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Journal of Physics A Mathematical and General

Universal scaling function for domain growth in the Glauber-Ising chain

Alan J. Bray

Published Jan 21, 1990• 77 citations
Publisher

Abstract

The equal-time correlation function is calculated at T=0 for the one-dimensional Ising model with Glauber dynamics. Random initial conditions, appropriate to a sudden quench from non-zero temperature, are imposed. Averaging over initial conditions yields the scaling form C(r, t)=f(r 2 /t). The scaling function is given by f(x)= integral 0 1 (dy/ pi )(y(1-y)) -1/2 exp(-x/4y), and is universal, i.e. independent of the probability distribution for the initial conditions.

DOI
10.1088/0305-4470/23/2/005

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The equal-time correlation function is calculated at T=0 for the one-dimensional Ising model with Glauber dynamics. Random initial conditions, appropriate to a sudden quench from non-zero temperature, are imposed. Averaging over initial conditions yields the scaling form C(r, t)=f(r 2 /t). The scaling function is given by f(x)= integral 0 1 (dy/ pi )(y(1-y)) -1/2 exp(-x/4y), and is universal, i.e. independent of the probability distribution for the initial conditions.
The equal-time correlation function for an Ising chain with alternating ferromagnetic coupling constants (J1{\mathit{J}}_{1}J1​>J2{\mathit{J}}_{2}J2​) and Glauber dynamics has been calculated. Averaging over random initial conditions shows that the correlation function is independent of the probability distribution for the initial conditions, but depends on the nontrivial combination of different coupling constants, which casts some doubt as to the validity of the scaling form C(r,t)=f(r2{\mathit{r}}^{2}r2/t). Furthermore, by calculating the spin-spin correlations for different times, we find that the power-law decay of the autocorrelation function has the same scaling form as in the simple uniform case C(0,t,t')\ensuremath{\sim}(2/\ensuremath{\pi})(2t/t')1/2{)}^{1/2})1/2, for t'\ensuremath{\gg}t. This may suggest that a different exponent, instead of the dynamic critical exponent z=1+J1{\mathit{J}}_{1}J1​/J2{\mathit{J}}_{2}J2​, is needed to characterize the scaling of the autocorrelation function.
The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T f =0 and T f =T c . The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t 1 2/, for T f =0, and the 'nonequilibrium correlation length', xi (t) approximately t 1 z/, for T f =T c . The equal-time correlation function has the expected scaling forms f(r/L(t)) and r -(d-2+ eta ) f c (r/ xi (t)) for T f =0 and T c respectively. The scaling function f c (x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S i (t))S i (O)) approximately L(t) - . Results for a quench from the equilibrium critical state to T f =0 are consistent with theoretical predictions.
Abstract This work is devoted to an in-depth analysis of arbitrary temperature protocols applied to the ferromagnetic Glauber-Ising chain launched from a disordered initial state and evolving in the low-temperature scaling regime. We focus our study on the density of domain walls and the reduced susceptibility. Both the inverse of the former observable and the latter one provide two independent measures of the typical size of the growing ferromagnetic domains. Their product is thus a dimensionless form factor characterising the pattern of growing ordered domains and providing a measure of the distance of the system to thermal equilibrium. We apply this framework to a variety of protocols: everlasting slow quenches, where temperature decreases continuously to zero in the limit of infinitely long times, slow quenches of finite duration, where temperature reaches zero at some long but finite quenching time, time-periodic protocols with weak and strong modulations, and the two-temperature protocol leading to the memory effect found by Kovacs.
We consider the one-dimensional random-field Ising model, where the spin-spin coupling J is ferromagnetic and the external field is chosen to be +h with probability p and -h with probability 1-p. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function 〈s0{\mathit{s}}_{0}s0​sn{\mathit{s}}_{\mathit{n}}sn​〉-〈s0{\mathit{s}}_{0}s0​〉〈sn{\mathit{s}}_{\mathit{n}}sn​〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈s0{\mathit{s}}_{0}s0​sn{\mathit{s}}_{\mathit{n}}sn​〉.