Some New Results on the 2D Stochastic Ising Model in the Phase Coexistence Region (original) (raw)
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Physical Review E, 2008
We study the dynamical response of a two-dimensional Ising model subject to a square-wave oscillating external field. In contrast to earlier studies, the system evolves under a so-called soft Glauber dynamic [P. A. Rikvold and M. Kolesik, J. Phys. A: Math. Gen. 35, L117 (2002)], for which both nucleation and interface propagation are slower and the interfaces smoother than for the standard Glauber dynamic. We choose the temperature and magnitude of the external field such that the metastable decay of the system following field reversal occurs through nucleation and growth of many droplets of the stable phase, i.e., the multidroplet regime. Using kinetic Monte Carlo simulations, we find that the system undergoes a nonequilibrium phase transition, in which the symmetry-broken dynamic phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. The critical point is located where the half-period of the external field is approximately equal to the metastable lifetime of the system. We employ finite-size scaling analysis to investigate the characteristics of this dynamical phase transition. The critical exponents and the fixed-point value of the fourth-order cumulant are found to be consistent with the universality class of the two-dimensional equilibrium Ising model. As this universality class has previously been established for the same nonequilibrium model evolving under the standard Glauber dynamic, our results indicate that this far-from-equilibrium phase transition is universal with respect to the choice of the stochastic dynamics.
Relaxation to equilibrium for two dimensional disordered Ising systems in the Griffiths phase
Communications in Mathematical Physics, 1997
We consider Glauber-type dynamics for two dimensional disordered magnets of Ising type. We prove that, if the disorder-averaged influence of the boundary condition is sufficiently small in the equilibrium system, then the corresponding Glauber dynamics is ergodic with probability one and the disorder-average C(t) of time-autocorrelation function satisfies C(t) e −m(log t) 2 (for large t). For the standard two dimensional dilute Ising ferromagnet with i.i.d. random nearest neighbor couplings taking the values 0 or J 0 > 0, our results apply even if the active bonds percolate and J 0 is larger than the critical value J c of the corresponding pure Ising model. For the same model we also prove that in the whole Griffiths' phase the previous upper bound is optimal. This implies the existence of a dynamical phase transition which occurs when J crosses J c .
Noise-driven dynamic phase transition in a one-dimensional Ising-like model
Physical Review E, 2010
The dynamical evolution of a recently introduced one dimensional model in (henceforth referred to as model I), has been made stochastic by introducing a parameter β such that β = 0 corresponds to the Ising model and β → ∞ to the original model I. The equilibrium behaviour for any value of β is identical: a homogeneous state. We argue, from the behaviour of the dynamical exponent z, that for any β = 0, the system belongs to the dynamical class of model I indicating a dynamic phase transition at β = 0. On the other hand, the persistence probabilities in a system of L spins saturate at a value Psat(β, L) = (β/L) α f (β), where α remains constant for all β = 0 supporting the existence of the dynamic phase transition at β = 0. The scaling function f (β) shows a crossover behaviour with f (β) = constant for β << 1 and f (β) ∝ β -α for β >> 1.
Metastability in the Two-Dimensional Ising Model with Free Boundary Conditions
Journal of Statistical Physics, 1998
We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime
Phase transitions and autocorrelation times in two-dimensional Ising model with dipole interactions
Physica B: Condensed Matter, 2010
The two-dimensional Ising model with nearest-neighbor ferromagnetic and long-range dipolar interactions exhibits a rich phase diagram. The presence of the dipolar interaction changes the ferromagnetic ground state expected for the pure Ising model to a series of striped phases as a function of the interaction strengths. Monte Carlo simulations and histogram reweighting techniques applied to multiple histograms are performed to identify the critical temperatures for the phase transitions taking place for stripes of width h = 2 on square lattices. In particular, we aim to study the intermediate nematic phase, which is observed for large lattice sizes only. For these lattice sizes, we calculate the critical temperatures for the striped-nematic and nematic-tetragonal transitions, critical exponents, and the bulk free-energy barrier associated with the coexisting phases. We also evaluate the long-term correlations in our time series near the finite-size critical points by studying the integrated autocorrelation time τ as a function of the lattice size. This allows us to infer how severe the critical slowing down for this system with long-range interaction and nearby thermodynamic phase transitions is.
