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arXiv:cond-mat/0703445 · Journal of Physics A Mathematical and Theoretical

Kinetics of the long-range spherical model

Florian Baumann, Sreedhar B Dutta, Malte Henkel

Published Jun 19, 2007• 22 citations
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Abstract

The kinetic spherical model with long-range interactions is studied after a quench to $T < T_c$ or to $T = T_c$. For the two-time response and correlation functions of the order-parameter as well as for composite fields such as the energy density, the ageing exponents and the corresponding scaling functions are derived. The results are compared to the predictions which follow from local scale-invariance.

DOI
10.1088/1751-8113/40/27/001

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The kinetic spherical model with long-range interactions is studied after a quench to T<TcT < T_cT<Tc​ or to T=TcT = T_cT=Tc​. For the two-time response and correlation functions of the order-parameter as well as for composite fields such as the energy density, the ageing exponents and the corresponding scaling functions are derived. The results are compared to the predictions which follow from local scale-invariance.
The scaling forms of the space- and time-dependent two-time correlation and response functions are calculated for the kinetic spherical model with a conserved order parameter and quenched to its critical point from a completely disordered initial state. The stochastic Langevin equation can be split into a noise part and into a deterministic part which has local scale transformations with a dynamical exponent z = 4 as a dynamical symmetry. An exact reduction formula allows us to express any physical average in terms of averages calculable from the deterministic part alone. The exact spherical model results are shown to agree with these predictions of local scale invariance. The results also include kinetic growth with mass conservation as described by the Mullins–Herring equation.
The dynamical scaling and ageing in the relaxational dynamics of the quenched directed spherical model is analysed. The exact two-time correlation and response functions display new regimes of ballistic or anisotropic ballistic scaling, at larger distances than probed in the usual regime of diffusive scaling. The rôle of long-ranged initial correlations on the existence of these scaling regimes is clarified. Their dynamical symmetries are described in terms of extensions of the Schrödinger algebra appropriate to non-equilibrium dynamics in that the anisotropic ballistic scaling regime can be interpreted in terms of meta-Schrödinger invariance while the regime of isotropic ballistic scaling is meta-conformally invariant.
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.
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.
Recently Baumann et al (2007 Preprint 0709.3228v1) studied the phase-ordering kinetics of the two-dimensional Ising model for T < T c with uniform spatially quenched disorder by Monte Carlo simulations. They found that the two-time response and correlation functions are in agreement with the predictions of local scale invariance generalized to . The present paper shows why this is actually not true and suggests an alternative approach which leads to a much better agreement with the numerical results.
For ageing systems, quenched onto a critical temperature T=TcT=T_cT=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{z}=2z=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 1D1D1D Glauber-Ising model at T=0T=0T=0 and the critical spherical model in d>2d>2d>2 dimensions.