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/0603386 · Physical Review E

Discrete scale invariance effects in the nonequilibrium critical behavior of the Ising magnet on a fractal substrate

M. A. Bab, G. Fabricius, Ezequiel Vicente Albano

Published Oct 26, 2006• 18 citations
arXivHTMLPDF

Abstract

The nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension d(H)=1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. We have obtained evidence showing that during these relaxation processes both the growth and the fragmentation of magnetic domains become influenced by the hierarchical structure of the substrate. In fact, the interplay between the dynamic behavior of the magnet and the underlying fractal leads to the emergence of a logarithmic-periodic oscillation, superimposed to a power law, which has been observed in the time dependence of both the decay of the magnetization and its logarithmic derivative. These oscillations have been carefully characterized in order to determine the critical temperature of the second-order phase transition and the critical exponents corresponding to the short-time regime. The effects of the substrate can also be observed from the dependence of the effective critical exponents on the segmentation step. The exponent theta of the initial increase of the magnetization has also been obtained and the results suggest that it would be almost independent of the fractal dimension of the substrate, provided that d(H) is close enough to d=2. The oscillations have been discussed within the framework of the discrete scale invariance of the substrate.

DOI
10.1103/physreve.74.041123

Related papers

Short-time dynamics of an Ising system on fractal structures

Guangping Zheng, Mo Li

The short-time critical relaxation of an Ising model on a Sierpinski carpet is investigated using Monte Carlo simulation. We find that when the system is quenched from high temperature to the critical temperature, the evolution of the order parameter and its persistence probability, the susceptibility, and the autocorrelation function all show power-law scaling behavior at the short-time regime. The results suggest that the spatial heterogeneity and the fractal nature of the underlying structure do not influence the scaling behavior of the short-time critical dynamics. The critical temperature, dynamic exponent z, and other equilibrium critical exponents beta and nu of the fractal spin system are determined accurately using conventional Monte Carlo simulation algorithms. The mechanism for short-time dynamic scaling is discussed.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 1, 2000•17 citations

Fractality in persistence decay and domain growth during ferromagnetic ordering: Dependence upon initial correlation

Saikat Chakraborty, Subir K. Das

The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.

arXiv:1509.07590•Physical review. E•Mar 22, 2016•14 citations

Critical behavior of the quantum Ising model on a fractal structure

Hangmo Yi

We study the critical behavior of the transverse-field quantum Ising model on a fractal structure, namely the Sierpinski carpet. When a magnetic field Δ is applied perpendicular to the Ising spin direction, quantum fluctuations affect the transition between the ferromagnetic and the paramagnetic phases. Employing the continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, we investigate the interplay between the quantum fluctuations and the exotic dimensionality of the fractal structure and its effect on the critical behavior. As the transverse magnetic field increases, the critical temperature monotonically decreases until it apparently vanishes at a critical field Δ(c), beyond which the system becomes paramagnetic at all temperatures. However, the critical exponents are independent of Δ and remain the same as in the purely classical(Δ=0) case.

Physical Review E•Jul 29, 2013•2 citations

Study of the Nonequilibrium Critical Quenching and Annealing Dynamics for the Long-Range Ising Model

D. E. Rodríguez, M. A. Bab, Ezequiel Vicente Albano

Extensive Monte Carlo simulations are employed in order to study the dynamic critical behavior of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form $\frac{1}{r^{d+σ}}$, with $σ=0.75$. The critical temperature, as well as the critical exponents, is evaluated from the power-law behavior of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different cutoffs of the interaction range. All the estimated static critical exponents ($γ/ν$, $β/ν$, and $1/ν$) are in good agreement with Renormalization Group (RG) predictions and previously reported numerical data obtained under equilibrium conditions. It is found that the dynamic exponent $z$ is different for quenching and annealing experiments, most likely due to the influence of the Kosterlitz-Thouless transition occurring at relatively similar algebraic decay of the interactions with $σ=1$. However, for annealing experiments the measured exponent $z$ is close to the RG predictions. On the other hand, the relevant exponents of the dynamic behavior ($z$ and $θ$) are slightly different than the RG predictions, most likely due to the fact that they may depend on the especific dynamics used (Metropolis in the present paper).

arXiv:0908.1518•Aug 11, 2009•1 citation

Monte Carlo study for the thermodynamic and dynamic phase transitions in the spin-S Ising model on Sierpiński carpet

Hoseung Jang, Mouhcine Azhari, Unjong Yu

Abstract We study the thermodynamic and dynamic phase transitions (TPT and DPT) of the spin- 1 / 2 and spin-1 Ising models on three graphs constructed on the Sierpiński carpet. This study employs Monte Carlo methods, specifically the Wolff and Metropolis algorithms, in conjunction with finite-size scaling analysis. By calculating the critical temperature and critical exponent ratio γ / ν associated with the TPT, we demonstrate that the three graphs exhibit an identical critical exponent ratio for both the spin- 1 / 2 and spin-1 Ising models within statistical error. Furthermore, we explore the kinetic Ising model by varying the period of the oscillating external magnetic field and verify the existence of the DPT. We find that the critical exponent ratio γ / ν for the DPT matches that of the TPT. Our results suggest that the critical exponents or their ratios for continuous phase transitions in equilibrium and non-equilibrium systems with short-range interactions are independent of the graph structure and interaction type, as long as the background space is the same.

