using Green׳s function techniques within a real space renormalization scheme. In particular, we focus on the transmission probability, on the noise power of current fluctuations and on the thermopower. Our results show different transport regimes for these molecular systems as a function of the coupling intensities and the length of chains.

Quantum transport through aromatic molecules

In this paper, we study the electronic transport properties through aromatic molecules connected to two semi-infinite leads. The molecules are in different geometrical configurations including arrays. Using a nearest neighbor tight-binding approach, the transport properties are analyzed into a Green's function technique within a real-space renormalization scheme. We calculate the transmission probability and the Current-Voltage characteristics as a function of a molecule-leads coupling parameter. Our results show different transport regimes for these systems, exhibiting metal-semiconductor-insulator transitions and the possibility to employ them in molecular devices.

Disordered contact process with asymmetric spreading

An asymmetric variant of the contact process where the activity spreads with different and independent random rates to the left and to the right is introduced. A real space renormalization scheme is formulated for the model by means of which it is shown that the local asymmetry of spreading is irrelevant on large scales if the model is globally (statistically) symmetric. Otherwise, in the presence of a global bias in either direction, the renormalization method predicts two distinct phase transitions, which are related to the spreading of activity in and against the direction of the bias. The latter is found to be described by an infinite randomness fixed point while the former is not.

Current and Shot noise in DNA chains

In this paper we studied the transport properties of a finite homogeneous fragments of DNA composed by N base pairs of Guanine (poly (G)) and Cytosine (poly (C)) connected to two semi-infinite leads. We study these molecules adopting different models within a nearest neighbour tight-binding approach. We proposed a semi-analytic method for the calculation of the transport properties of DNA molecules by using Green’s function techniques within a real-space renormalization scheme. We studied the transmission probability, the I–V characteristics and the Noise power of current fluctuations as a function of intrasite and DNA-leads coupling parameters. Our results show different transport regimes for these molecular systems as a function of the coupling intensities, exhibiting metal–semiconductor transitions.

Duality symmetries and effective dynamics in disordered hopping models

We identify a duality transformation in one-dimensional hopping models that relatespropagators in general disordered potentials linked by an up–down inversion of the energylandscape. This significantly generalizes previous results on a duality between trapand barrier models. We use the resulting insights into the symmetries of thesemodels to develop a real-space renormalization scheme that can be implementedcomputationally and allows rather accurate prediction of propagation in these models.We also discuss the relation of this renormalization scheme to earlier analyticaltreatments.

Ground state of a two-dimensional quasiperiodic quantum antiferromagnet

We consider the spin 1/2 Heisenberg antiferromagnet on a two dimensional quasiperiodic tiling. The T=0 state is long range ordered, with an inhomogeneous distribution of the staggered moment. An approximate real space renormalization scheme first developed for the square lattice is generalized and used to obtain information on the ground state energy as well as the distribution of local correlations in the quasicrystal.

Dissipation in weakly polarizable electrodes across irregular boundaries

The spectroscopic impedance associated with a localized linear redox reaction distributed across irregular boundaries is studied. Taking advantage of recent results in the literature, we present a real space renormalization scheme for the calculation of the coefficients of the impedance in the Makarov regime. It is predicted that the impedance diminishes for successive prefractal generations of the same fractal generator, so that the final fractal object is expected to dissipate less than all previous ones.

Dynamical real space renormalization group applied to sandpile models.

A general framework for the renormalization group analysis of self-organized critical sandpile models is formulated. The usual real space renormalization scheme for lattice models when applied to nonequilibrium dynamical models must be supplemented by feedback relations coming from the stationarity conditions. On the basis of these ideas the dynamically driven renormalization group is applied to describe the boundary and bulk critical behavior of sandpile models. A detailed description of the branching nature of sandpile avalanches is given in terms of the generating functions of the underlying branching process.

Phases of random antiferromagnetic spin-1 chains

We formulate a real-space renormalization scheme that allows the study of the effects of bond randomness in the Heisenberg antiferromagnetic spin-1 chain. There are four types of bonds that appear during the renormalization flow. We implement numerically the decimation procedure. We give a detailed study of the probability distributions of all these bonds in the phases that occur when the strength of the disorder is varied. Approximate flow equations are obtained in the weak-disorder regime as well as in the strong disorder case where the physics is that of the random singlet phase.

