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:1809.09523

Universal behavior in finite 2D kinetic ferromagnets

James Denholm, Ben Hourahine

Published Sep 25, 2018• 0 citations
arXivHTMLPDF

Abstract

We study the time evolution of the two-dimensional kinetic Ising model in finite systems with a non-conserved order parameter, considering nearest-neighbour interactions on the square lattice with periodic and open boundary conditions. Universal data collapse in spin product correlation functions is observed which, when expressed in rescaled units, is valid across the entire time evolution of the system at all length scales, not just within the time regime usually considered in the dynamical scaling hypothesis. Consequently, beyond rapidly decaying finite size effects, the evolution of correlations in small finite systems parallels arbitrarily larger cases, even at large fractions of the size of these finite systems.

DOI
10.48550/arxiv.1809.09523

Related papers

Critical percolation in the dynamics of the 2D ferromagnetic Ising model

Thibault Blanchard, Leticia F. Cugliandolo, Marco Picco +1 more

Abstract We study the early time dynamics of the 2 d ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. We evolve the system with kinetic Monte Carlo rules that do not conserve the order parameter. We confirm the rapid approach to random critical percolation in a time-scale that diverges with the system size but is much shorter than the equilibration time. We study the scaling properties of the evolution towards critical percolation and we identify an associated growing length, different from the curvature driven one. By working with the model defined on square, triangular and honeycomb microscopic geometries we establish the dependence of this growing length on the lattice coordination. We discuss the interplay with the usual coarsening mechanism and the eventual fall into and escape from metastability.

arXiv:1705.06508•Journal of Statistical Mechanics Theory and Experiment•Nov 1, 2017•29 citations

The correlation function in two dimensional Ising model on the finite size lattice. I

A. I. Bugrij

The correlation function of two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor representation are obtained both for the ferromagnetic and paramagnetic regions of coupling constant. The peculiarities caused by the finite size of the system are discussed. The asymptotic behaviour of the corresponding quantities as functions of size is calculated. Some generalization of the obtained result is conjectured.

arXiv:hep-th/0011104•Nov 13, 2000•11 citations

Persistence in the two dimensional ferromagnetic Ising model

Thibault Blanchard, Leticia F. Cugliandolo, Marco Picco

We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the 2d ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value θ that depends upon the initial condition. More precisely, we find that θ takes one universal value 0.199(2) for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value 0.033(1) for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with a square and a triangular lattice, and by imposing free and periodic boundary conditions. We found that the effective exponent suffers from stronger finite size effects in the former case.

arXiv:1410.4007•Journal of Statistical Mechanics Theory and Experiment•Dec 22, 2014•21 citations

Response of the two-dimensional kinetic Ising model under a stochastic field

Asim Kumar Ghosh, Bikas K. Chakrabarti

We study, using Monte Carlo dynamics, the time ( t ) dependent average magnetization per spin m ( t ) behavior of the 2D kinetic Ising model under a binary (± h 0 ) stochastic field h ( t ). The time dependence of the stochastic field is such that its average over each successive time interval τ is assured to be zero (without any fluctuation). The average magnetization is considered as an order parameter of the system. The phase diagram in ( h 0 ,τ) plane is obtained. Fluctuations in the order parameter and their scaling properties are studied across the phase boundary. These studies indicate that the nature of the transition is Ising like (static Ising universality class) for field amplitudes h 0 below some threshold value (dependent on τ values; as τ → ∞ across the phase boundary). Beyond these , the transition is no longer continuous.

arXiv:1305.7108•Journal of Statistical Mechanics Theory and Experiment•Nov 27, 2013•4 citations

Finite-size behavior of the Ising square lattice

David Paul Landau

An importance-sampling Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N$ Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of $N$ and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.

