The nonequilibrium response of a two-dimensional, paramagnetic system to a rapid and arbitrarily large change (quench) in a symmetry-breaking field is investigated. The complete time dependence of the conjugate order parameter is calculated with a real-space renormalization-group (RG) method. The nonlinear relaxation "rate" is found to be time dependent. Its approach to the corresponding prediction of linear-response theory is characterized. The nonlinear response is found to go over to linear response via an early, exponential regime followed by a crossover to an asymptotic, algebraic approach to linear response at long times. The initial transient decay is quench-strength dependent in contrast with the final behavior, which is largely "universal" and quench insensitive. We study a nearest-neighbor Ising model in a magnetic field in the paramagnetic, "disordered" phase. The dynamics are controlled by a kinetic Ising model with simultaneous spin-flip (adsorption-desorption) and spin-exchange (diffusion) kinetics. The model is applicable to non-equilibrium "dosing" experiments on adsorbed systems. Diffusion is shown not to affect the growth rate of the total order parameter (magnetization or coverage). General recursion relations for the temperature and magnetic field are derived for the entire thermodynamic plane. The RG analysis reproduces linear-response theory for arbitrary magnetic field and is consistent with known scaling laws concerning nonlinear relaxation. The time-dependent magnetization in the critical region has a power-law regime before crossing over into its final exponential decay.
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 consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time
teq governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue
E1=1/teq of the quantum Hamiltonian
H. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time
teq can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in
d=1 is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio
K, we compute the exponential growth of
teq(N) with the number
N of generations.
Based on the obtained exact results we systematically study the quench dynamics of a one-dimensional spin-1/2 transverse field Ising model with zero- and finite-temperature initial states. We focus on the magnetization of the system after a sudden change of the external field and a coherent time-evolution process. With a zero-temperature initial state, the quench magnetic susceptibility as a function of the initial field strength exhibits strongly similar scaling behaviors to those of the static magnetic susceptibility, and the quench magnetic susceptibility as a function of the final field strength shows a discontinuity at the quantum critical point. This discontinuity remains robust and always occurs at the quantum critical point even for the case of finite-temperature initial systems, which indicates a great advantage of employing quench dynamics to study quantum phase transitions.
We consider the nonequilibrium, elastic-scattering structure factor S(q,t) (q denotes the wave vector, t the time), for the Kawasaki spin-conserving kinetic Ising model of a one-dimensional system with nearest-neighbor interactions, initially in equilibrium at temperature
TI, that is suddenly placed in contact with a heat bath at temperature
TF, with
TF\ensuremath{\gg}
TI. We present detailed results for the case of
TF=\ensuremath{\infty}, for which we have succeeded in calculating the exact form of S(q,t). For finite
TF, we present an approximation scheme for the higher-order nonequilibrium correlation functions that leads to closure of the hierarchy of equations of motion. The merits of this approximation are that (i) S(q,t) is guaranteed to satisfy an exact sum rule over the Brillouin zone (BZ) of wave vectors q, and (ii) S(q,t) evolves to the correct value in the long-time limit. For antiferromagnetic coupling, the structure factor, initially dominated by the Bragg peak associated with
TI at the edge of the BZ, decays exponentially with time, ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{t}/{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}}$ while approximately preserving its shape in q space, since the lifetime ${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$ is nearly independent of q. Except near the center of the BZ, after the Bragg peak has decayed sufficiently, the dependence of S(q,t) on q can be characterized as though the spins rapidly quasiequilibrate to the equilibrium structure factor associated with
TF, \ensuremath{\chi}(q,
TF), in that S(q,t)/\ensuremath{\chi}(q,
TF) is independent of q, but is time dependent, slowly approaching unity as ${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$ for large t. For q\ensuremath{\approxeq}0 the initial form of S remains in effect until the value of t is of order ${\mathit{q}}^{\mathrm{\ensuremath{-}}2}$. For ferromagnetic coupling, the initial Bragg peak for q\ensuremath{\approxeq}0 does not preserve its shape while decaying exponentially, since the lifetime ${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$ strongly depends on the wave-vector q, diverging as ${\mathit{q}}^{\mathrm{\ensuremath{-}}2}$ for q\ensuremath{\rightarrow}0, and, in particular, it is as though the spins for q\ensuremath{\approxeq}0 remain ``frozen'' at
TI. Analogous to the behavior for antiferromagnetic interactions, away from the center of the BZ, we find that S(q,t)/\ensuremath{\chi}(q,
TF) is independent of q and is a function of t/
tw, very slowly approaching unity. The characteristic ``waiting time''
tw is anomalously long, proportional to ${\ensuremath{\xi}}^{2}$, where \ensuremath{\xi} is the equilibrium correlation length at temperature
TI. This behavior of
tw can be related to the random walk of domain boundaries. \textcopyright{} 1996 The American Physical Society.