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arXiv:1506.01098 · Physical Review E

Interrupted coarsening in the zero-temperature kinetic Ising chain driven by a periodic external field

Su Do Yi, Seung Ki Baek

Published Jun 5, 2015• 2 citations
arXivHTMLPDF

Abstract

If quenched to zero temperature, the one-dimensional Ising spin chain undergoes coarsening, whereby the density of domain walls decays algebraically in time. We show that this coarsening process can be interrupted by exerting a rapidly oscillating periodic field with enough strength to compete with the spin-spin interaction. By analyzing correlation functions and the distribution of domain lengths both analytically and numerically, we observe nontrivial correlation with more than one length scale at the threshold field strength.

DOI
10.1103/physreve.91.062107

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If quenched to zero temperature, the one-dimensional Ising spin chain undergoes coarsening, whereby the density of domain walls decays algebraically in time. We show that this coarsening process can be interrupted by exerting a rapidly oscillating periodic field with enough strength to compete with the spin-spin interaction. By analyzing correlation functions and the distribution of domain lengths both analytically and numerically, we observe nontrivial correlation with more than one length scale at the threshold field strength.
We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance r r decaying as r^{-\alpha} r − α . For \alpha =0 α = 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 >1 α > 1 there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. 0<\alpha <1 0 < α < 1 , we show that the system shows both features, with probability P_\alpha (N) P α ( N ) of having the latter one, with the different limiting behaviours \lim _{N\to \infty}P_\alpha (N)=0 lim N → ∞ P α ( N ) = 0 (at fixed \alpha<1 α < 1 ) and \lim _{\alpha \to 1}P_\alpha (N)=1 lim α → 1 P α ( N ) = 1 (at fixed finite N N ). 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^\alpha τ α ( N ) ∼ N α .
Abstract While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.
We investigate the zero-temperature coarsening dynamics of a chain of Ising spins with a nearest-neighbour ferromagnetic and an nth-neighbour antiferromagnetic interactions. For sufficiently large antiferromagnetic interaction, the ground state consists of n consecutive up spins followed by n down spins, etc. We show that the asymptotic coarsening into this ground state can be described in terms of a multispecies reactive gas of elementary excitations. The basic elementary excitations are identified and each decays at a different power-law rate in time. The dominant excitations are domains of n + 1 spins which diffuse freely and disappear through processes which are effectively governed by (n + 1)-particle annihilation. This leads to a slow temporal approach to the ground state.
The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.
We investigate the zero-temperature coarsening dynamics of a chain of Ising spins with a nearest-neighbour ferromagnetic and an n th-neighbour antiferromagnetic interactions. For sufficiently large antiferromagnetic interaction, the ground state consists of n consecutive up spins followed by n down spins, etc. We show that the asymptotic coarsening into this ground state can be described in terms of a multispecies reactive gas of elementary excitations. The basic elementary excitations are identified and each decays at a different power-law rate in time. The dominant excitations are domains of n + 1 spins which diffuse freely and disappear through processes which are effectively governed by ( n + 1)-particle annihilation. This leads to a slow temporal approach to the ground state.