Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chinese Physics Letters

Dynamic Critical Behavior of Sierpinski Gasket Type Fractal

Wang Xiaomin, Ruibao Tao

Published Aug 1, 1991• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/8/8/004

Related papers

Critical spin-wave dynamics for two-dimensional Sierpinski-type fractal

S Bhattacharya

The dynamic critical exponent z for a system of nearest-neighbour interacting Heisenberg spins on a two-dimensional Sierpinski-type fractal is obtained by applying a real space renormalisation group transformation of the equations of motion.

Journal of Physics A Mathematical and General•May 11, 1985•4 citations

Critical Dynamics at the Percolation Threshold by Fractal and Scaling Approaches

C K Harris, Robin B. Stinchcombe

The critical dynamics of Heisenberg spin systems at the percolation threshold is obtained from the transformation of the equations of motion under a length scaling acheived by climination of sites. The method is applied exactly to $d$-dimensional nonrandom fractals representing the infinite-cluster backbone and approximatcly to a truly random twodimensional system to yield dynamic scaling forms for response function and characteristic frequency, and associated static and dynamic critical exponents.

Physical Review Letters•May 2, 1983•60 citations

Quantum critical behavior of the quantum Ising model on fractal lattices

Hangmo Yi

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.

arXiv:1408.5595•Physical Review E•Jan 9, 2015•9 citations

Dynamic scaling near the percolation threshold in the diluted Heisenberg chain

Robin B. Stinchcombe, C K Harris

Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.

Journal of Physics A Mathematical and General•Dec 1, 1983•18 citations

Critical dynamics of Heisenberg spins on self-avoiding-walk chains

S. Bhattacharya, Bikas K. Chakrabarti

The dynamical exponent z for the critical spin-wave dynamics of nearest-neighbour interacting Heisenberg spins on a self-avoiding-walk (SAW) chain is estimated here using a scaling picture and also applying a real-space renormalisation group technique to some quasilinear fractal lattices. The results indicate z=2Dt, where D is the fractal dimensionality of the SAW chain lattice and t is the exponent for the length of the shortest nearest-neighbour connecting path of the SAW.

Journal of Physics A Mathematical and General•Oct 1, 1984•3 citations

Critical dynamics in Heisenberg ferromagnets and antiferromagnets near the percolation threshold

Iveta R. Pimentel, Robin B. Stinchcombe

The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.

Journal of Physics A Mathematical and General•Sep 21, 1989•4 citations
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.
The critical dynamics of Heisenberg spin systems at the percolation threshold is obtained from the transformation of the equations of motion under a length scaling acheived by climination of sites. The method is applied exactly to ddd-dimensional nonrandom fractals representing the infinite-cluster backbone and approximatcly to a truly random twodimensional system to yield dynamic scaling forms for response function and characteristic frequency, and associated static and dynamic critical exponents.
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.
Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.
The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.