Exact Real-space Renormalization Group Research Articles

Studies of the spectral dimension of the Vicsek snowflake fractal

By an exact real space renormalization group method, the authors obtain the spectral dimension of the Vicsek snowflake fractal. The result seems to cast some doubt on the range of validity of the Einstein relation on the deterministic fractal.

Phase Transitions of Ultradiffusion Model on Vicsek Snowflakes Fractal

Through an exact real space renormalisation group approach, we analysed the ultradiffusion with a hierarchical pattern of hopping rates defined on Vicsek snowflackes fractal. The transition from anomalous to normal diffusion is shown to occur in the arbitrary n-dimension (n-D) case. In the presence of external field, we find another unusual transition which is essentially different from the one for biased ultradiffusion of 1-D multifurcating hierarchical system.

Dynamic Critical Behavior of Sierpinski Gasket Type Fractal

We have applied the exact real space renormalisation group method to analyse the critical dynamics of Heisenberg spin system at the percolation threshold embedded in the two dimensional Sierpinski Gasket type fractal. Our analysis yields the dynamic critical exponent Z and the scaling behaviors of response function and characteristic frequency. We also show that the scaling exponent relation is accurately satisfied by the values of the Z and two other exponents.

Local phonon properties of the Fibonacci-chain quasicrystal

A new exact real-space renormalization group approach is presented for the local phonon properties of any given site in an infinite Fibonacci chain. It is found that the local phonon densities of states for different sites are different from each other, although they have the same structure. Results also suggest that the local phonon spectrum is a Cantor set.

Local electronic properties of one-dimensional quasiperiodic systems.

An exact real-space renormalization-group approach is developed to calculate the local Green's function and the local density of states at any site in an infinite Fibonacci chain, in which three basic renormalization transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathit{T}}_{\ensuremath{\gamma}}$ are introduced. Transformations ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ and ${\mathit{T}}_{\ensuremath{\gamma}}$ divide the Fibonacci chains of different generations into two distinct subclasses with key sites ${\mathit{S}}_{\mathrm{\ensuremath{\beta}}}$ and ${\mathit{S}}_{\ensuremath{\gamma}}$, respectively. The local Green's function and the local density of states at the key site ${\mathit{S}}_{\mathrm{\ensuremath{\beta}}}$ or ${\mathit{S}}_{\ensuremath{\gamma}}$ can be obtained by successive iterations of the transformation ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ or ${\mathit{T}}_{\ensuremath{\gamma}}$. Any other site can be transferred to the key site of a renormalized Fibonacci chain by suitable combinations of the transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathit{T}}_{\ensuremath{\gamma}}$. In this approach, the relevant scale concerning the key site is ${\mathrm{\ensuremath{\tau}}}^{2}$, where \ensuremath{\tau} is the golden mean.

Exact real-space renormalization-group approach for the local electronic Green's functions on an infinite Fibonacci chain.

Exact real-space renormalization-group approach for the local electronic Green's functions on an infinite Fibonacci chain.

The trail problem on finitely ramified fractals

Using an exact real-space renormalisation group technique. The authors show that self-avoiding trails (SAT) and self-avoiding walks (SAW) on the Sierpinski gasket obey the same critical exponent nu for the 'correlation length' and therefore belong to the same universality class. On the other hand, it is shown that SAT on branching Koch curves with maximum ramification number Rmax>3 is in a different universality class from that of SAW.

ADSORPTION OF A SELF-AVOIDING WALK ON FRACTAL SPACES

An exact real space renormalisation group is applied to study the adsorption of a single self-avoiding walk on fractal spaces. Exact results of critical exponent ν and crossover exponent φ show that de Gennes' conjecture φ =1-ν is invalid. The phase diagram has qualitatively the same shape as it has on Euclidean space, except for a unique and important difference that the adsorption transition would take place for a single polymer chain at a hard wail exerting a infinitesimal short-range attractive force on the fractal space instead of a finite one as on Euclidean space.

Resistor-diode percolation on the hierarchical diamond lattice

The resistor-diode percolation problem on the hierarchical diamond lattice is studied by using an exact real space renormalisation group transformation. This problem exhibits several critical and multicritical points. The flow diagram obtained is topologically equivalent to the one found by Redner (1982) in the framework of an approximate renormalisation group transformation on regular lattices.

Exact real-space renormalization group and new truncation algorithms for lattice theories

This paper discusses the theoretical basis of variational algorithms for thinning degrees of freedom in a lattice Hamiltonian theory. We show that such a thinning can, in principle, be an exact real-space renormalization-group transformation to a lattice with fewer sites. This exact transformation can only be constructed given the exact ground state of the theory. However, this insight teaches us how to significantly improve previous variational algorithms. Further improvements in ability to calculate quantities of interest are achieved by introducing "look-ahead" algorithms which allow us to maintain a renormalization-group interpretation while introducing long-distance physics into the determination of variational parameters. We find that algorithms incorporating both these features are much more powerful than previous truncation algorithms. For pedagogical reasons we present our algorithms using as examples two well-understood theories, namely, free scalar field theory and the Ising model in a transverse field in 1 + 1 dimensions. Discussion of higher dimensions and less trivial theories is included in the final section. An appendix of generalized mean-field theory is also included.