Journal of Statistical Mechanics Theory and Experiment•Jan 1, 2024•4 citations

Local and global magnetization on the Sierpiński carpet

Jozef Genzor, Andrej Gendiar, Tomotoshi Nishino

The phase transition of the classical Ising model on the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet, which has the fractal dimension ${log}_{3}^{\phantom{\rule{0.16em}{0ex}}}8\ensuremath{\approx}1.8927$, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature ${T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}\ensuremath{\approx}1.478$. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent $\ensuremath{\beta}$ associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas ${T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}$ is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of $\ensuremath{\beta}\ensuremath{\approx}0.135$.

arXiv:1904.10645•Physical review. E•Apr 17, 2023•5 citations
The nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension d(H)=1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. We have obtained evidence showing that during these relaxation processes both the growth and the fragmentation of magnetic domains become influenced by the hierarchical structure of the substrate. In fact, the interplay between the dynamic behavior of the magnet and the underlying fractal leads to the emergence of a logarithmic-periodic oscillation, superimposed to a power law, which has been observed in the time dependence of both the decay of the magnetization and its logarithmic derivative. These oscillations have been carefully characterized in order to determine the critical temperature of the second-order phase transition and the critical exponents corresponding to the short-time regime. The effects of the substrate can also be observed from the dependence of the effective critical exponents on the segmentation step. The exponent theta of the initial increase of the magnetization has also been obtained and the results suggest that it would be almost independent of the fractal dimension of the substrate, provided that d(H) is close enough to d=2. The oscillations have been discussed within the framework of the discrete scale invariance of the substrate.
The short-time critical relaxation of an Ising model on a Sierpinski carpet is investigated using Monte Carlo simulation. We find that when the system is quenched from high temperature to the critical temperature, the evolution of the order parameter and its persistence probability, the susceptibility, and the autocorrelation function all show power-law scaling behavior at the short-time regime. The results suggest that the spatial heterogeneity and the fractal nature of the underlying structure do not influence the scaling behavior of the short-time critical dynamics. The critical temperature, dynamic exponent z, and other equilibrium critical exponents beta and nu of the fractal spin system are determined accurately using conventional Monte Carlo simulation algorithms. The mechanism for short-time dynamic scaling is discussed.
The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.
We study the critical behavior of the transverse-field quantum Ising model on a fractal structure, namely the Sierpinski carpet. When a magnetic field Δ is applied perpendicular to the Ising spin direction, quantum fluctuations affect the transition between the ferromagnetic and the paramagnetic phases. Employing the continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, we investigate the interplay between the quantum fluctuations and the exotic dimensionality of the fractal structure and its effect on the critical behavior. As the transverse magnetic field increases, the critical temperature monotonically decreases until it apparently vanishes at a critical field Δ(c), beyond which the system becomes paramagnetic at all temperatures. However, the critical exponents are independent of Δ and remain the same as in the purely classical(Δ=0) case.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behavior of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1rd+σ\frac{1}{r^{d+σ}}rd+σ1​, with σ=0.75σ=0.75σ=0.75. The critical temperature, as well as the critical exponents, is evaluated from the power-law behavior of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different cutoffs of the interaction range. All the estimated static critical exponents (γ/νγ/νγ/ν, β/νβ/νβ/ν, and 1/ν1/ν1/ν) are in good agreement with Renormalization Group (RG) predictions and previously reported numerical data obtained under equilibrium conditions. It is found that the dynamic exponent zzz is different for quenching and annealing experiments, most likely due to the influence of the Kosterlitz-Thouless transition occurring at relatively similar algebraic decay of the interactions with σ=1σ=1σ=1. However, for annealing experiments the measured exponent zzz is close to the RG predictions. On the other hand, the relevant exponents of the dynamic behavior (zzz and θθθ) are slightly different than the RG predictions, most likely due to the fact that they may depend on the especific dynamics used (Metropolis in the present paper).
Abstract We study the thermodynamic and dynamic phase transitions (TPT and DPT) of the spin- 1 / 2 and spin-1 Ising models on three graphs constructed on the Sierpiński carpet. This study employs Monte Carlo methods, specifically the Wolff and Metropolis algorithms, in conjunction with finite-size scaling analysis. By calculating the critical temperature and critical exponent ratio γ / ν associated with the TPT, we demonstrate that the three graphs exhibit an identical critical exponent ratio for both the spin- 1 / 2 and spin-1 Ising models within statistical error. Furthermore, we explore the kinetic Ising model by varying the period of the oscillating external magnetic field and verify the existence of the DPT. We find that the critical exponent ratio γ / ν for the DPT matches that of the TPT. Our results suggest that the critical exponents or their ratios for continuous phase transitions in equilibrium and non-equilibrium systems with short-range interactions are independent of the graph structure and interaction type, as long as the background space is the same.
The phase transition of the classical Ising model on the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet, which has the fractal dimension ${log}_{3}^{\phantom{\rule{0.16em}{0ex}}}8\ensuremath{\approx}1.8927$, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature ${T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}\ensuremath{\approx}1.478$. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent $\ensuremath{\beta}$ associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas Tc{T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}Tc​ is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of $\ensuremath{\beta}\ensuremath{\approx}0.135$.