On logarithmic extensions of local scale-invariance

Abstract

Ageing phenomena far from equilibrium naturally present dynamical scaling and in many situations this may generalised to local scale-invariance. Generically, the absence of time-translation-invariance implies that each scaling operator is characterised by two independent scaling dimensions. Building on analogies with logarithmic conformal invariance and logarithmic Schrödinger-invariance, this work proposes a logarithmic extension of local scale-invariance, without time-translation-invariance. Carrying this out requires in general to replace both scaling dimensions of each scaling operator by Jordan cells. Co-variant two-point functions are derived for the most simple case of a two-dimensional logarithmic extension. Their form is compared to simulational data for autoresponse functions in several universality classes of non-equilibrium ageing phenomena.

Logarithmic correlators or responses in non-relativistic analogues of conformal invariance

“…Very large values of the waiting time s are required, and one may observe from figure 3a that even data with s < 10 3 are not yet fully in the scaling regime. While non-logarithmic lsi gives an overall agreement with an accuracy up to about 5%, when a ′ = −0.5 is assumed, clear [48,51]. and systematic deviations remain.…”

“…Logarithmic representation of age(d), analogously to section 2, can be constructed by replacing both scaling dimensions x and ξ by matrices [51] x…”

“…(3.2) in section 3. Therefore, an extension to the parabolic sub-algebra age(1) will produce the analogous causality constraints on the two-point functions, when then should be interpreted as responses, and not as correlators [49,51].…”

“…We now briefly review discuss two candidate models for an application of logarithmic lsi (llsi) in physical ageing [51]. The universality classes of both the Kardar-Parisi-Zhang equation and directed percolation are widely considered to be the most simple models for the non-equilibrium phase transitions they describe.…”

From dynamical scaling to local scale-invariance: a tutorial

“…and where the exponent a ′ and the amplitudes h 0 , g 0 , f 0 must be fitted to the data [72]. At present, four universality classes are known where the prediction (3.8) of ageing invariance is no longer enough, but where (3.15) describes the data well in the entire region where dynamical scaling is found: kpz for d = 1 [34] and very recently also for d = 2 [73,74], critical directed percolation for d = 1 [72] and the critical 2D Glauber-Ising model [75].…”