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arXiv:cond-mat/0302483 · International Journal of Modern Physics A

AGEING, DYNAMICAL SCALING AND CONFORMAL INVARIANCE

Malte Henkel

Published Jun 10, 2004• 1 citation
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Abstract

Building on an analogy with conformal invariance, local scale transformations consistent with dynamical scaling are constructed. Two types of local scale invariance are found which act as dynamical space–time symmetries of certain nonlocal free field theories. The scaling form of two-point functions is completely fixed by the requirement of local scale invariance. These predictions are confirmed through tests in the 3D ANNNI model at its Lifshitz point and in ageing phenomena of simple ferromagnets, here studied through the kinetic Ising model with Glauber dynamics.

DOI
10.1142/s0217751x04018440

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Building on an analogy with conformal invariance, local scale transformations consistent with dynamical scaling are constructed. Two types of local scale invariance are found which act as dynamical space–time symmetries of certain nonlocal free field theories. The scaling form of two-point functions is completely fixed by the requirement of local scale invariance. These predictions are confirmed through tests in the 3D ANNNI model at its Lifshitz point and in ageing phenomena of simple ferromagnets, here studied through the kinetic Ising model with Glauber dynamics.
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 zzz is described and the two distinct types of local scale invariance are presented. The special case z=2z=2z=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.
In situations such as phase ordering or non-equilibrium critical dynamics, the two-time autoresponse function scales as R(t,s)approxs−1−aF2(t/s)R(t,s)\\approx s^{-1-a} F_2(t/s)R(t,s)approxs−1−aF2​(t/s), where aaa is a known exponent. For any value of the dynamical exponent zzz, we argue that this dynamical scaling can be extended towards conformal invariance. Thus the scaling function F2F_2F2​ is determined. This prediction is confirmed by several examples, both for T<TcT<T_cT<Tc​ and at T=TcT=T_cT=Tc​: 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.
Local scale-invariance for ageing systems without detailed balance is tested through studying the dynamical symmetries of the critical bosonic contact process and the critical bosonic pair-contact process.Their field-theoretical actions can be split into a Schrödinger-invariant term and a pure noise term. It is shown that the two-time response and correlation functions are reducible to certain multipoint response functions which depend only on the Schrödinger-invariant part of the action. For the bosonic contact process, the representation of the Schrödinger group can be derived from the free diffusion equation, whereas for the bosonic pair-contact process, a new representation of the Schrödinger group related to a non-linear Schrödinger equation with dimensionful couplings is constructed. The resulting predictions of local scale-invariance for the two-time responses and correlators are completely consistent with the exactly-known results in both models.
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 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.