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arXiv:cond-mat/0307649

Local scale invariance and its applications to strongly anisotropic critical phenomena

Malte Henkel, Alan Picone A, Michel Pleimling, Jérémie M. Unterberger

Published Jul 25, 2003• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0307649

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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.
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.
A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/N$, with N=1,2,3…N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}N=1,2,3…. Differential equations for the two-point function are derived and explicitly solved for all values of NNN. Known special cases are conformal invariance ( N=2N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2N=2) and Schr\"odinger invariance ( N=1N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1N=1). For N=4N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4N=4 and N=6N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6N=6, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.
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.
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.
The scaling functions of single-time and two-time correlators in systems undergoing non-equilibrium critical dynamics with dynamical exponent z=2{z}=2z=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.