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arXiv:cond-mat/0603369 · Journal of Statistical Mechanics Theory and Experiment

Is local scale invariance a generic property of ageing phenomena?

Haye Hinrichsen

Published Jun 21, 2006• 16 citations
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Abstract

It is shown by means of numerical simulations that, in contradiction to recent claims by Enss et al (2004 J. Phys. A: Math. Gen. 37 10479), the autoresponse function of the critical (1+1)-dimensional contact process is not in agreement with the predictions of local scale invariance.
DOI
10.1088/1742-5468/2006/06/l06001

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We investigate some aspects of the ageing behaviour observed in the contact process after a quench from its active phase to the critical point. In particular we discuss the scaling properties of the two-time response function and we calculate it and its universal ratio to the two-time correlation function up to first order in the field-theoretical -expansion ( = 4− d ). The scaling form of the response function does not fit the prediction of the theory of local scale invariance. Our findings are in good qualitative agreement with recent numerical results.
The long-time dynamics of the 1D contact process suddenly brought out of an uncorrelated initial state is studied through a light-cone transfer-matrix renormalisation group approach. At criticality, the system undergoes ageing which is characterised through the dynamical scaling of the two-times autocorrelation and autoresponse functions. The observed non-equality of the ageing exponents a and b excludes the possibility of a finite fluctuation-dissipation ratio in the ageing regime. The scaling form of the critical autoresponse function is in agreement with the prediction of local scale-invariance.
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.
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.
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.