For ageing systems, quenched onto a critical temperature
T=Tc such that the dominant noise comes from the thermal bath, with a non-conserved order-parameter and in addition with dynamical exponent
z=2, the form of the two-time auto-correlator as well as the time-space form of the single-time correlator are derived from Schr\"odinger-invariance, generalised to non-equilibrium ageing. These findings reproduce the exact results in the
1D Glauber-Ising model at
T=0 and the critical spherical model in
d>2 dimensions.
Abstract For ageing systems, quenched onto a critical temperature T = T c such that the dominant noise comes from the thermal bath, with a non-conserved order-parameter and in addition with dynamical exponent z = 2 , the form of the two-time autocorrelator as well as the time-space form of the single-time correlator are derived from Schrödinger-invariance, generalised to non-equilibrium ageing. These findings reproduce the exact results in the 1 D Glauber–Ising model at T = 0 and the critical spherical model in d > 2 dimensions.
We obtain the exact critical relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi}), where \ensuremath{\xi} is the bulk correlation length, for the Glauber kinetic Ising model of spins on a one-dimensional lattice of finite length L for both periodic and free boundary conditions (BC's). We show that, independent of the BC's, the dynamic critical exponent has the well-known value z=2, and we comment on a recent claim that z=1 for this model. The ratio ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi})/${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\infty}}}$(\ensuremath{\xi}), in the double limit L,\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty} for fixed x=L/\ensuremath{\xi}, approaches a limiting functional form, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(L/\ensuremath{\xi}), the finite-size scaling function. For free BC's we derive the exact scaling function ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\omega}(x)/x
)2${]}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\omega}(x) is the smallest root of the transcendental equation \ensuremath{\omega}tan(\ensuremath{\omega}/2)=x. We provide expansions of \ensuremath{\omega}(x) in powers of x and ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ for the regimes of small and large x, respectively, and establish their radii of convergence. The scaling function shows anomalous behavior at small x, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}x, instead of the usual ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}
xz, as x\ensuremath{\rightarrow}0. This is because, even for finite L, the lifetime of the slowest dynamical mode diverges for T\ensuremath{\rightarrow} 0 K. For periodic BC's, with the exception of one system, ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$ is independent of L, and hence ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$=1. The exceptional system, that with an odd number of spins and antiferromagnetic couplings, exhibits frustration at T=0 K, and the scaling function is given by ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\pi}/x
)2${]}^{\mathrm{\ensuremath{-}}1}$. \textcopyright{} 1996 The American Physical Society.
Systems undergoing phase-ordering kinetics after a quench into the ordered phase with 0 < T < T c from a fully disordered initial state and with a non-conserved order-parameter have the dynamical exponent z = 2 . The long-time behaviour of their single-time and two-time correlators, determined by the noisy initial conditions, is derived from Schrödinger-invariance. It is shown that the generic ageing scaling forms of the single-time and two-time correlators follow from the Schrödinger-covariance of the four-point response functions. The autocorrelation exponent λ is related to the passage exponent ζ p which describes the time-scale for the cross-over into the ageing regime. Both Porod’s law and the bounds d / 2 ≤ λ ≤ d are reproduced in a simple way. The dynamical finite-size scaling in fully finite systems is established. The dynamical scaling of global correlators are found and the low-temperature generalisation λ = d − 2 Θ of the Janssen-Schaub-Schmittmann scaling relation is derived.
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.