Dynamically driven renormalization group

We present a detailed discussion of a novel dynamical renormalization group scheme: the dynamically driven renormalization group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady state. The method is based on a real-space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open nonlinear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scalling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed.

Rectangular renormalization

A generalized real-space renormalization scheme is developed for geometrical critical phenomena. The renormalization group is parametrized by the standard length-scaling factor and a new rectangular area-fraction factor. This rectangular renormalization scheme utilizes relatively small rectangular sublattices to effectively renormalize large square lattices. With the area-fraction factor, one can systematically study rectangular generalizations of the conventional square-cell renormalization theories. Application to self-avoiding random walks yields critical descriptors that are comparable to, and in most cases better than previous results obtained from more complex renormalization schemes.

Real-space renormalization scheme for calculating the dynamical response of disordered chains.

The method of real-space renormalization has been extended to calculate the dynamical response of disordered chains. It is shown that the dependence of the structure function on wave vector and energy can be mapped over the entire Brillouin zone and the entire energy band. Dynamical properties such as the density of states and the dispersion relation can be obtained from the map, and the system's response to a probing neutron can be measured. The scheme works for all ranges of concentration. It has been applied to two situations of interest: a harmonic chain with mass defect and a diluted one-dimensional antiferromagnet

Real-space renormalization scheme for reaction-limited cluster aggregation.

A real-space renormalization scheme for reaction-limited cluster-cluster aggregation is presented based on the properties of the reaction surface. For the soluble monodisperse case both the fractal dimensions and reaction surface exponents can be calculated in good agreement with simulation and experiment. The associated scaling functions are also found. Taking the limits d=1 and d\ensuremath{\rightarrow}\ensuremath{\infty} in the scheme, we recover linear and ghost aggregates exactly as expected. The polydisperse case is also studied in some detail, although a numerical solution has not been obtained.

Polymer adsorption : bounds on the cross-over exponent and exact results for simple models

We provide an upper and a lower bound for the cross-over exponent Φ in polymer adsorption on an impenetrable attractive surface. These bounds are shown to be verified for exactly solvable models where the polymer resides on a fractal lattice. In one of these models we also find a multicritical point where a bulk collapse transition and a surface adsorption transition coexist. A simple real-space renormalization scheme is presented for the square lattice, it gives Φ = 0.48, in close agreement with the presumably exact value 1/2, which coincides with our upper bound.

Real-space renormalization for structural phase transitions in the Ising universality class

A real-space renormalization scheme for calculating the thermodynamic properties of models with continuous variables on a lattice having phase transitions in the Ising universality class is described. The first renormalization transformation maps the initial model onto a spin-\textonehalf{} Ising model with generalized interactions, which is then analyzed by an Ising renormalization transformation or by other means. Two versions of this procedure are used to calculate the phase diagram of a double-well model for structural phase transitions, which is identical with the ${\ensuremath{\varphi}}^{4}$ field theory on a lattice, in two dimensions. The results are in excellent agreement with the molecular-dynamics studies of Schneider and Stoll.

Applications of an exact duality-decimation transformation for two-dimensional spin systems on a square lattice

An exact duality-decimation transformation is derived for a large class of two-dimensional spin systems on a square lattice. It maps a model with interactions in every second square (checkerboard interactions) onto a model with interactions in every square and vice versa. The transformation is applicable to Ising, Potts, solid-on-solid (SOS), and $x\ensuremath{-}y$ models. It may be used to reformulate the site percolation problem in terms of the $s\ensuremath{\rightarrow}1$ limit of a Potts model with four-spin interactions in every elementary square. The transformation maps the van Beijeren SOS model, which is equivalent to the $F$ model, onto an $x\ensuremath{-}y$ model with checkerboard interactions. Thus one has an exactly soluble $x\ensuremath{-}y$ model, which, however, bears little resemblance to the ordinary $x\ensuremath{-}y$ model. The transformation may be combined with a bond-shifting approximation to obtain a real-space renormalization scheme which in some respects represents an improvement over the Migdal-Kadanoff formula. No nearest-neighbor bonds are shifted. Better values of the exponent $\ensuremath{\nu}$ are obtained for the Ising and Potts models. Apart from exponentially vanishing corrections, the renormalization transformation for SOS and $x\ensuremath{-}y$ models has a surface of fixed points corresponding to generalizations of the usual discrete Gaussian and Villain models.