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

arXiv:cond-mat/9812118 · International Journal of Modern Physics C

CROSSOVER PROPERTIES FROM RANDOM PERCOLATION TO FRUSTRATED PERCOLATION

Luigi Cannavacciuolo, ANTONIO DE CANDIA, Antonio Coniglio A

Published Jun 1, 1999• 2 citations
arXivHTMLPDF

Abstract

We investigate the crossover properties of the frustrated percolation model on a two-dimensional square lattice, with asymmetric distribution of ferromagnetic and antiferromagnetic interactions. We determine the critical exponents ν, γ, and β of the percolation transition of the model, for various values of the density of antiferromagnetic interactions π in the range 0≤π≤0.5. Our data is consistent with the existence of a crossover from random percolation behavior for π=0, to frustrated percolation behavior, characterized by the critical exponents of the ferromagnetic 1/2-state Potts model, as soon as π>0.

DOI
10.1142/s0129183199000425

Related papers

Quantum Phase Transition of the Randomly Diluted Heisenberg Antiferromagnet on a Square Lattice

Kiyoshi Katō, Synge Todo, Kenji Harada +3 more

Ground-state magnetic properties of the diluted Heisenberg antiferromagnet on a square lattice are investigated by means of the quantum Monte Carlo method with the continuous-time loop algorithm. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the two-dimensional percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. Furthermore, we found that the transition is not universal, i.e., the critical exponents significantly depend on S.

arXiv:cond-mat/9905379•Physical Review Letters•May 1, 2000•68 citations

Critical properties of a two-dimensional Ising magnet with quasiperiodic interactions

Gladstone Alencar Alves, M.S. Vasconcelos, T. F. A. Alves

We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards-Anderson order parameter q_{EA}, the magnetic susceptibility χ, and the specific heat c in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β,γ, and ν. In the low-temperature limit we obtained a spin-glass phase with critical temperature around T_{c}≈2.274, and the critical exponents β,γ, and ν, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.

Physical review. E•Apr 12, 2016•4 citations

Weak Universality in the Disordered Two-Dimensional Antiferromagnetic Potts Model on a Triangular Lattice

Albert Babaevich Babaev, Akai Kurbanovich Murtazaev

The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality $$\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = d$$ . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.

Journal of Experimental and Theoretical Physics Letters•May 1, 2018•5 citations

Comment on “Quantum Phase Transition of the Randomly Diluted Heisenberg Antiferromagnet on a Square Lattice”

Anders W. Sandvik

In Phys. Rev. Lett. 84, 4204 (2000) (cond-mat/9905379), Kato et al. presented quantum Monte Carlo results indicating that the critical concentration of random non-magnetic sites in the two-dimensional antiferromagnetic Heisenberg model equals the classical percolation density; pc=0.407254. The data also suggested a surprising dependence of the critical exponents on the spin S of the magnetic sites, with a gradual approach to the classical percolation exponents as S goes to infinity. I here argue that the exponents in fact are S-independent and equal to those of classical percolation. The apparent S-dependent behavior found by Kato et al. is due to temperature effects in the simulations as well as a quantum effect that masks the true asymptotic scaling behavior for small lattices.

arXiv:cond-mat/0010433•Physical Review Letters•Apr 2, 2001•24 citations

Nonlocal percolation in an antiferromagnetic Potts model

Henry Phillip Fried, Michael Schick C

We devise an algorithm to determine the critical site-occupation probability, ${x}_{c}$, at which antiferromagnetic order in the three-state Potts model first propagates on a triangular lattice. We find ${x}_{c}=0.735$, larger than the value of 0.5 necessary to percolate ferromagnetic order. Even though the means to propagate the order in this problem can be nonlocal as well as local, our results for critical exponents indicate that the percolation transition is in the same universality class as that of ferromagnetic percolation.

Physical review. B, Condensed matter•Jul 1, 1988•15 citations

Ising Antiferromagnet with Nearest-Neighbor and Next-Nearest-Neighbor Interactions on a Square Lattice

Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович, М. К. Бадиев

The critical properties of two-dimensional antiferromagnetic Ising model in square lattice are investigated using the replica Monte-Carlo method with account of interactions of second nearest neighbors. The diagram of critical temperature dependence on an interaction value of second nearest neighbors is plotted. Static critical exponents of the heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated for this model using the finite-size scaling theory.

Diffusion and defect data, solid state data. Part B, Solid state phenomena/Solid state phenomena•Apr 1, 2014•0 citations
We investigate the crossover properties of the frustrated percolation model on a two-dimensional square lattice, with asymmetric distribution of ferromagnetic and antiferromagnetic interactions. We determine the critical exponents ν, γ, and β of the percolation transition of the model, for various values of the density of antiferromagnetic interactions π in the range 0≤π≤0.5. Our data is consistent with the existence of a crossover from random percolation behavior for π=0, to frustrated percolation behavior, characterized by the critical exponents of the ferromagnetic 1/2-state Potts model, as soon as π>0.
Ground-state magnetic properties of the diluted Heisenberg antiferromagnet on a square lattice are investigated by means of the quantum Monte Carlo method with the continuous-time loop algorithm. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the two-dimensional percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. Furthermore, we found that the transition is not universal, i.e., the critical exponents significantly depend on S.
We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards-Anderson order parameter q_{EA}, the magnetic susceptibility χ, and the specific heat c in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β,γ, and ν. In the low-temperature limit we obtained a spin-glass phase with critical temperature around T_{c}≈2.274, and the critical exponents β,γ, and ν, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.
The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality 2βν+γν=d\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = dν2β​+νγ​=d . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.
In Phys. Rev. Lett. 84, 4204 (2000) (cond-mat/9905379), Kato et al. presented quantum Monte Carlo results indicating that the critical concentration of random non-magnetic sites in the two-dimensional antiferromagnetic Heisenberg model equals the classical percolation density; pc=0.407254. The data also suggested a surprising dependence of the critical exponents on the spin S of the magnetic sites, with a gradual approach to the classical percolation exponents as S goes to infinity. I here argue that the exponents in fact are S-independent and equal to those of classical percolation. The apparent S-dependent behavior found by Kato et al. is due to temperature effects in the simulations as well as a quantum effect that masks the true asymptotic scaling behavior for small lattices.
We devise an algorithm to determine the critical site-occupation probability, xc{x}_{c}xc​, at which antiferromagnetic order in the three-state Potts model first propagates on a triangular lattice. We find xc=0.735{x}_{c}=0.735xc​=0.735, larger than the value of 0.5 necessary to percolate ferromagnetic order. Even though the means to propagate the order in this problem can be nonlocal as well as local, our results for critical exponents indicate that the percolation transition is in the same universality class as that of ferromagnetic percolation.