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arXiv:cond-mat/0703226 · Journal of Statistical Mechanics Theory and Experiment

Autocorrelation functions in phase-ordering kinetics from local scale invariance

Malte Henkel, Florian Baumann

Published Jul 12, 2007• 9 citations

Abstract

The explicit calculation of the scaling form of the two-time autocorrelation function in phase-ordering kinetics and in those cases of non-equilibrium critical dynamics where the dynamical exponent z = 2 through the extension of dynamical scaling to local scale invariance is reviewed. Conceptually, this mainly requires an extension from the usually considered d -dimensional ageing or Schrödinger algebras to a new kind of representation of the conformal algebra in d + 2 dimensions. Explicit tests in several exactly solved models of simple magnets and through simulations in the 2D Ising and q -states Potts models ( q = 2, 3, 8) quenched to T < T c are presented and the extension to systems with non-equilibrium steady states is discussed through two exactly solvable models as well. In the context of surface growth models, possible generalizations for a dynamical exponent z = 4 and beyond are discussed.

DOI: 10.1088/1742-5468/2007/07/p07015

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The explicit calculation of the scaling form of the two-time autocorrelation function in phase-ordering kinetics and in those cases of non-equilibrium critical dynamics where the dynamical exponent z = 2 through the extension of dynamical scaling to local scale invariance is reviewed. Conceptually, this mainly requires an extension from the usually considered d -dimensional ageing or Schrödinger algebras to a new kind of representation of the conformal algebra in d + 2 dimensions. Explicit tests in several exactly solved models of simple magnets and through simulations in the 2D Ising and q -states Potts models ( q = 2, 3, 8) quenched to T < T c are presented and the extension to systems with non-equilibrium steady states is discussed through two exactly solvable models as well. In the context of surface growth models, possible generalizations for a dynamical exponent z = 4 and beyond are discussed.

The time-dependent scaling of the two-time autocorrelation function of spin systems without disorder undergoing phase-ordering kinetics is considered. Its form is shown to be determined by an extension of dynamical scaling to a local scale invariance which turns out to be a new version of conformal invariance. The predicted autocorrelator is in agreement with Monte Carlo data on the autocorrelation function of the 2D kinetic Ising model with Glauber dynamics quenched to a temperature below criticality.

The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent

{z}=2

are predicted from a new time-dependent non-equilibrium representation of the Schrödinger algebra. These explicit predictions are tested and confirmed in the ageing of several exactly solvable models.

In situations such as phase ordering or non-equilibrium critical dynamics, the two-time autoresponse function scales as

R(t,s)\\approx s^{-1-a} F_2(t/s)

, where

a

is a known exponent. For any value of the dynamical exponent

z

, we argue that this dynamical scaling can be extended towards conformal invariance. Thus the scaling function

F_2

is determined. This prediction is confirmed by several examples, both for

T<T_c

and at

T=T_c

: the two- and three-dimensional kinetic Ising model with Glauber dynamics is studied numerically, while exact results are available for the kinetic spherical model with a non-conserved order parameter, both for short-ranged and long-ranged interactions, as well as for the mean-field spherical spin glass.

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 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.

The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent

z

is described and the two distinct types of local scale invariance are presented. The special case

z=2

and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.