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arXiv:cond-mat/0605211 · Journal of Physics A Mathematical and General

On the identification of quasiprimary scaling operators in local scale-invariance

Malte Henkel, Tilman Enss, Michel Pleimling

Published Oct 4, 2006• 40 citations
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Abstract

The relationship between physical observables defined in lattice models and the associated (quasi-)primary scaling operators of the underlying field theory is revisited. In the context of local scale-invariance, we argue that this relationship is only defined up to a time-dependent amplitude and derive the corresponding generalizations of predictions for two-time response and correlation functions. Applications to non-equilibrium critical dynamics of several systems, with a fully disordered initial state and vanishing initial magnetization, including the Glauber–Ising model, the Frederikson–Andersen model and the Ising spin glass are discussed. The critical contact process and the parity-conserving non-equilibrium kinetic Ising model are also considered.

DOI
10.1088/0305-4470/39/42/l01

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The relationship between physical observables defined in lattice models and the associated (quasi-)primary scaling operators of the underlying field theory is revisited. In the context of local scale-invariance, we argue that this relationship is only defined up to a time-dependent amplitude and derive the corresponding generalizations of predictions for two-time response and correlation functions. Applications to non-equilibrium critical dynamics of several systems, with a fully disordered initial state and vanishing initial magnetization, including the Glauber–Ising model, the Frederikson–Andersen model and the Ising spin glass are discussed. The critical contact process and the parity-conserving non-equilibrium kinetic Ising model are also considered.
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.
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.
Motivated by recent numerical findings of Henkel et al (2006 J. Phys. A: Math. Gen. 39 L589) we re-examine via Monte Carlo simulations the linear response function of the two-dimensional Ising model with Glauber dynamics quenched to the critical point. At variance with the results of Henkel et al , we detect discrepancies between the actual scaling behavior of the response function and the prediction of local scale invariance. Such differences are clearly visible in the impulse autoresponse function, whereas they are drastically reduced in the integrated response functions. Accordingly, the scaling form predicted on the basis of local scale invariance simply provides an accurate fitting form for some quantities but cannot be considered to be exact.
Recent studies on the phenomenology of ageing in certain many-particle systems which are at a critical point of their non-equilibrium steady states are reviewed. Examples include the contact process, the parity-conserving branching-annihilating random walk, two exactly solvable particle reaction models and kinetic growth models. While the generic scaling descriptions known from magnetic systems can be taken over, some of the scaling relations between the ageing exponents are no longer valid. In particular, there is no obvious generalization of the universal limit fluctuation–dissipation ratio. The form of the scaling function of the two-time response function is compared with the prediction of the theory of local scale-invariance.
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.