Kinetics of the two-dimensional long-range Ising model at low temperatures

Agrawal, Ramgopal; Corberi, Federico; Lippiello, Eugenio; Politi, Paolo; Puri, Sanjay

doi:10.1103/physreve.103.012108

Cited by 15 publications

(11 citation statements)

References 31 publications

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“…It should also be stressed that our result τ α (N ) ∼ N α is found as well at finite temperature for 1 < α ≤ 2, both in one dimension [14] and in higher dimension [23][24][25][26]. Based on our current understanding this seems a coincidence for a couple of reasons.…”

Section: Stability Of Individual Domains and Their Evolutionsupporting

confidence: 73%

“…Irrelevant means that the overall qualitative behavior is the same for all T < T c and universal quantities such as exponent do not depend on T . Instead, with algebraic interactions and α > 1 there is a difference between quenches to T = 0 and to 0 < T < T c , both in d = 1 [13,14] and in d = 2 [24,30]. The situation in the present 1d case with SLR is shown in the inset of Fig.…”

Section: Discussionmentioning

confidence: 71%

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Coexistence of coarsening and mean field relaxation in the long-range Ising chain

et al. 2021

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We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance rr decaying as r^{-\alpha}r−α. For \alpha =0α=0, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with \alpha >1α>1 there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. 0<\alpha <10<α<1, we show that the system shows both features, with probability P_\alpha (N)Pα(N) of having the latter one, with the different limiting behaviours \lim _{N\to \infty}P_\alpha (N)=0limN→∞Pα(N)=0 (at fixed \alpha<1α<1) and \lim _{\alpha \to 1}P_\alpha (N)=1limα→1Pα(N)=1 (at fixed finite NN). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time \tau _\alpha (N)\sim N^\alphaτα(N)∼Nα.

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Section: Stability Of Individual Domains and Their Evolutionsupporting

confidence: 73%

Section: Discussionmentioning

confidence: 71%

Coexistence of coarsening and mean field relaxation in the long-range Ising chain

et al. 2021

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“…According to the Bray-Rutenberg predictions [29,30], for a quench to non-zero T , the model (1) has growth law R(t) ∼ t 1/z of domains, where the exponent z = 2 for σ > 1 and z = 1 + σ for σ ≤ 1, with logarithmic corrections at σ = 1. Right at T = 0 a different exponent z = 4/3 has been observed for any σ [20,21]. For low but finite T a crossover from this behavior to the Bray-Rutenberg law is observed [20].…”

mentioning

confidence: 82%

“…For σ > 0, instead, in the socalled weak long-range regime, one observes a coarsening phenomenon qualitatively similar to the one of shortrange systems. Despite that, it is reported in recent works [20,21] that the striped metastable states (typical of those systems) are absent or significantly suppressed in the presence of long-range interactions. This can be understood at a qualitative level because the extended interaction allows single spins to probe the local magnetization far away, and this may provide a global drift to their update, even if not as efficient as in the mean-field.…”

mentioning

confidence: 94%

“…Initially, the system is in equilibrium at infinite temperature, which is suddenly quenched to T < T c (σ) at the time t = 0 and afterward evolved at this temperature with the above transition rates. The Ewald summation technique [20,27,28] is used to handle the pairwise long-range interaction between any two spins on the periodic lattice. To assure the accuracy of data, the number of initial configurations used for each L is typically 10 4 − 10 6 .…”

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confidence: 99%

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Asymptotic States of Ising Ferromagnets with Long-range Interactions

Agrawal,

Corberi,

Insalata

et al. 2021

Preprint

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It is known that, after a quench to zero temperature (T = 0), two-dimensional (d = 2) Ising ferromagnets with short-range interactions do not always relax to the ordered state. They can also fall in infinitely long-lived striped metastable states with a finite probability. In this paper, we study how the abundance of striped states is affected by long-range interactions. We investigate the relaxation of d = 2 Ising ferromagnets with power-law interactions by means of Monte Carlo simulations at both T = 0 and T = 0. For T = 0 and finite system size, the striped metastable states are suppressed by long-range interactions. In the thermodynamic limit, their occurrence probabilities are consistent with the short-range case. For T = 0, the final state is always ordered. Further, the equilibration occurs at earlier times with an increase in the strength of the interactions.

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Aging properties of the voter model with long-range interactions

Corberi,

Smaldone

2024

J. Stat. Mech.

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We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, S i = ± 1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P ( r ) ∝ r − α . Employing both analytical and numerical methods, we compute the two-time correlation function G ( r ; t , s ) ( t ⩾ s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A ( t , s ) = G ( r = 0 ; t , s ) , decays algebraically for α > 1 as [ L ( t ) / L ( s ) ] − λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ = 1 / ( α − 1 ) for 1 < α ⩽ 2 . For α ⩽ 1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G ( r ; t , s ) obeys a scaling symmetry G ( r ; t , s ) = g [ r / L ( s ) ; L ( t ) / L ( s ) ] for α > 2. Similarly, for 1 < α ⩽ 2 , one has G ( r ; t , s ) = g [ r / L ( t ) ; L ( t ) / L ( s ) ] , where the length L regulating two-time correlations now differs from the coherence length as L ∝ L δ , with δ = 1 + 2 ( 2 − α ) .

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Kinetics of the two-dimensional long-range Ising model at low temperatures

Cited by 15 publications

References 31 publications

Coexistence of coarsening and mean field relaxation in the long-range Ising chain

Coexistence of coarsening and mean field relaxation in the long-range Ising chain

Asymptotic States of Ising Ferromagnets with Long-range Interactions

Aging properties of the voter model with long-range interactions

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