Slow quench dynamics in classical systems: kinetic Ising model and zero-range process

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.

Distribution of Kinks in an Ising Ferromagnet After Annealing and the Generalized Kibble-Zurek Mechanism

“…The nonlinear passage across a critical point has been proposed to suppress the mean number of defects generated in a phase transition, as it yields a power-law dependence on the quench time with a tunable exponent [21][22][23]. This feature is also found in the finite-time cooling of an Ising ferromagnet under Glauber dynamics [10,12,13] and we next study its effect on the distribution of kinks and the cumulant scaling. To this end, we consider the algebraic cooling schedule…”

“…(up to a logarithmic factor in τ Q ) where C is a constant dependent on the cooling schedule specifics, τ Q is the time taken in total for the temperature to pass from an effectively infinite value to T = 0 and the power-law exponent δ is set by the dynamic critical exponent z, the cooling schedule, and system dimensionality; see as well [12,13].…”

“…Using this result, together with those derived in the preceding section, we next describe the kink statistics resulting from different cooling schedules. Specifically, under Glauber dynamics the flipping rate is dictated by the parameter γ = tanh(2βJ) and we shall consider different functional forms for the variation of this parameter in time [10][11][12][13].…”

“…Specifically, a universal power-law scaling is predicted when the control parameter is driven linearly in time. The finite-time cooling of an Ising ferromagnet has been used as a paradigmatic testbed to explore KZM physics [9][10][11][12][13], which provides useful heuristics in adiabatic quantum optimization and quantum annealing [14][15][16][17][18]. Generalizations of KZM have been established that account for disorder [19,20], nonlinear driving protocols [21][22][23] as well as inhomogeneous systems [23][24][25][26][27][28][29][30][31][32][33], see Refs.…”

“…Specifically, we consider the onedimensional Ising model with no magnetic field and evolving under Glauber dynamics. The mean number of defects in this setting has been studied by Krapivsky [10], see as well [12,13] for related work.…”