Ising Transition Driven by Frustration in a 2D Classical Model with Continuous Symmetry
Physical Review Letters, 2003
We study the thermal properties of the classical antiferromagnetic Heisenberg model with both nearest (J 1 ) and next-nearest (J 2 ) exchange couplings on the square lattice by extensive Monte Carlo simulations. We show that, for J 2 =J 1 > 1=2, thermal fluctuations give rise to an effective Z 2 symmetry leading to a finite-temperature phase transition. We provide strong numerical evidence that this transition is in the 2D Ising universality class, and that T c ! 0 with an infinite slope when J 2 =J 1 ! 1=2.
Long-time behavior for the 1-D stochastic Ising model with unbounded random couplings
2003
We consider the ferromagnetic Ising model with Glauber spin flip dynamics in one dimension. The external magnetic field vanishes and the couplings are i.i.d. random variables. If their distribution has compact support, the disorder averaged spin auto-correlation function has an exponential decay in time. We prove that, if the couplings are unbounded, the decay switches to either a power law or a stretched exponential, in general.
Phase Transition in 2D Ising Model with Next-Neighbor Interaction
Journal of physics, 2019
We used a Monte Carlo method to analyze a planar Ising model taking into account a ferromagnetic interaction with next neighbors. We found that independent of the value of the additional interaction, the character of singularity of the heat capacity did not change. Namely, at the critical point the heat capacity has a logarithmic singularity. We obtained a semitheoretical formula relating the value of the critical temperature with the value of the additional ferromagnetic interaction.
Two dimensional kicked quantum Ising model: dynamical phase transitions
New Journal of Physics, 2014
Using an efficient one and two qubit gate simulator, operating on graphical processing units, we investigate ergodic properties of a quantum Ising spin 1/2 model on a two dimensional lattice, which is periodically driven by a δ-pulsed transverse magnetic field. We consider three different dynamical properties: (i) level density and (ii) level spacing distribution of the Floquet quasienergy spectrum, as well as (iii) time-averaged autocorrelation function of components of the magnetization. Varying the parameters of the model, we found transitions between ordered (non ergodic) and quantum chaotic (ergodic) phases, but the transitions between flat and non-flat spectral density do not correspond to transitions between ergodic and non-ergodic local observables. Even more surprisingly, we found nice agreement of level spacing distribution with the Wigner surmise of random matrix theory for almost all values of parameters except where the model is essentially noninteracting, even in the regions where local observables are not ergodic or where spectral density is non-flat. These findings put in question the versatility of the interpretation of level spacing distribution in many-body systems and stress the importance of the concept of locality.
Universality in the time correlations of the long-range 1d Ising model
Journal of Statistical Mechanics: Theory and Experiment, 2019
The equilibrium and nonequilibrium properties of ferromagnetic systems may be affected by the long-range nature of the coupling interaction. Here we study the phase separation process of a one-dimensional Ising model in the presence of a power-law decaying coupling, J(r) = 1/r 1+σ with σ > 0, and we focus on the two-time autocorrelation function C(t, tw) = s i (t)s i (tw). We find that it obeys the scaling form C(t, tw) = f (L(tw)/L(t)), where L(t) is the typical domain size at time t, and where f (x) can only be of two types. For σ > 1, when domain walls diffuse freely, f (x) falls in the nearest-neighbour (nn) universality class. Conversely, for σ ≤ 1, when domain walls dynamics is driven, f (x) displays a new universal behavior. In particular, the so-called Fisher-Huse exponent, which characterizes the asymptotic behavior of f (x) ≃ x −λ for x ≫ 1, is λ = 1 in the nn universality class (σ > 1) and λ = 1/2 for σ ≤ 1.