Physical review. B, Solid state•Apr 1, 1976•411 citations

Universality and logarithmic corrections in two-dimensional random Ising ferromagnets

Fabio D. A. Aarao Reis, S. L. A. de Queiroz, Raimundo R. dos Santos

We address the question of weak versus strong universality scenarios for the random-bond Ising model in two dimensions. A finite-size scaling theory is proposed, which explicitly incorporates $\mathrm{ln} L$ corrections $(L$ is the linear finite size of the system) to the temperature derivative of the correlation length. The predictions are tested by considering long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings, along which free energy, spin-spin correlation functions, and specific heats are calculated by transfer-matrix methods. The ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is calculated and has the same value as in the pure case; consequently conformal invariance predictions remain valid for this type of disorder. Semilogarithmic plots of correlation functions against distance yield average correlation lengths ${\ensuremath{\xi}}^{\mathrm{av}}$, whose size dependence agrees very well with the proposed theory. We also examine the size dependence of the specific heat, which clearly suggests a divergency in the thermodynamic limit. Thus our data consistently favor the Dotsenko-Shalaev picture of logarithmic corrections (enhancements) to pure system singularities, as opposed to the weak universality scenario.

arXiv:cond-mat/9706202•Physical review. B, Condensed matter•Sep 1, 1997•42 citations
We study the time evolution of the two-dimensional kinetic Ising model in finite systems with a non-conserved order parameter, considering nearest-neighbour interactions on the square lattice with periodic and open boundary conditions. Universal data collapse in spin product correlation functions is observed which, when expressed in rescaled units, is valid across the entire time evolution of the system at all length scales, not just within the time regime usually considered in the dynamical scaling hypothesis. Consequently, beyond rapidly decaying finite size effects, the evolution of correlations in small finite systems parallels arbitrarily larger cases, even at large fractions of the size of these finite systems.
Abstract We study the early time dynamics of the 2 d ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. We evolve the system with kinetic Monte Carlo rules that do not conserve the order parameter. We confirm the rapid approach to random critical percolation in a time-scale that diverges with the system size but is much shorter than the equilibration time. We study the scaling properties of the evolution towards critical percolation and we identify an associated growing length, different from the curvature driven one. By working with the model defined on square, triangular and honeycomb microscopic geometries we establish the dependence of this growing length on the lattice coordination. We discuss the interplay with the usual coarsening mechanism and the eventual fall into and escape from metastability.
The correlation function of two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor representation are obtained both for the ferromagnetic and paramagnetic regions of coupling constant. The peculiarities caused by the finite size of the system are discussed. The asymptotic behaviour of the corresponding quantities as functions of size is calculated. Some generalization of the obtained result is conjectured.
We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the 2d ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value θ that depends upon the initial condition. More precisely, we find that θ takes one universal value 0.199(2) for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value 0.033(1) for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with a square and a triangular lattice, and by imposing free and periodic boundary conditions. We found that the effective exponent suffers from stronger finite size effects in the former case.
We study, using Monte Carlo dynamics, the time ( t ) dependent average magnetization per spin m ( t ) behavior of the 2D kinetic Ising model under a binary (± h 0 ) stochastic field h ( t ). The time dependence of the stochastic field is such that its average over each successive time interval τ is assured to be zero (without any fluctuation). The average magnetization is considered as an order parameter of the system. The phase diagram in ( h 0 ,τ) plane is obtained. Fluctuations in the order parameter and their scaling properties are studied across the phase boundary. These studies indicate that the nature of the transition is Ising like (static Ising universality class) for field amplitudes h 0 below some threshold value (dependent on τ values; as τ → ∞ across the phase boundary). Beyond these , the transition is no longer continuous.
An importance-sampling Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N$ Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of NNN and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.
We address the question of weak versus strong universality scenarios for the random-bond Ising model in two dimensions. A finite-size scaling theory is proposed, which explicitly incorporates lnL\mathrm{ln} LlnL corrections (L(L(L is the linear finite size of the system) to the temperature derivative of the correlation length. The predictions are tested by considering long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings, along which free energy, spin-spin correlation functions, and specific heats are calculated by transfer-matrix methods. The ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is calculated and has the same value as in the pure case; consequently conformal invariance predictions remain valid for this type of disorder. Semilogarithmic plots of correlation functions against distance yield average correlation lengths ${\ensuremath{\xi}}^{\mathrm{av}}$, whose size dependence agrees very well with the proposed theory. We also examine the size dependence of the specific heat, which clearly suggests a divergency in the thermodynamic limit. Thus our data consistently favor the Dotsenko-Shalaev picture of logarithmic corrections (enhancements) to pure system singularities, as opposed to the weak universality scenario.